%%%%%%%%% TIENE 26 PAGINAS COMPILADAS --- PAPER 3 --- %%%%%%%%%
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\begin{document}

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\markboth{\sc \centerline{Proceedings of the IWOP'96} }{\sc \centerline{Proceedings of the IWOP'96} }
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\begin{center}
{\large \bf CONTINUOUS SOLUTIONS OF DIFFERENCE EQUATIONS 
AND THEIR ORTHOGONALITY RELATIONS.}
\end{center}
\begin{center}
{\bf Natig M. Atakishiyev}
\end{center}

\begin{abstract}
Some methods for solving hypergeometric--type difference equations are 
reviewed. In particular, a method of constructing solutions in the form of a 
development in "generalized powers" of the lattice $x(s)$, is discussed in 
detail. Various forms of the continuous orthogonalities for these solutions 
are also presented.
\end{abstract}


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\vspace{24pt}
%\centerline
\noindent{\large \bf 1 CLASSICAL SPECIAL FUNCTIONS}
\vspace{12pt}
 
Classical special functions of mathematical physics arise in solving 
many theoretical and applied problems in various branches of natural science.
They form a fundamental part of our mathematical language, helping us to
express precisely our knowledge about the world in which we live. Their 
importance is therefore difficult to overestimate.

Well-known examples of special functions are the classical Jacobi, Laguerre
and Hermite polynomials \cite{erdely53}, satisfying the second-order linear 
differential equation of Sturm-Liouville type:
$$ \sigma(x) y^{\prime\prime}(x) + \tau (x) y^{\prime} (x) + \lambda y(x) = 0
\, , \eqno (1.1) $$
where primes denote differentiation with respect to the independent variable 
$x$, $\sigma (x)$ and $\tau (x)$ are polynomials of at most second and first
degree, respectively, and $\lambda $ is a constant. Observe that the generic 
case of (1.1) with $\sigma (x) = a x^2 + b x + c$ is reduced to the 
differential equation 
$$ x(1-x) y^{\prime\prime}(x) + [\gamma - (\alpha + \beta +1)x] y^{\prime}(x) 
- \alpha \beta y(x) = 0 \eqno (1.2) $$
 for the Gaussian hypergeometric function $F(\alpha, \beta; \gamma; x)$,
upon making the linear change of the variable $ x \mapsto (x_2 - x_1 ) x 
+ x_1$,  where  $x_1 $ and $x_2$ are distinct zeros of $\sigma (x)$ 
({\it i.e.}, $\sigma (x_1) = \sigma (x_2) = 0 $ and $ x_1 \neq x_2 $). 
The explicit form of $ F(\alpha , \beta ; \gamma; x)$ is given by a power 
series in the variable $x$, namely
$$    F(\alpha , \beta ; \gamma; x) = \sum _{n=0}^{\infty} \frac{(\alpha )_n
\, (\beta )_n}{(\gamma )_n} \, \frac{x^n}{n!} \, , \eqno (1.3) $$
where $(\alpha )_n = \Gamma (\alpha + n) / \Gamma (\alpha) \, , n = 0, 1, 2,
...$, is the shifted factorial.

The Jacobi, Laguerre and Hermite polynomials share the following common 
properties:
\begin{itemize}
\item
 they satisfy three-term recurrence relations, 
\item
they can be
represented by a Rodrigues formula ({\it i.e.}, as an {\it $n$}th derivative 
of an elementary function),
\item
 they are orthogonal with respect to continuous 
weight functions (on the intervals  $[-1,1],\, [0, \infty)$, and $(-\infty, 
\infty)$, respectively), and 
\item
the derivative of each of them is again a 
polynomial of the same type.
\end{itemize}

Recently it has become clear that there exists a more general set of 
orthogonal polynomials which are basic hypergeometric extensions of the
classical orthogonal polynomials \cite{ask85}. We shall discuss them in 
section 3.2. 

The standard way of solving the equation (1.1) is to look for its solution in 
the form of power series in the independent variable $x$. Indeed, if $x_1$ is
a root of the equation $\sigma (x)= 0 $, then the equation (1.1) has a 
particular solution of the form 
$$ y(x) = \sum_{n=0}^{\infty} \, c_n (x - x_1)^n\, , \eqno (1.4) $$
where the coefficients $c_n$ satisfy the two-term recurrence relation
\cite{ats93}
$$ (n+1) [\tau (x_1) + n \sigma^{ \prime} (x_1)] c_{n+1} + \{\lambda + 
n[\tau ^ \prime + \textstyle\frac{1}{2}(n-1) \sigma ^{\prime\prime}]\} c_n = 0 
\, . \eqno (1.5) $$
For proving this, it is more convenient to rewrite the equation (1.1) in the 
self-adjoint form
$$ [\sigma (x) \rho (x) y^{\prime} (x) ]^{\prime} + \lambda \rho (x) y(x) = 0 
\,,   \hspace{1cm}
[\sigma (x) \rho (x) ]^{\prime} = \tau (x) \rho (x) \, , \eqno (1.6)$$
where the second line is the first-order differential equation for defining a
function $\rho (x) $. Now the proof that (1.4) and (1.5) represent a solution 
to (1.1) follows from the identity
$$ \rho ^{-1} (x) \frac{d}{dx} \left [ \sigma (x) \rho (x) \frac{d}{dx} (x -
\xi )^n \right ] = n (n-1) \sigma (\xi) ( x -  \xi )^{n-2}  $$
$$ + n [\tau (\xi ) + (n-1) \sigma^ \prime (\xi) ] (x - \xi )^{n-1} - 
\lambda _n ( x - \xi )^n \, , \eqno (1.7)$$
which can be easily verified; here $ \lambda _n = - n \tau^{\prime} - n (n-1) 
\sigma^{\prime\prime}/2$ and $\xi$ is an arbitrary value of the 
independent variable $x$ (not necessarily a zero of $ \sigma (x) $!). 
Futhermore, as is evident from (1.7) , in the case when $\sigma (x) $ has no 
zeros, the series (1.4) will satisfy the equation (1.1) if
$$ (n+1) (n+2) \sigma c_{n+2} + (\lambda + n \tau ^{\prime}) c_n = 0 
\eqno (1.8) $$
 and $x_1$ is a root of the equation $\tau (x) = 0 $.

\vspace{24pt}
%\centerline 
\noindent{\large \bf 2  DIFFERENCE OPERATORS}
\vspace{12pt}

Those of you who are versed in non-relativistic quantum mechanics 
probably remember that the notion of difference operator appears at an early 
stage. Indeed, consider a quantum-mechanical system of $N$ particles with 
positions in three-dimensional space, that are characterized by the radius 
vectors $\vec r_n , 1\leq n \leq N$ \cite{landau91}. Then an infinitely
small parallel displacement (or translation) in this space over a distance
$\delta \vec r$ signifies a transformation under which the radius vectors
$\vec r_n$ of all the particles receive the same increment $\delta \vec r$:
$\vec r_n \Rightarrow \vec r_n + \delta \vec r $. The operator of a parallel 
displacement over any finite (not only infinitesimal) distance $\vec a$ can 
be expressed in terms of the momentum operator $\hat {\vec p}  = - i \hbar 
\frac{\partial}{\partial \vec r} = - i \hbar \vec \bigtriangledown  $. By
the definition of this operator $\hat {T}_{\vec a}$ we must have
$$ \hat {T}_{\vec a} \, \psi (\vec r) = \psi (\vec r + \vec a) \, . 
\eqno (2.1) $$
 Expanding the function $ \psi (\vec r + \vec a)$ in a Taylor series gives 
$$ \psi (\vec r + \vec a )  = \psi ( \vec r ) + \vec a \, \frac{\partial \psi
(\vec r)}{\partial \vec r}+ \cdots \, , \eqno (2.2) $$
or, in terms of the momentum $\hat {\vec p}$,
$$ \psi (\vec r + \vec a )  = [1 + \frac{i}{\hbar} \vec a \, \hat {\vec p}
 + \frac{1}{2} (\frac{i}{\hbar} \vec a \,\hat {\vec p} )^2 + \cdots]
 \psi ( \vec r )\, . \eqno (2.3) $$
 The expression in brackets, $\hat {T}_{\vec a} = \exp (\frac{i}{\hbar} \vec a 
\, \hat {\vec p})$, is the required operator of the finite displacement. From 
(2.2) or (2.3) it is also evident, that the momentum $\hat {\vec p} $ itself 
( multiplied by the constant factor $i/\hbar$ ) represents the operator of 
infinitesimal displacements, or, in the terminology of group theory, it is 
the generator of displacements. 

Once the action of the simple difference operator $ \exp (a\, \partial _x), \,
\partial _x \equiv {d}/{d x}$, is clear from (2.1), we can try to solve a 
difference equation of the form
$$ [\alpha _l (x) \, e^{- \partial _x} + \alpha _l (x+1)\, e^{\partial _x}] \,
f_n (x; l) = \lambda _n (l)\, f_n (x; l) \, , \eqno (2.4) $$
where $\alpha _l (x) = \sqrt {(x+l)(l+1-x)}$ and $-l\leq x \leq l$.
Observe that in view of the relation $\alpha _l (x) = \alpha _l (1-x)$, the
 equation (2.4) is invariant with respect to the change of the sign of $x$.
As usual, one starts with a ``ground state" $f_0 (x; l)$, which defines
``asymptotic" behaviour of solutions $f_n(x; l)$ at the ends of the interval 
$[-l, l]$. Since
$$e^{\partial _x} \, \Gamma (x+a) = \Gamma (x+a+1) = (x+a)\, \Gamma(x+a)\, , $$
$$e^{\partial _x} \, \Gamma (a-x) = \Gamma (a-x-1) = (a-x-1)^{-1}\, 
\Gamma(a-x)\, ,  \eqno (2.5) $$
and the action of the second operator $\exp (-\partial _x)$ on the same
gamma-functions is readily obtained from (2.5) by reversing the sign of the
variable $x$, we can look for a solution
$$ f_0 (x; l) = \Gamma ^{\beta} (x+a)\, \Gamma ^{\beta} (a-x)\, , \eqno(2.6)$$
where $a$ and $\beta$ are some constants to be found. Substituting (2.6) into 
the left-hand side of (2.4) gives
$$\alpha  _l (x)\, f_0(x-1; l) + \alpha _l (x+1) \, f_0 (x+1; l) = $$
$$ [\alpha _l (x) \, (a-x)^{\beta} (a+x-1)^{-\beta} +
 \alpha _l (x+1)\,
(x+a)^\beta \, (a-x-1)^{-\beta}]\, f_0 (x; l) \, .  \eqno (2.7) $$
The equation (2.4) for $f_0(x; l)$ will be satisfied if the expression in the 
square brackets in (2.7) turns out to be a constant. So if we choose $a=l+1$
and $\beta = -1/2$, this expression becomes $x + l + l - x = 2 l $. Thus,
$$ f_0 (x;l)  = [\Gamma (x+l+1)\, \Gamma (l+1-x) ]^{-1/2} \eqno (2.8) $$
and the corresponding eigenvalue $\lambda _0 (l) $ is equal to $2l$.
Having defined $f_0 (x; l) $, one can successively find all $2l+1$ linearly 
independent "excited states"
$$ f_n (x; l) = d^{-1}_n \, p_n (x; l)\, f_0 (x; l)\,, \, \qquad \, 
0\leq n \leq 2l \, , \eqno (2.9) $$
which correspond to the eigenvalues $ \lambda _n(l) = 2 (l-n) $.
The polynomials $p_n(x;l)$ in (2.9) satisfy the three-term recurrence
relation
$$ 2 x p_n (x; l) = (n+1) \, p_{n+1} (x; l) + (2l +1 -n)\, p_{n-1} (x; l) 
\eqno (2.10)$$
and they are particular case of the well-known Kravchuk polynomials 
$$
 k_n^{(p)} (x, N) = (-1)^n C_N^n \, p^n\,F(-n, -x; -N; p^{-1}) 
$$
with $p = 1/2$ and $N= 2l $ \cite{nikifo91} . The normalization constant 
$d_n = [(2l)!/n! (2l-n)!]^{1/2} $ in (2.9) is chosen in such a way, that the
 functions $f_n(x;l)$ satisfy the discrete orthogonality relation
$$ \sum_{j=0}^{2l} \, f_m (x_j; l)\, f_n (x_j; l) = \delta _{m,n} \eqno (2.11) $$ 
at the points $x_j = j - l$.

The reason for picking up this particular example of difference equation (2.4)
 is that it actually represents a discrete model of the harmonic oscillator 
\cite{ataw94}.

We recall in this connection that the quantum-mechanical analogue of Newton's 
equation is $ m \dot v = -{\partial U}/{\partial x}$, where $m$ is the 
mass, $v = \dot x = i \hbar ^{-1} [H, x] $ is the velocity operator, and 
$H(x) = p^2/2m + U(x)$ is the Hamiltonian \cite{landau91}. For the harmonic 
oscillator, {\it i.e.}, when we have $U(x) = {m \, \omega ^2 x^2} / 2$, 
this equation takes the form
$$[H(x), [H(x), x]] = (\hbar \omega )^2 x\, . \eqno (2.12) $$
Now it is easy to check that the difference operator
$$ D_l (x) = \frac{1}{2} [\alpha _l (x) \, e^{-\partial _x} + \alpha _l(x+1) \,
e^{\partial_x}] \eqno (2.13) $$
 satisfies the same commutation relation
$$ [D_l (x) , [D_l (x), x ]] = x \eqno (2.14) $$
 with the independent variable $x$ as in (2.12) .

\vspace{24pt}
%\centerline
\noindent{\large \bf 3 HYPERGEOMETRIC-TYPE DIFFERENCE EQUATIONS}
%\vspace{6pt}
%\centerline{\hspace{5mm} \large \bf DIFFERENCE EQUATIONS}
\vspace{12pt}

Having discussed some particular examples of difference operators and
difference equations in section 2, we consider now a general approach
to the problem of constructing a hypergeometric-type difference equation 
by approximating the differential equation (1.1) on certain lattices 
\cite{nikifo91}. The standard way of approximating any differential equation 
is based on the use of the definition of the derivative
$$ y^{\prime} (x) = \lim_{h \to 0} \frac{y(x+h) - y(x)}{(x+h) - x}
\eqno(3.1)$$
and involves the substitution
$$ y^{\prime} (x) \Rightarrow \frac{1}{h} [ y(x+h) - y(x) ] = \frac{1}{h} 
( e^{ h \partial _x} - 1 ) \, y(x) \eqno (3.2) $$
for each derivative of the unknown function $y(x)$. In the case of the 
differential equation (1.1) the corresponding difference equation of 
hypergeometric type
$$ \frac{\sigma (x)}{h} \, \left [ \frac{y(x+h) - y(x)}{h} - 
\frac{y(x) - y(x-h)}{h} \right ] + $$
$$ + \frac{\tau (x)}{2} \, \left[ \frac{y(x+h) - y(x)}{h} + \frac{y(x) - 
y(x-h)}{h} \right ] + \lambda y(x) = 0   \eqno (3.3) $$
approximates (1.1) up to the second-order of accuracy with respect to the
constant $h$. We remind the reader that the difference operator $L_h$  
approximates the differential operator $L$ at point $x$ to the order of
accuracy $n$ with respect to $h$ if $ (L_h - L)\, y(x) = O (h^n) $
when $h \to 0$ \cite{nikifo91}.

This scheme is often employed for obtaining (up to some fixed 
order of accuracy in the constant $h$) approximate solutions to  
differential equations, which are difficult to solve exactly. But a
more general case
arises when one introduces a new indepent variable $s$ and approximates a 
differential equation on an arbitrary lattice $x(s)$, {\it i.e.},
by replacing
$$ y^{\prime} (x)  \Rightarrow  \frac{y[x(s+h)] - y[x(s)]}{x(s+h) - x(s)}
\, . \eqno (3.4)$$
The difference equation of hypergeometric type corresponding to (1.1) and 
(3.3)  on a lattice $x(s)$ has the form

$$ \frac{\sigma [x(s)]}{x(s+\frac{h}{2}) - x(s - \frac{h}{2})}\,
\left [\frac {y(s+h) - y(s)}{x(s+h) - x(s)} - \frac{y(s)-y(s-h)}{x(s)-x(s-h)}  \right ] + $$
$$+ \frac{\tau[x(s)]}{2} \left[ \frac {y(s+h) - y(s)}{x(s+h) - x(s)} 
+ \frac{y(s)-y(s-h)}{x(s)-x(s-h)}  \right ] + \lambda y(s) = 0 \, . 
\eqno (3.5) $$

Naturally, in the particular case of the linear lattice $ x(s) = s$, (3.5) 
reduces 
to the difference equation (3.3). Observe that the linear change of the 
independent variable $s \to h s$ and the replacement of the lattice 
$x(hs) \to  x(s)$ lead to the more compact form of equation (3.5) :
$$ \sigma [x(s)] \frac {\bigtriangleup}{\bigtriangledown x (s+\onehalf )}
\, \left [ \frac{\bigtriangledown y(s) }{\bigtriangledown x(s)} \right]
+ \frac{\tau [x(s)]}{2} \left [ \frac{\bigtriangleup y(s) }{\bigtriangleup 
x(s)} + \frac{\bigtriangledown y(s) }{\bigtriangledown x(s)} \right] +
\lambda y(s)  = 0 \, , \eqno (3.6) $$
where the difference operators $\bigtriangleup$ and $\bigtriangledown$
are defined as
$$\bigtriangleup y(s) = y(s+1) - y(s) = ( e^{\partial _s} - 1 ) \, y(s) \,,
\eqno(3.7a) $$ 
$$\bigtriangledown y(s) = y(s) - y(s-1) = (1 - e^{-\partial _s}) \, y(s) \,. 
\eqno (3.7b) $$


Assume that the difference equation (3.6) has some polynomial solution 
$y[x(s)]$ and we require (by analogy with the classical case of the 
Jacobi, Laguerre and Hermite polynomials) that the difference-derivative
$\bigtriangleup y[x(s)] / \bigtriangleup x(s) $ of this solution satisfies 
an equation of the same kind. This requirement holds if and only if the 
lattice $x(s)$ is a quadratic or $q$-quadratic lattice of the form
\cite{nikifo91,ars95}
\begin{center}
$$%\[
x(s) = \left \{
\begin{array}{rr} 
  C_1 s^2 + C_2 s \,,\\
  C_1 q^{-s} + C_2 q^s \,. 
\end{array}
\right.
\eqno(3.8) $$
%\]
\end{center}

Difference equations, which approximate differential ones on certain lattices $x(s)$, can be solved exactly, 
{\it i.e.}, without expanding  solutions in powers of the lattice mesh $h$. 
Evidently, this important fact means that there exists a more general set 
of solutions for each such difference equation, which coincides with the 
set of solutions for the corresponding differential equation only in 
the limit of vanishing mesh $h$.

\vspace{24pt}
%\centerline
\noindent{\large \bf 3.1 ``Power" series developments}
\vspace{12pt}

 It is possible to construct solutions of the difference equation (3.6) 
in the form of a development in "generalized powers" of the lattice 
$x(s)$, defined as \cite{atsu88} 
$$ [x(s) - x(s_0) ]^{(n)} = \prod _{k=0}^{n-1} [ x(s) - x(s_0 - k)] \,.
\eqno(3.9) $$
 But, as in the case of the differential equation (1.1), it is more convenient
first to represent (3.6) in the self-adjoint form \cite{nikifo91}
$$\frac{\bigtriangleup}{\bigtriangledown x (s +\onehalf)} \,
\left [ \widetilde \sigma (s) \, \rho (s) \, \frac{\bigtriangledown y(s)}
{\bigtriangledown x(s)}\right ] + \lambda \rho (s) y(s) = 0\, , $$
 
$$\frac{\bigtriangleup}{\bigtriangledown x (s + \onehalf)} \,
\left [ \widetilde \sigma (s) \, \rho (s)\right ] = \tau (s) \, \rho (s) \,, 
\eqno (3.10)$$
where $ \widetilde \sigma (s) = \sigma [x(s)] - \onehalf \tau [x(s)] 
\bigtriangledown x(s+\onehalf) $ . 

The series of the form
$$ y(s) = y[x(s)] = \sum_{n=0}^\infty \, c_n [ x(s) - x(r+n - 1)]^{(n)}\, ,
\eqno (3.11)$$
 where $r$ is a finite root of the equation
$$\widetilde \sigma (s) = \sigma [x(s)] - \onehalf \tau [x(s)] 
\bigtriangledown x(s+\onehalf) = 0  \eqno (3.12)  $$
and
$$ \frac{c_{n+1}}{c_n} = - \frac{\{\lambda + \gamma (n) [\alpha (n-1)
\tau^{\prime} + \gamma (n-1) \sigma^{\prime\prime}/2 \}\, \bigtriangledown x
[r+(n-1)/2]}{\gamma (n+1) [\widetilde \sigma (r+n) + \tau (r+n)
\bigtriangledown x (r + n + 1/2)]} \, , \eqno (3.13) $$
satisfies a non-homogeneous equation of the type (3.6) on the non-uniform 
lattices (3.8) with the right-hand side
$$G(s) = \lim_{N \to \infty} (\lambda - \lambda_N) [x(s) - x(r+N-1)]^{(N)}\,. 
\eqno(3.14)$$

This can be verified by using the equality
$$\rho ^{-1} (s) \frac{\bigtriangleup}{\bigtriangledown x_1(s)} \left \{ 
\widetilde \sigma (s) \rho (s) \frac{\bigtriangledown}{\bigtriangledown
x(s)} [x(s) - x(s_0) ]^{(n)} \right \}= $$
$$= \gamma (n)  \gamma (n-1) \widetilde \sigma (s_0-n+1 )\, [x(s) - 
x(s_0-1) ]^{(n-2)} + \eqno (3.15) $$

$$ + \gamma (n) \tau_{n-1} (s_0-n+1) [x(s) - x(s_0-1)]^{(n-1)} -
\lambda _n [x(s) - x(s_0)]^{(n)} \, , $$
 where the definitions 
$$ \tau _{\nu} (s) \bigtriangledown x_{\nu+1} (s) = \widetilde \sigma
(s+\nu) + \tau (s+\nu) \bigtriangledown x_1 (s+\nu) - \widetilde \sigma (s) \,,
$$
$$\lambda _{\mu} = - \gamma (\mu) \, [\alpha (\mu -1) \tau^{\prime} + 
\gamma (\mu -1) \sigma^{\prime\prime}/2 ]\, , \eqno (3.16) $$
$$ x_\nu (s) = x(s+\nu/2) $$
 and the notations
$$
\alpha (\mu) = \left \{
\begin{array}{ll}
1, \\
\onehalf (q^{\mu/2} + q^{-\mu/2} )\, , 
\end{array}
\right.
\qquad
\gamma(\mu) = \left \{
\begin{array}{rr}
{} &\mu \, ,\\
{} &\displaystyle {\frac{q^{\mu/2} - q^{-\mu/2}}{q^{1/2}-q^{-1/2}}} 
\end{array}
\right.
\eqno (3.17)
$$
have been used for the quadratic and $q$-quadratic lattices, respectively.

Indeed, let us write the difference equation of hypergeometric type (3.6)
in the self-adjoint form (3.10) and look for its solutions in the form of
the following development in generalized powers (3.9) 
$$ y(s) = \lim _{N\to  \infty} \sum_{n=0}^N c_n [x(s) - x(s_0)]^{(n)} \,,
\eqno (3.18) $$
 where $s_0$ is a constant. Substituting (3.18) into (3.10), with the
aid of identity (3.15) with $ s_0 = r + n - 1$ and $\sigma (r) = 0 $,
we obtain
$$ \left [\rho ^{-1} (s) \frac{\bigtriangleup}{\bigtriangledown x_1(s)} \, 
\left (
\sigma (s) \rho (s) \frac{\bigtriangledown}{\bigtriangledown x(s)} \right  ) 
+ \lambda  \right ]\, y(s) = $$
$$=\lim_{N \to \infty} \sum_{n=0}^N c_n \left [ \rho^{-1} (s) 
\frac{\bigtriangleup}{\bigtriangledown x_1(s)} \left ( \sigma (s) \rho (s) \,
\frac{\bigtriangledown}{\bigtriangledown x(s)}\right ) + \lambda \right ]\,
 [x(s) - x(s_0)]^{(n)} = $$
$$=\lim_{N\to \infty} \left \{ \sum_{n=0}^N c_n \gamma (n) \tau_{n-1} (r) 
[x(s) - x(r+n-2)]^{(n-1)} +\right. $$
$$\left.+ \sum_{n=0}^N c_n (\lambda - \lambda_n ) 
[x(s) - x (r+n-1)]^{(n)} \right \} = G(z) \, ,  \eqno(3.19) $$
provided that the coefficients $c_n$ satisfy the two-term recurrence relation
$$ \gamma (n+1) \tau _n (r) c_{n+1} = (\lambda _n - \lambda )\, c_n \,. 
\eqno (3.20) $$
From (3.20), in view of (3.16), (3.13) follows.

Observe that when $\lambda = \lambda _\nu$ and $\nu = n = 0, 1, 2, \dots $,
solution (3.11) is a polynomial of degree $n$ and $G(s) \equiv 0$. 

\vspace{24pt}
%\centerline
\noindent{\large \bf 3.2 The Askey-Wilson polynomials}
\vspace{12pt}

Particular choices of the lattice $x(s)$, generically defined by (3.8), 
lead to various families of polynomial solutions of the hypergeometric-type 
difference equation (3.6). The most general one is the 
five-parameter set of the Askey--Wilson polynomials, which contains as special
or limiting cases all the classical polynomials, satisfying
orthogonality  relations with continuous weight functions. They are 
defined as \cite{ask85}
$$ p_n (x; a, b, c, d|q) 
 = (ab, ac, ad; q)_n \, a^{-n} \, {}_4 \phi _3 
\left [{{{q^{-n}, abcdq^{n-1}, ae^{i \theta}, a e^{-i \theta}} \atop
{ab, ac, ad}} ; q, q} \right ] \,,\eqno (3.21) $$
where $ x = \cos \theta, \, (a; q)_0 = 1$ and $(a; q)_n = \prod _{k=0}^{n-1}
(1 - a q^k), \, n = 1, 2, 3, \ldots, $ define the $q$-shifted factorial, 
$(a_1, \ldots, a_k; q)_n = \prod _{j=1}^k (a_j; q)_n $ is the contracted
notation for the multiple $q$-shifted factorials and the basic hypergeometric 
series ${}_4 \phi _3 $ is
$$ {}_4 \phi _3 \left [{{{a_1, a_2, a_3, a_4}\atop{b_1, b_2, b_3}}; q, z}
 \right ] = \sum_{k=0}^\infty \frac{(a_1, a_2, a_3, a_4; q)_k}{(b_1, b_2, b_3, q; q)_k} \, z^k\,. \eqno(3.22)$$
To verify that the formula (3.21) constitutes a polynomial of {\it n}th 
degree in the variable $x = \cos \theta $ we have to take into account the 
following two facts. First, since the $q$-shifted factorial $(q^{-n}; q)_k = 
\prod _{j=0}^{k-1} (1 - q^{j-n})$ vanishes for any positive integer 
$ k\geq n+1$ by its
definition, from (3.22) it follows that the basic hypergeometric series
${}_4 \phi _3$ in (3.21) represents the finite sum
$$ {}_4 \phi _3 \left [{{{q^{-n}, abcdq^{n-1}, ae^{i \theta}, a e^{-i \theta}} 
\atop {ab, ac, ad}} ; q, q} \right ] =
 \sum_{k=0}^n 
\frac{(q^{-n}, abcdq^{n-1}, a e^{i \theta}, a e^{-i  \theta}; q)_k}
{(ab,ac, ad, q; q)_k} \, q^k \,. \eqno (3.23)$$ 
Second, since the product of two $\theta$-dependent factors in (3.23)
can be written as
$$ ( a e^{i \theta}, ae^{-i \theta} ; q)_k = \prod _{j=0}^{k-1} \,
(1 - a q^j e^{i \theta})\, (1 - a q^j e^{-i \theta}) = 
 \prod _{j=0}^{k-1} \, [ 1 - 2 a q^j \cos \theta + a^2 q^{2j} ] \, ,
\eqno (3.24)$$
it certainly forms a polynomial of {\it k}th degree in $ x = \cos \theta $.

The Askey--Wilson polynomials (3.21) satisfy the orthogonality relation 
\cite {ask85}
$$ \int\limits_{-1}^1 p_m (x; a, b, c, d|q)\, p_n (x; a, b, c, d|q)\,
w(x; a, b, c, d|q)\, dx =
 \delta _{mn}\, I_n (a, b, c, d|q) \eqno (3.25)$$
on the finite interval $ -1 \leq x = \cos \theta \leq 1 $ with respect to the 
continuous weight function
$$ w(x; a, b, c, d|q) = 4 {\rm sin} \theta \, \frac{\prod _{v = a,b,c,d} \,
e_q(v e^{i  \theta})\, e_q(v e^{-i \theta})}{e_q (q e^{2i \theta})\, 
e_q (q e^{-2i \theta})}\,, \eqno (3.26)$$
where $e_q(z) $ is the $q$-exponential function, defined as \cite{gasper90}
$$ e_q (z) = \sum_{n=0}^\infty \frac{z^n}{(q; q)_n}\,, \eqno (3.27)$$
and $I_n (a, b, c, d|q)$ is equal to 
$$ I_n (a, b, c, d|q) = 2 \pi e_q (q^{n+1}) \frac{e_q(abq^n)\,e_q (acq^n)\,
e_q(adq^n)\, e_q(bcq^n)\, e_q(bdq^n)\, e_q(cdq^n)}
{(1 - abcdq^{2n-1})\, e_q (abcdq^{n-1})} \, . \eqno (3.28) $$

\vspace{24pt}
%\centerline
\noindent{\large \bf 3.3 The transformation $q \to q^{-1} $ }
\vspace{12pt}

Jacobi's triple product identity
$$ \theta _1 (z, q) = 2 q^{\frac{1}{4}}{\rm sin} z \,
(q^2, q^2 e^{2iz}, q^2 e^{-2iz} ; q^2 )_\infty  \eqno (3.29) $$
 and Euler's formula
 $$ e_q (z) = (z; q)^{-1}_\infty \eqno (3.30) $$
enable the weight function (3.26) for the Askey--Wilson polynomials to be 
written as
$$ w(x;a,b,c,d|q) = 2 q^{-\frac{1}{8}}\, e_q(q)\, \vartheta _1 
(\theta, q^{1/2} ) 
\, \prod_{v=a,b,c,d,}\, e_q (v e^{i \theta})\, e_q(v e^{-i \theta})\,  . 
\eqno (3.31) $$
Since the theta-function in (3.31) is defined as 
$$ \vartheta _1 (z,q) = 2 \sum _{n=0}^\infty (-1)^n q^{{(n+1/2)}^2}\,
{\rm sin}(2n+1) \, x \, , \eqno (3.32) $$
 it is evident that this series is convergent only if the parameter $q$
 satisfies $|q| < 1$, therefore the common notation in the theory of
special functions is $q \equiv e^{\pi i \tau}$, $\Im \tau > 0 $. This means
 that the representation (3.25) does not admit the transformation 
$ q \to q^{-1}$. But using the modular and periodicity properties of the 
theta-function $\vartheta _1(\theta, q^{1/2})$ it turns out to be possible 
to replace them by quadratically decreasing exponential functions.
The Ramanujan-type measures thus obtained (with support on the infinite
interval) admit the transformation $q \to q^{-1}$.

Indeed, let us consider first the Askey--Wilson $q$-beta integral
$$ I_0  (a, b, c, d|q) = \int\limits_{-1}^1 w (x; a, b, c, d|q) \, d x\,,
\eqno (3.33) $$
 which corresponds to the orthogonality relation (3.25) with
$ m = n = 0$. From the point of view of symmetry it is more convenient
to make the change of variable $ \theta = \frac{\pi}{2} - \varphi, \,
-\frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2}$ in (3.33). Consequently, the 
integral in (3.33) can be written as
$$ I_0 (a,b,c,d|q) = 2 q^{-\frac{1}{8}} e_q(q) 
\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\vartheta _2(\varphi, q^{1/2})\, \rho  ({\rm sin} \varphi; a,b,c,d|q)
cos \varphi d\varphi \,, \eqno (3.34) $$
 where, according to (3.31),
$$ \rho ({\rm sin} \varphi; a,b,c,d|q) =  \prod_{v=a,b,c,d} 
e_q (ive^{-i\varphi}) e_q(-ive^{i\varphi}) \,.\eqno (3.35)$$
With the aid of the modular transformation \cite{whitta90}
$$ \vartheta_2 (z|\tau) = \frac{\exp (-i z^2/\pi \tau)}{(-i \tau )^{1/2}} 
\, \vartheta _4 (z \tau^{-1} |- \tau^{-1}) \,,  \qquad
\tau = \frac{i \kappa^2}{\pi}\,, \eqno (3.36) $$
 and the change of variable $\varphi = \kappa x$, the integral (3.34) is 
represented as  
$$ I_0 (a,b,c,d|q) = 2 \sqrt {\pi} q^{-\frac{1}{8}} e_q (q) 
\int\limits_{-\pi /2 \kappa}^{\pi /2 \kappa} \vartheta _4 (\pi i x / \kappa ,
e^{{-\pi ^2} /\kappa^2})\,\times $$

$$ \rho ({\rm sin }\kappa x; a,b,c,d|q)\, e^{-x^2}\, \cos \kappa x d x \, .
\eqno(3.37) $$
Using the expansion
$$\vartheta _4(z,q) = \sum_{k=-\infty}^{\infty} (-1)^k q^{k^2} e^{2ikz} 
\eqno (3.38) $$
and taking into account the uniform  convergence of the series (3.38)
in any bounded domain of values of $z$ \cite{whitta90}, we substitute (3.38) 
into (3.37) and integrate this series termwise.
This gives
$$ I_0 (a,b,c,d|q) = 2 \sqrt {\pi} q^{-\frac{1}{8}} e_q (q) 
\sum_{n=-\infty}^{\infty} (-1)^n
\int\limits_{-\pi /2 \kappa}^{\pi /2 \kappa} 
\rho({\rm sin}\kappa x; a,b,c,d|q) 
e^{-(x + \pi/n\kappa)^2} \cos \kappa x d x\,. \eqno(3.39)$$
 The change of variable $ x_n = x + \frac{\pi}{\kappa} n,\, 
x_n^{\min} = \frac{\pi}{\kappa} (n - \frac{1}{2}) \leq x_n \leq 
\frac{\pi}{\kappa} (n + \frac{1}{2}) = x_n^{\max} $ allows to  sum the 
right-hand side of (3.39) with respect to $n$. This yields the following 
representation for the Askey--Wilson $q$-beta integral \cite{ata95}
$$  I_0 (a,b,c,d|q) = 2 \sqrt {\pi} q^{-\frac{1}{8}} e_q (q) 
\int\limits_{-\infty}^{\infty} \rho({\rm sin}\kappa x; a,b,c,d|q) 
 e^{-x^2} \cos \kappa x d x\,. \eqno(3.40) $$
 
The weight function (3.26) with the parameters $a= -b = \beta ^{1/2}$, and 
$ c = -d = a q^{1/2} = (\beta q)^{1/2} $, corresponds to the continuous 
$q$-ultraspherical polynomials $C_n(x; \beta|q)$ \cite{asis83}.
The relations \cite{gasper90}
$$ (a; q)_\infty = ( a, aq; q^2 )_\infty\,, \qquad 
(a, -a;q)_\infty = (a^2; q^2)_\infty  $$
enable the representation (3.40) for this particular case to be simplified to
$$ \int\limits_{-\infty}^{\infty} e_q(-\beta e^{2i\kappa x})  
e_q(-\beta e^{-2i\kappa x}) e^{-x^2+i\kappa x} dx = 
\sqrt \pi q^{1/8} e_q (\beta^2) E_q(-\beta) E_q(-q \beta)\,. \eqno(3.41) $$
where $E_q (z) $ is the $q$-exponential function, {\it i.e.},
$$
E_q (z) = \sum _{n=0}^{\infty} \, \frac{q^{{n(n-1)} \over {2}}}{(q; q)_n } \, 
z^n = (-z; q)_\infty \, . \eqno (3.42)
$$ 

If one compares (3.41) with the Ramanujan integral \cite{ram88,ask82}
$$ \int\limits_{-\infty}^{\infty} e^{-x^2+2mx} e_q(a q^{1/2} e^{2i\kappa x})
e_q(b q^{1/2} e^{-2i \kappa x}) dx =  
 \sqrt \pi e^{m^2}  e_q(abq) E_q(aqe^{2im\kappa}) E_q (bqe^{-2im\kappa}), 
\eqno (3.43) $$ 
it is easy to verify that (3.41) agrees with (3.43) if one sets
$ 2m = i \kappa$ and $ a = b = -\beta q^{1/2} $.

\newpage 
\vspace{24pt}
%\centerline
\noindent{\large \bf 4 RAMANUJAN-TYPE  MEASURES FOR THE ASKEY--WILSON POLYNOMIALS}
%\vspace{6pt}
%\centerline{\hspace{0.5cm} \large \bf THE ASKEY--WILSON POLYNOMIALS}
\vspace{12pt}

In this section it will be shown that Ramanujan-type measures for a hierarchy 
of Askey--Wilson polynomials can be systematically 
built from simple cases of the continuous $q$-Hermite and 
$q^{-1}$-Hermite polynomials by using the Berg-Ismail 
procedure of attaching generating functions to measures \cite{berg94}. 
Starting with the orthogonality relation at the lower lever of 
the hierarchical ladder, namely, for the $q$-Hermite polynomials, 
one can move up with the aid of the corresponding generating functions 
and eventually obtain the orthogonality for the most general case, i.e.,
Askey-Wilson polynomials.  

\vspace{24pt}
%\centerline 
\noindent{\large \bf 4.1 The continuous $q$-Hermite ladder $(0 < q < 1)$} 
\vspace{12pt}

We begin with the Ramanujan-type orthogonality relation
$$\int\limits_{-\infty}^{\infty}  \, H_m(\sin \kappa x \vert q)
H_n (\sin\kappa x \vert q)\, e^{-x^2}\, \cos \kappa x \, d x  =
\sqrt{\pi} \, q^{1/8} \, (q; q)_n \, \delta_{mn} \eqno (4.1.1)$$
for the continuous $q$-Hermite polynomials, assuming at first that 
 $0 < q <1$ and, therefore, using Ramanujan's notation $ q = \exp 
(-2 \kappa ^2)$. The relation (4.1.1) was derived in 
\cite{atanag94} by the transformation of the standard orthogonality relation 
on the finite interval $- 1 \leq x \leq 1$ for the continuous $q$-Hermite 
polynomials $H_n ( x \vert q )$ \cite{asis83}, based on the use of the 
modular and periodicity properties of the theta-function, 
entering in this measure for $H_n (x \vert q )$. The validity of (4.1.1) 
can be verified with the aid of recurrence relation and the 
explicit form of the difference lowering operator for the $q$-Hermite
polynomials $H_n ( \sin \kappa x \vert q ) $ \cite{atanag94}.
We will give here another derivation of the orthogonality relation (4.1.1). In 
view of the symmetry of the integrand in (4.1.1) with respect to  $n$ and $m$ 
we may assume for definiteness that $m\leq n$.
The Rogers linearization formula \cite{rogers95,bres80}
$$ 
H_m (x \vert q) \, H_n (x \vert q) = \sum_{k=0}^m \frac{(q; q)_n}{(q; q)_{n-m+k}} \left [ {m} \atop {k} \right]_q \, H_{n-m+2k} (x \vert q) \eqno (4.1.2) $$
 and the Fourier expansion \cite{asis83}
$$ 
H_n ( \sin \kappa x \vert q ) = i^n \sum_{k=0}^n \, (-1)^k \left [ {n} \atop 
{k} \right ]_q e^{i(2k-n)\kappa x}   \eqno(4.1.3) $$
for the $q$-Hermite polynomials, where 
$$ \left [{n} \atop {k} \right ]_q =
\frac{(q; q)_n}{(q; q)_k \, (q, q)_{n-k}} $$ 
is the $q$-binomial coefficient,
enable  the left side of (4.1.1) to be represented as
$$I_{mn} (q) = i^{n-m} \sum_{j=0}^m \frac{(-1)^j (q; q)_n}{(q; q)_{n-m+j}}\,
\left [ {m} \atop {j} \right ]_q \, 
\sum_{k=0}^N \, (-1)^k \left [ {N} \atop
{k} \right ]_q\,
\int\limits_{-\infty}^{\infty} \, e^{-x^2+i(2k-N)\kappa x} \,\cos {\kappa x} \, dx
 \eqno (4.1.4) $$
with $N = n - m + 2 j$. The integration over $x$ in (6.1.4) is easily 
performed 
by  the formula (Fourier transformation for the ground state of the linear 
harmonic oscillator)
$$ \int\limits_{-\infty}^{\infty} \, d x \, e^{-x^2+2ixy} = \sqrt {\pi} \, 
e^{-y^2} \eqno (4.1.5) $$
and therefore (4.1.4) takes the form
 $$ 
I_{mn} (q) = i^{n-m}\, \frac{\sqrt{\pi}}{2}\, \sum_{j=0}^m 
\frac{(-1)^j (q; q)_n}{(q; q)_{n-m+j}}\,
\left [ {m} \atop {j} \right ]_q  \sum_{k=0}^N \, (-1)^k \left[ {N} \atop
{k} \right ]_q \,
\left [ q^{\frac{1}{2} (k - \frac{N-1}{2})^2} + q^{\frac{1}{2}(k - \frac{N+1}
{2})^2} \right ]\, .  \eqno(4.1.6) $$
The sum over $k$ in (4.1.6) is calculated with the aid of the identity (see, 
for example, \cite{gasper90})
$$ 
(z; q)_n = \sum_{k=0}^n \left [{n} \atop {k} \right ]_q (-z)^k \,    
q^{\frac{k(k-1)}{2}}\,. \eqno (4.1.7) $$
This leads to the expression
$$
q^{\frac{1}{8} (N^2+1)} \left [ q^{\frac{N}{4}} ( q^{-\frac{N}{2}} ; q )_N +
q^{-\frac{N}{4}} ( q^{1-\frac{N}{2}} ; q )_N \right ] \,  . \eqno (4.1.8) $$
Since $(1 - a) (a  q; q)_n = ( 1 - a q^n ) ( a; q )_n $, the square bracket
in (4.1.8) vanishes for any positive number $N$, whereas for 
$N = n - m + 2 j =
0 $ it is equal to two. Thus, we arrive at the right side of (4.1.1).

After the change of variable $\theta = \frac{\pi}{2} - \kappa x$, the 
conventional generating function for the continuous $q$-Hermite polynomials 
\cite{rogers95}
$$
e_q (t e^{i \theta} ) \, e_q (t e^{-i\theta}) =  ( t e^{i \theta}, 
t e^{-i \theta}; q)_{\infty}^{-1} = 
\sum_{n=0}^{\infty} \, \frac{H_n (\cos \theta \vert q)}{(q; q)_n } \, t^n , 
\qquad \vert t \vert < 1 \, , \eqno (4.1.9) $$
can be written as 
$$
e_q (i t e^{ - i \kappa x})\,e_q ( - i t e^{i \kappa x}) =   
(i t e^{- i \kappa x}, \, -i t e^{i \kappa x}; q)_{\infty}^{-1} = 
\sum_{n=0}^{\infty} \, \frac{H_n (\sin \kappa  x \vert q)}{(q; q)_n } \, t^n 
\, .  \eqno(4.1.10) $$
Now let us multiply both sides of the orthogonality relation (4.1.1) 
for the $q$-Hermite polynomials by $ t^m \tau ^n (q; q)_m^{-1} \, 
(q; q)_n^{-1} $ and then sum over $m$ and $n$ from 
zero to infinity, using the generating function (4.1.10)  and definition (3.27)
$$ e_q (z) = \sum_{n=0}^\infty \frac{z^n}{(q; q)_n}\, . \eqno (4.1.11)$$
The result is
$$
\int \limits_{-\infty}^{\infty} \, \prod_{v=t, \tau} e_q (i v e^{-i \kappa x}) \, e_q (- i v e^{i \kappa x})\, e^{- x^2} \cos \kappa x \, d x =
\sqrt {\pi} \, q^{1\over 8} \, e_q (t \tau) \eqno (4.1.12) $$
and the integrand in the representation (4.1.12) may serve as a weight 
function for a family of polynomials, depending on the two more 
parameters $t$ and $\tau $ in addition to $q$. The possibility of finding such 
a family is simplified by the circumstance \cite{ask85,berg94}, that the 
multiplication of the integrand 
in (4.1.12) by the factor $ (i v e^{-i \kappa x}, \, 
- i v e^{i \kappa x}; q)_k 
\,,\, v = t, \tau $, just transforms the parameters $v$ to $ v q^k $, since
$$ 
e_q (a q^k ) = (a; q)_k \, e_q (a) \, , \qquad  k = 0,1,2,\ldots 
\eqno (4.1.13)$$
Indeed, the same reasoning as in Ref. \cite{berg94} {\it mutatis 
mutandis} shows that the Al-Salam-Chihara polynomials \cite{salam76}
$$ p_n (\sin \kappa x ; t, \tau ; q) =  \frac{(t \tau ; q)_n}{t^n} \,\, 
{_3 \phi _{2}} \left [ {{q^{-n}, i t e^{-i \kappa x}, - i t e^{i \kappa x}}
\atop {t \tau , 0}}; q, q \right ]  \eqno (4.1.14) $$   
are orthogonal with respect to the weight integrand in (4.1.12), namely,
$$
\int\limits_{-\infty}^{\infty} \, p_m (\sin \kappa x ; t, \tau ; q) \, 
p_n (\sin {\kappa x} ; t, \tau ; q) \times \eqno (4.1.15) $$
$$ \prod _{v=t, \tau} \, e_q (i v e^{-i \kappa x}) 
\, e_q (- i v e^{i \kappa x}) \, e^{- x^2} \cos \kappa x \, d x = 
\sqrt {\pi} \, q^{\frac{1}{8}} \, ( q; q)_n \, e_q (t \tau q^n )\, 
\delta_{mn} \,.   $$
The basic hypergeometric series $_3 \phi _2 $ in (4.1.14) is defined as 
\cite{gasper90}
$$
_3 \phi _{2} \left [ {{ a_1, a_2, a_3 } \atop { b_1, b_2}} ; q, z \right ]
 = \sum _{k=0}^{\infty} \,
\frac{(a_1, a_2, a_3; q)_k}{(b_1, b_2, q; q)_k }\, z^k \eqno (4.1.16) $$
and the Al-Salam-Chihara polynomials are symmetric in the parameters $t$ and 
$\tau $ \cite{berg94}. Observe that the normalization in (4.1.14) slightly 
differs from the one, employed in Refs.~\cite{salam76} and \cite{berg94}, and 
is chosen in a such way, that the polynomials (4.1.14) are  a particular case 
of the more general five-parameter Askey-Wilson polynomials, {\it i.e.}, they 
coincide with $p_n ( \sin {\kappa x}; t, \tau , 0, 0 \vert q )$ ( see the
formula (4.1.22) below ).
  
A useful consequence of the first step is the following integral for the 
Al-Salam-Chihara polynomials (4.1.14)
$$
\int\limits_{-\infty}^{\infty} \, p_n (\sin \kappa x ; t, \tau ; q) \, 
 e_q (i t e^{-i \kappa x}) 
\, e_q (- i t e^{i \kappa x}) \, e^{- x^2} \cos \kappa x \, d x = 
\sqrt {\pi} \, q^{\frac{1}{2} (n - \frac{1}{2})^2} (- \tau )^n  \,.
\eqno (4.1.17) $$
To verify (4.1.17), use the formulas (4.1.13) -- (4.1.16) and the 
Chu-Vandermonde $q$-sum
$$
_2\phi _{1} ( a, q^{-n} ; c; q, q) = \frac{(c a^{-1} ; q)_n}{(c; q)_n} \, a^n
\, . \eqno (4.1.18) $$

To repeat the same procedure with the Al-Salam-Chihara polynomials as the 
starting point, we can use the generating function \cite{berg94}
$$ \sum _{n=0}^{\infty} \frac{ t^n}{(q; q)_n} \, p_n (\sin 
\kappa x ; t_1, t_2; q) = \frac{(t t_1 , t t_2 ; q)_{\infty}}
{(i t e^{-i \kappa x}, - i t e^{i \kappa x} ; q)_{\infty}} \,. \eqno(4.1.19) $$
Indeed, multiplying both sides of the orthogonality relation (4.1.15) ( with 
$ t = t_1$ and $\tau = t_2 $ )
by the factor $  (q ; q )_m^{-1}  \, (q; q)_n^{-1}\, t_3^m \, t_4^n $ and then summing over $m$ and $n$ from 
zero to infinity with the aid of (4.1.19) and the $q$-binomial formula 
\cite{gasper90}
$$ 
\sum_{n=0}^{\infty} \, \frac{(a; q)_n}{(q; q)_n} \, z^n =  
\frac{(a z; q)_\infty}{(z; q)_\infty} \, , \eqno (4.1.20)$$
we obtain the Ramanujan-type representation for the Askey-Wilson $q$-beta 
integral \cite{ata95,ata94}
$$
\int\limits_{-\infty}^{\infty} \,\prod _{j=1}^{4} \, 
e_q (i t_j e^{-i \kappa x}) 
\, e_q (- i t_j e^{i \kappa x}) \, e^{- x^2} \cos \kappa x \, d x =
$$
$$
\sqrt {\pi} \, q^{\frac{1}{8}} \, E_q (-  t_1 t_2 t_3 t_4 ) \, 
\prod _{1\leq j < k \leq 4} \, e_q (t_j t_k ) \, ,
  \eqno (4.1.21)    $$
where $E_q (z) $ is the $q$-exponential function  (3.42). 
The polynomials, orthogonal with respect to the weight function, represented 
by the integrand in the left side of (4.1.21), belong to the well-known 
family of the Askey-Wilson polynomials \cite{ask85} ({\it cf.} (3.21))

$$\begin{array}{l}
\hspace{-.7cm} p_n ( \sin \kappa  x ; t_1, t_2, t_3, t_4 \vert q) =  \\
\\
=\displaystyle  ( t_1 t_2, t_1 t_3, t_1 t_4; q )_n\, t_1^{-n} {}_4
  \phi _3 \left [ {{q^{-n} ,\,\, t_1 t_2 t_3 t_4 q^{n-1}, \,\, i t_1 
e^{-i \kappa x}, \,\, - i t_1 e^{i \kappa x}} \atop { t_1 t_2 \, , \,\, 
t_1 t_3 \, ,\, \, t_1 t_4}} ; q,  q \right ] \, . 
\end{array}\eqno (4.1.22)    
$$
The use of the formulas (4.1.13), (4.1.21) and the Pfaff-Saalsch\"utz $q$-sum 
\cite{gasper90}
$$ _3 \phi _{2} \left [ {{q^{-n},\,\, a,\,\, b} \atop { c, \,\, a b c^{-1} 
q^{1-n}}} ; q\,, \, q \right ] = \frac{( c / a , \, c / b \, ; q)_n}
{(c,\, c / {a b}; q)_n} \eqno (4.1.23)$$
leads to the Ramanujan-type orthogonality relation 
$$ \int\limits_{-\infty}^{\infty} \, p_m ( \sin {\kappa x}; t_1 , t_2 , t_3 , 
t_4 \vert q) \, p_n ( \sin {\kappa x}; t_1 , t_2 , t_3 , t_4 \vert q) \times $$
$$\prod _{j=1}^{4} \, e_q ( i t_j e^{-i \kappa x} ) \, e_q ( - i t_j e^{i 
\kappa x} ) \, e^{- x^2} \, \cos {\kappa x} \, d x = \eqno (4.1.24)$$
$$ \sqrt {\pi}\, q^{1\over 8} \, (q; q)_n \,  
\frac {E_q (- t_1 t_2 t_3 t_4 q^{n-1})}
{ 1 - t_1 t_2 t_3 t_4 q^{2n-1} } \,
\displaystyle{\prod _{1 \leq j < k \leq 4}} \, e_q  (q^n\, t_j \, t_k ) \,\, 
\delta_{mn}  $$    
for the Askey-Wilson polynomials (4.1.22), obtained by a different method in 
the Refs. \cite{ata95,atak95}.

\vspace{24pt}
%%\centerline 
\noindent{\large \bf 4.2 The continuous $q^{-1}$-Hermite ladder $(q > 1)$ }
\vspace{12pt}

The existence of the orthogonality relation of Ramanujan-type
$$
\int\limits_{-\infty}^{\infty}  \, h_m (\sinh {\kappa x} \vert q)\, 
h_n ( \sinh {\kappa x} \vert q) \, e^{-x^2} \, \cosh {\kappa x} \, d x = 
\sqrt {\pi} \, q^{-\frac{1}{2} ( n + \frac{1}{2})^2} \, (q ; q)_n \, 
\delta_{mn} \eqno (4.2.1) $$
for the continuous $q^{-1}$-Hermite polynomials \cite{askey89}
$$ h_n ( x \vert q) = i^{-n} H_n ( i x \vert q^{-1})  \eqno(4.2.2) $$
became clear even before that for the continuous $q$-Hermite 
polynomials $ H_n ( x \vert q) $. It has emerged in the course of 
constructing a simple difference realization of the Heisenberg $q$-algebra on 
the full real line \cite{afw94}. From the definition (4.2.2) and inversion 
formula
$$
( q^{-1} ;  q^{-1}  )_n = (-1)^n \, q^{-\frac{n(n+1)}{2}} \, (q ; q)_n \, ,
\qquad n = 0,1,2, \ldots \eqno(4.2.3)$$
it follows that the orthogonality relations (4.1.1) and (4.2.1) are in fact 
related to each other by the substitution $q \to q^{-1} \, 
( \kappa \to i \kappa )$.
In other words, derivation of the Ramanujan-type orthogonality relation 
(4.2.1)  goes through in exactly the same way as of (4.1.1) in the previous 
section, namely, by using the linearization formula $(m \leq n)$
$$
h_m (x \vert q) \, h_n (x \vert q) = \sum_{j=0}^{m} \, (-q^j)^{j-m} \, \left [ {m} \atop {j} \right ] _q \, (q^{-n} ; q)_{m-j} \, h_{n-m+2j} (x \vert q)\, , 
 \eqno (4.2.4) $$
the explicit representation for the continuous $q^{-1}$-Hermite polynomials
$$
h_n (\sinh {\kappa} x \vert q) =  \sum_{k=0}^{n} \, (-1)^k \, q^{k(k-n)} 
\left [ {n} \atop {k} \right ] _q \, e^{(n-2k) \kappa x} \,  , \eqno (4.2.5) $$
and the integral (4.1.5) with $ y = i t $. 

In complete analogy with the case $0 < q < 1$, we can combine 
the orthogonality relation (4.2.1) with the generating function 
$$
\sum  _{n=0}^{\infty} \frac{(-it)^n \, q^{\frac{n(n-1)}{2}}}{(q ; q)_n} \,
h_n (\sinh {\kappa x} \vert q) = ( i t e^{\kappa x} \, , - i t e^{- \kappa x}
\, ; q)_{\infty} = E_q (i t e^{- \kappa x})\, E_q (  - i t e^{\kappa x})
\eqno (4.2.6)  $$
to obtain the representation ( {\it cf}. (4.1.12))
$$
\int \limits_{-\infty}^{\infty} \, \prod _{v=t,\tau} \, 
E_q (i v e^{-\kappa x})\,
E_q (- i v e^{\kappa x})\, e^{-x^2} \, \cosh \kappa x \, d x =
\sqrt {\pi} q^{-\frac{1}{8}}\, E_q (- q^{-1} t \tau ) \, . \eqno (4.2.7) $$
Polynomials, orthogonal with respect to the integrand in (4.2.7), are again 
the Al-Salam-Chihara polynomials ({\it cf.} the definition (4.2.2))
$$ \widetilde p_n (\sinh {\kappa x} ; t , \tau ; q ) =
 i^{-n} p_n ( i \sinh \kappa x ; t , \tau ; q^{-1} ) =$$
$$ \frac{(t \tau ; q^{-1})_n} {( i t )^n} \, _3 \phi _{2} \left [ 
{{q^n , i t e^{\kappa x} \,, \, - i t e^{ - \kappa x}} \atop {t \tau ,\,\, 0}} ; 
q^{-1}, q^{-1} \right ]\,, \eqno (4.2.8)  $$
but corresponding to the case when the parameter $q >1$. To verify that one 
needs to reverse the order of summation in (4.2.8)  , leading to the explicit 
sum of the form ({\it cf.} \cite{berg94} )
$$
\widetilde p_n ( \sinh {\kappa x}; t , \tau ; q ) = q^{-\frac{n(n-1)}{2}} \, 
( i t )^n \, (i t^{-1} e^{\kappa x}, - i t^{-1} e^{-\kappa x} ; q)_n 
\times \,$$

$$\sum _{n=0}^{n} 
\frac{(q^{-n} , t \tau q^{1-n} ; q)_k \, q^k}{(i t q^{1-n} e^{\kappa x}, \,
 - i t q^{1-n} e^{-\kappa x}\,, \, q ; q)_k} \, .  \eqno (4.2.9) $$
In fact, let us consider the product
$$ \widetilde p_n ( \sinh {\kappa x} ; t , \tau ; q ) \,
 E_q (i q t e^{-\kappa x}) \, E_q(- i q t e^{\kappa x})  \, .\eqno (4.2.10) $$
First, the factor $ (i t^{-1} e^{\kappa x}, - i t^{-1} e^{-\kappa x} ; q)_n$
from the representation (4.2.9) is easily absorbed into the 
$q$-exponential functions in the above product, since 
$$
E_q (z) \, (- \frac{q}{z} ; q)_n =  q^{\frac{n(n+1)}{2}} \, z^{-n} \, 
E_q (z q^{-n} )\, . \eqno (4.2.11) $$
Then, the resulting $q$-exponential functions $E_q (i t q^{1-n} \, 
e^{- \kappa x}) $
and  $E_q (-i t q^{1-n} \, e^{\kappa x}) $ can be combined with the factors 
$ (-i t q^{1-n} \, e^{- \kappa x}; q)_k\, $  and $\, (i t q^{1-n} \, 
e^{\kappa x}; q )_k $ in the denominator of the summand in (4.2.9), 
respectively, by using the formula
$$ E_q (z) = ( -z; q)_k \, E_q (z q^k) \, . \eqno (4.2.12) $$
Thus, the product (4.2.10) may be written as
$$
\widetilde p_n( \sinh {\kappa x}; t , \tau ; q )  \, 
E_q ( i q t e^{-\kappa x}) \, E_q (- i q t e^{\kappa x}) =$$
$$ \frac{q^{\frac{n(n-1)}{2}}}{( -i t )^n} \,\sum _{k=0}^{n} \frac{(q^{-n},
t \tau q^{1-n}; q)_k \, q^k}{(q ; q)_k } \, E_q (i t q^{k+1-n} \, 
e^{-\kappa x}) \, E_q (- i t q^{k+1-n} \, e^{\kappa x})\, , \eqno (4.2.13)  $$
where the entire dependence of this product on the variable $x$ 
is accumulated in 
the arguments of two $q$-exponential functions only. In other words, the 
derivation of an orthogonality relation for the Al-Salam-Chihara polynomials 
(4.2.8) reduces to computing the known integral (4.2.7), namely,
$$
I_{mn} (t, \tau ; q) = \int \limits_{-\infty}^{\infty} 
\tilde p_m ( \sinh {\kappa x} ; t , \tau; q ) \, 
\tilde p_n ( \sinh {\kappa x} ; t , \tau ; q)  \times $$
$$ \prod _{v=t, \tau}
\, E_q (i q v e^{-\kappa x}) \, E_q ( - i q v e^{\kappa x} )\, e^{-x^2}\,
\cosh {\kappa x}\, d x =   \sqrt {\pi} \,\, \frac{q^{\frac{n(n-1)}{2 }+
\frac{m(m-1)}{2} - \frac{1}{8}}}
{( - i t )^n \, ( - i \tau )^m } \, \times  $$
$$
\sum_{k=0}^{n} \, \frac{(q^{-n}, t \tau q^{1-n}; q)_k\, q^k}{(q; q)_k} \,
\sum_{j=0}^{m} \, \frac{(q^{-m}, t \tau q^{1-m}; q)_j\, q^j}{(q; q)_j} \,
E_q (-t \tau q^{k+j-n-m+1} ) \,.   \eqno (4.2.14)  $$
With the aid of the formulas (4.2.12) and (4.1.18) we can sum over $j$ , which 
yields
$$ I_{mn} (t, \tau; q) = \sqrt{\pi} \,\, i^{n+m} q^{\frac{n(n-1)}{2}-
\frac{m(m-1)}{2} - \frac{1}{8}} \, t^{m-n} \times $$
$$ E_q (-t \tau q^{1-n}) \, 
(q^{-n} ;q)_m \, \sum _{k=0}^{n} \frac{(q^{m-n}; q)_k}{(q; q)_k} \, . 
\eqno (4.2.15) $$
The factor $(q^{-n}; q)_m $ is non-vanishing only if $m \leq n$; but the 
integral (4.2.14) is symmetric in $n$ and $m$, therefore it is nonzero for 
$ m = n$ only, i.e.,
$$ I_{mn} (t, \tau; q) = \sqrt {\pi}\,\, q^{-\frac{1}{2} (n + \frac{1}{2})^2}
\,\, (q; q)_n \,\, E_q (- t \tau q^{1-n}) \, \,\delta _{mn} \,, 
\eqno (4.2.16) $$
where we have used the formula \cite{gasper90}
$$ (q^{-n}; q)_n = (-1)^n \, q^{-\frac{n(n+1)}{2}} \, (q; q)_n \, . $$ 
A counterpart of the integral (4.1.17) for the Al-Salam-Chihara polynomials 
(4.2.8) in the case under consideration has the form
$$ \int\limits _{-\infty}^{\infty} \, \widetilde p_n (\sinh {\kappa x}  ; t , 
\tau ; 
q) \, E_q ( i q t e^{-\kappa x}) \, E_q (- i q t e^{\kappa x}) \, e^{- x^2} 
\cosh {\kappa x } \, d x =\sqrt {\pi} \, ( i \tau )^n \, q^{- \frac{1}{2} 
(n - \frac{1}{2}) ^2} \, . \eqno (4.2.17)$$ 

Having obtained at this intermediate stage the Ramanujan-type orthogonality
relation (4.2.16) for the Al-Salam-Chihara polynomials (4.2.8), for the next 
step to be made we need to determine an analogue of the generating function 
(4.1.19) for the case 
when $q>1$. To this end one can reverse the order of summation in (4.2.8), 
apply the $q$-analogue of the Pfaff-Kummer transformation \cite{gasper90} 
(these two steps are discussed in detail in Ref.\cite{berg94}), and,
finally, use the inversion formula (with respect to the transformation 
$ q \to q^{-1}$ )
$$ ( a; q^{-1})_n = (a^{-1} ; q )_n \, ( - a )^n \, q^{-\frac{n(n-1)}{2}}
\, , \eqno (4.2.18) $$
to express a multiple of the polynomials (4.2.8) in the form of a Cauchy 
product of two sequences , namely,
$$ \widetilde p_n ( \sinh {\kappa x}; t , \tau ; q ) =   
 q^{- \frac{n(n-1)}{2}} \,(q ; q)_n \, ( i t )^n \sum_{k=0}^n \, \frac{(- i \tau ^{-1} 
e^{ - \kappa x} ; q )_k}{(q; q)_k} \, \frac {( i t^{-1} e^{\kappa x} ; 
q )_{n-k}}{(q; q)_{n-k}}\, \left ( \frac{\tau}{t} \right )^k  \, . 
\eqno(4.2.19)  $$  
From the $q$-binomial theorem (4.1.20) and the representation (4.2.19)
 it follows 
that the required generating function has the form
$$ \sum _{n=0}^{\infty} \, \frac{(- i s)^n q^{\frac{n(n-1)}{2}}}{(q ; q)_n} \,
\widetilde p_n ( \sinh {\kappa x} ; t , \tau ; q )  = 
\frac{(i s e^{\kappa x} \, , - i s e^{- \kappa x} ; q)_\infty}
{( s t ,  s \tau ; q )_\infty} \,   \eqno (4.2.20) $$
and it contains (4.2.6) as the particular case when $ t = \tau = 0 $.
The orthogonality relation (4.2.16) and the generating function (4.2.20) 
imply the Ramanujan-type representation \cite{ata95,atak95}
$$ \int\limits _{-\infty}^{\infty} \, \prod _{j=1}^{4} \, E_q (i q t_j 
e^{-\kappa x} ) \, E_q ( - i q t_j e^{\kappa x} ) \, e^{-x^2} \, 
\cosh {\kappa x} \, d x = $$

$$\sqrt {\pi} q^{-1 / 8} \, e_q ( q t_1 t_2 t_3 t_4 ) \, \prod _{1\leq j < k
\leq 4} \, E_q ( - q t_j t_k )    \eqno (4.2.21)   $$
for the Askey-Wilson $q$-beta integral \cite{ask85} with $ q>1$.
The corresponding orthogonal polynomials 
$$\widetilde p_n (\sinh {\kappa x} ; t_1 , t_2 , t_3 , t_4 \vert q ) = 
i^{-n} p_n (i \sinh {\kappa x}  ; t_1 , t_2 , t_3 , t_4 \vert q^{-1} ) 
\eqno (4.2.22) $$
can be interpreted as the continuous $q^{-1}$-Askey-Wilson polynomials ({\it 
cf.} the formulas (4.1.22) and (4.2.2)).Indeed, with the aid of 
(4.2.11), (4.2.21) and the Pfaff-Saalsch\"utz $q$-sum (4.1.23) it is not 
difficult to verify that  \cite{ata95,atak95}
$$ \int \limits_{-\infty}^{\infty} \widetilde p_m ( \sinh {\kappa x} ; t_1 , 
t_2 ,t_3 , t_4 \vert q) \, \widetilde p_n ( \sinh {\kappa x} ; t_1 , t_2 , 
t_3 , t_4 \vert q) \times $$
$$ \prod _{j=1}^{4} \, E_q ( i q t_j e^{-\kappa x} ) \,
E_q (  - i q t_j e^{\kappa x} )  
 e^{- x^2} \cosh \kappa x \,\, d x = 
\sqrt {\pi} \, q^{ - \frac{1}{2}(n + \frac{1}{2})^2} \times $$
$$ (q ; q)_n \,\,
\frac{e_q ( t_1 t_2 t_3 t_4 q^{2-n})}{1 - t_1 t_2 t_3 t_4 q^{1-2n}} \, \,
\prod_{1 \leq j < k \leq 4} E_q ( - t_j t_k q^{1-n}) 
  \delta _{mn} \,, \eqno (4.2.23)     $$ 
provided that $ t_1 t_2 t_3 t_4 \neq q^k $ , where $k$ is an arbitrary 
integer number. When any two of the parameters $ t_j , \, j = 1, 2, 3, 4$, 
are equal to zero, the Ramanujan-type orthogonality relation (4.2.23) 
reproduces the orthogonality relation (4.2.16) for the Al-Salam-Chihara 
polynomials (4.2.8),
whereas when all of them vanish, it reduces to that for the continuous 
$q^{-1}$-Hermite polynomials (4.2.1).  

\vspace{24pt}
%\centerline
\noindent{\large \bf 5 BIORTHOGONAL RATIONAL FUNCTIONS}
\vspace{12pt}

The purpose of this section is to apply the same technique of constructing 
Ramanujan-type measures, as in the previous one, to a family of ${}_4 \phi _3$
biorthogonal rational functions \cite{salam94}, which are unit 
circle analogues of the Askey-Wilson polynomials. It will be shown that
Ramanujan-type continuous measures for a hierarchy of ${}_4 \phi _3$
biorthogonal rational functions can be systematically built
from simple cases of the Rogers-Szeg\H{o} ( for $0 < q < 1$) and 
Stieltjes--Wigert ( for $q > 1$) polynomials by using the Berg--Ismail 
procedure of attaching generating functions to measures.

\vspace{24pt} 
%%%\centerline 
\noindent{\large \bf 5.1 The Rogers-Szeg\H{o} ladder $(0<q<1)$.}
\vspace{12pt}

The $q$-analogue of Hermite polynomials on the unit circle are the 
Rogers-Szeg\H{o} polynomials \cite{szego26,salam57,carl58}
$$  H_n (z; q) = \sum_{k=0}^n \left[ {{n} \atop {k}} \right ]_q \,z^k =
 {}_2\phi \, _{0} ( q^{-n} , 0 ; q, z q^n ) \, , \eqno (5.1.1) $$
where $ \left[ {{n} \atop {k}} \right ]_q $ is the $q$-binomial coefficient.
The basic hypergeometric function ${}_2\phi \,_0 $ in (5.1.1) 
represents a particular case $(r=2,\, s=0,\, a_1 = q^{-n},\, a_2 = 0 )$ of 
the general definition
$$ {}_r\phi \,_s \left [ {{a_1, \ldots, a_r}\atop {b_1, \ldots, b_s}}; q, z
\right ] =  \sum _{n=0}^\infty \, \frac{(a_1, \ldots, a_r; q)_n}{(b_1, 
\ldots, b_s, q; q)_n } \, [(-1)^n\, q^{\frac{n(n-1)}{2}}]^{1+s-r}\, z^n  \,.
\eqno (5.1.2) $$

Szeg\H{o} \cite{szego26} has proved that the polynomials (5.1.1) satisfy the 
orthogonality relation

$$ \frac{1}{2\pi i} \oint _{\vert z \vert = 1}\, H_m(-q^{-\frac{1}{2}}
 z^* ; q)\, H_n (-q^{-\frac{1}{2}} z; q)\, \vartheta_3 (\frac{\log z}{2 i} , 
q^{1/2}) \,
\frac{d z}{z} = \frac{(q; q)_m}{q^m} \, \delta_{mn} \,,  \eqno (5.1.3) $$
where $\vartheta _3 (z, q) $ is the theta-function, {\it i.e.},
$$ \vartheta _3 (z, q) \equiv \vartheta _3 ( z \vert \tau ) = 
\sum _{k=-\infty}  ^\infty \, q^{k^2}\, e^{2 i k z} \eqno (5.1.4) $$
 and $ q = \exp (\pi i \tau ) $ \cite{whitta90}. The orthogonality relation 
(5.1.3) on the unit circle can be transformed into the Ramanujan-type 
orthogonality of the form \cite{atana94}
$$ \int\limits _{-\infty}^\infty \, H_m (-q^{-\frac{1}{2}}\, 
e^{2 i \kappa x}; q)
H_n (-q^{-\frac{1}{2}}\, e^{- 2 i \kappa x}; q)\, e^{- x^2}\, d x =
\sqrt {\pi} \, \frac{(q;q)_n}{q^n} \, \delta _{mn} \,. 
\eqno (5.1.5) $$
Once the relation (5.1.5) is established, it is readily verified by a direct 
evaluation with the aid of the explicit representation (5.1.1), the Fourier
transformation
$$ \int \limits_{-\infty}^\infty \, e^{-x^2+2ixy}\, d x = \sqrt {\pi}\, 
e^{-y^2}\, , \eqno(5.1.6) $$
and Gauss' identity (4.1.7).

It follows from (5.1.1) and the $q$-binomial theorem (4.1.20)
that the generating function for the Rogers-Szeg\H{o} polynomials 
has the form \cite{carl58}
$$\sum _{n=0}^\infty \frac{t^n}{(q;q)_n} \, H_n (x; q) = e_q (t)\, e_q (xt)
\, . \eqno (5.1.7) $$
 Hence, combining (5.1.7) with the orthogonality relation (5.1.5) leads to the 
particular case
$$ \int _{-\infty}^{\infty} \, e_q (-t e^{2i\kappa x}) \, e_q ( - \tau
e^{-2i\kappa x}) \, e^{-x^2} \, d x = \sqrt \pi \, e_q (t \tau)\, 
E_q (- q^{\frac{1}{2}} t) \, E_q (- q^{\frac{1}{2}} \tau)  \eqno (5.1.8) $$  
of a Ramanujan's $q$-extension of Cauchy's form of the beta integral 
\cite{ram88,ask82,pastro85}, namely
$$ \int\limits_{-\infty}^{\infty} \, e_q ( \alpha e^{2i \kappa x} ) \,
 e_q( \beta e^{-2i \kappa x} ) \, e^{-x^2 + 2mx} \, d x = $$
$$=  \sqrt \pi \, 
 e^{m^2} \, e_q (\alpha \beta)
\, E_q (\alpha q^{1/2} e^{2im \kappa}) \, E_q (\beta q^{1/2} e^{- 2im \kappa}) 
\,.  \eqno(5.1.9) $$
Two points demand comments, however. First, it is common to express (see, for 
example, Refs.\cite{gasper90,ram88,ask82,pastro85}) two integrals of 
Ramanujan, 
{\it i.e.}, (5.1.9) and a $q$-extension of Euler's beta integral
$$ \int\limits _{-\infty}^{\infty} \, E_q (\alpha q^{1/2} e^{2 \kappa x}) \, 
E_q (\beta q^{1/2} e^{- 2 \kappa x}) \, e^{-x^2 + 2mx} \, d x =$$
$$=\sqrt \pi \,
e^{m^2} \, E_q (- \alpha \beta ) e_q (\alpha e^{2m\kappa }) \,
e_q (\beta e^{-2m\kappa})\, ,    \eqno(5.1.10) $$
in terms of the infinite product $(z; q)_\infty = \prod _{n=0}^{\infty}\,
(1 - z q^n)$, rather than the $q$-exponential functions (3.27) and (3.42).
The advantage of these forms (5.1.9) and (5.1.10) is that they
 exhibit the fact that they are  related to each other by the 
transformation $q \to q^{-1} $ ( {\it i.e.}, $\kappa \to i \kappa$) 
(see Refs.\cite{nat94,nat94a,nata95} for a more detailed discussion of this point). 
Second, 
since the term $2mx$ in the argument of exponential function 
$\exp (-x^2 + 2mx)$ can be removed by the
translation $x \to x + m$ and redefinition of the parameters $\alpha$ and 
$\beta$, to simplify a proof of (5.1.9) and (5.1.10) it is usually assumed 
that $m = 0$. Thus the parameter $m$ in (5.1.9) and (5.1.10) is 
supposed to  be real. 
In fact, these formulas are also valid for arbitrary complex $m$ 
\cite{ataf95}. This can be proved by a direct evaluation of integrals (5.1.9) 
and (5.1.10) with the aid of the formulas (5.1.6), (4.1.20) and
$$ ( -\frac{q}{z} ; q)_n \, E_q(z) =  q^{\frac{n(n+1)}{2}} \, z^{-n} \, E_q (z 
q^{-n}) \, . \eqno (5.1.11) $$                                             
 As the real part of a complex number $m$ can 
be recovered by the shift $x \to x - \Re \, m $, without loss of generality 
it may be assumed that $m$ is imaginary, {\it i.e.}, $ m = i y $. Consequently, the representations (5.1.9) and (5.1.10) are also instances of Fourier 
transformations and this circumstance will be used in what follows.

Now we return to the integrand in (5.1.8), which represents a weight function 
for the biorthogonal polynomials
$$p_n (z; t, \tau; q) := (q^{-\frac{1}{2}} \tau; q)_n \,\, 
{}_2\phi \,_1 ( q^{-n}, q^{\frac{1}{2}} t; \,
 q^{\frac{3}{2}-n}/ \tau; q, q z/\tau) = $$

$$\frac{(t \tau ; q)_n}{(q^{\frac{1}{2}} t)^n} \, {}_3\phi \,_2 \left [
{{q^{-n}, \, t z , \, q^{\frac{1}{2}} t} \atop {t \tau, \, 0}}; \,q , \, q
\right ]   \, ,  \eqno (5.1.12) $$
studied by P.I.Pastro \cite{pastro85}. Observe that the second line in (5.1.12)
follows from the Jackson transformation formula \cite{gasper90}
$${}_2\phi \,_1 (q^{-n}, \, b; \, c; \, q,\, z) = \frac{(c/b; q)_n}{(c; q)_n} 
\,
{}_3\phi \,_2 \left [{{q^{-n},\, b,\, bz/cq^n} \atop {b q^{1-n} /c ,\, 0}};\,
q,\, q \right ] \eqno (5.1.13) $$
for terminating basic hypergeometric series ${}_2\phi \,_1  $, upon employing
 the relation 
$$ \frac{(a q^{-n}; q)_n}{(b q^{-n};  q)_n } = 
\frac{(q / a; q)_n}{(q / b; q)_n } \, \left ( \frac{a}{b} \right ) ^n \, . 
\eqno (5.1.14) $$
 The normalization of polynomials (5.1.12)
is chosen in such a way, that when $t = \tau = 0 $ they coincide with 
the Rogers-Szeg\H{o} polynomials (5.1.1), {\it i.e.},
$$ p_n (z; 0, 0; q) = H_n (q^{-\frac{1}{2}} z; q) \, . \eqno (5.1.15) $$
This formula follows from the readily verified limit relation
$$ \lim _{c \to \infty} {}_2\phi \,_1 (q^{-n}, a; c; q, cz)  = {}_2 \phi \,_0 
( q^{-n}, a; q, z)\,. \eqno (5.1.16) $$
An explicit evaluation of the Ramanujan-type biorthogonality integral
$$ \int\limits_{-\infty}^{\infty} \, 
p_m ( - e^{2i\kappa x} ; \, t , \tau ; q) \, p_n ( - e^{-2i\kappa x}; \tau,
\, t;\, q) \, e_q (-t e^{2i\kappa x}) \, 
e_q (-\tau e^{-2i\kappa x }) \, e^{- x^2} \, d x = $$

$$ \sqrt \pi \, \frac{(q; q)_n}{ q^n} \, e_q (q^n t \tau) \, 
E_q (-q^{\frac{1}{2}} t) \, E_q (-q^{\frac{1}{2}} \tau) \, \delta _{mn} 
\eqno (5.1.17)   $$
is given in the section 5.3.

Multiplying both sides of the integral (5.1.17) by a 
factor $ t_3^m\, t_4^n \, (q;q)_m^{-1} \,(q;q)_n^{-1}$ and summing over 
indices $m$ and $n$ from 
zero to infinity with the aid of the generating function \cite{salam94}
$$ \sum_{n=0}^\infty \frac{t^n}{(q; q)_n} \, p_n  (z; a, b; q) =
\frac{( a t z, q^{-\frac{1}{2}} b t; q)_\infty}{(q^{-\frac{1}{2}} t z, t; q)_
\infty} = \frac{e_q (t) \, e_q (q^{-\frac{1}{2}} t z )}{e_q (q^{-\frac{1}{2}}
b t ) \, e_q (a t z)} \eqno(5.1.18)$$
for the polynomials (5.1.12), we get $( t = t_1, \tau = t_2 )$
$$ \int\limits_{-\infty}^{\infty} \, e^{-x^2}\, 
E_q (q^{\frac{1}{2}} t_1 t_3 e^{2i\kappa x})  \,
E_q (q^{\frac{1}{2}} t_2 t_4 e^{-2i\kappa x})\, \prod _{j=1}^4 \, 
e_q(-t_j e^{(-)^{j+1}2 i \kappa x})\, d x = $$
$$\sqrt \pi \, e_q (t_1 t_2)\,\, e_q (t_2 t_3)\, e_q (t_3 t_4)\, e_q(t_1 t_4)\,
E_q (- t_1 t_2 t_3 t_4) \,
\prod_{j=1}^4 \, E_q (- q^{\frac{1}{2}} t_j)\,.  \eqno (5.1.19) $$

The integrand in (5.1.19) represents a weight function for the biorthogonality 
relation
$$ \int \limits_{-\infty}^\infty\, r_m ( -e^{2i\kappa x}; t_1, t_2, t_3, t_4; 
q)\, r_n ( -e^{-2i\kappa x}; t_2, t_1, t_4, t_3; q)\,e^{-x^2} 
 E_q (q^{\frac{1}{2}} t_1 t_3 e^{2i\kappa x})  \times $$
$$ E_q (q^{\frac{1}{2}} t_2 t_4 e^{-2i\kappa x})\, 
 \prod _{j=1}^4 \, 
e_q(-t_j e^{(-)^{j+1} 2 i \kappa x})\, d x = 
 \sqrt \pi \, 
\frac{(q, t_1 t_2, t_3 t_4, t_1 t_2 t_3 t_4 q^{n-1}; q)_n}
{ q^n \,(t_1 t_2 t_3 t_4; q)_{2n}} \times $$

$$  \delta _{mn}\, e_q (t_1 t_2)\,\, e_q (t_2 t_3)\, 
e_q (t_3 t_4)\, e_q(t_1 t_4) \, 
E_q (-t_1 t_2 t_3 t_4 ) \, \prod_{l=1}^4 E_q (- q^{\frac{1}{2}} t_l)  \,, 
\eqno (5.1.20) $$
which is satisfied by ${}_4\phi \,_3 $ rational functions of Al-Salam and 
Ismail \cite{salam94}
$$ r_n (z; t_1, t_2, t_3, t_4; q) = \frac{(t_1 t_2; q)_n}{(q^{\frac{1}{2}} t_1)^n}\,  {}_4 \phi \,_3 \left [
{{q^{-n},\, t_1 z, \, q^{\frac{1}{2}} t_1, \, t_1 t_2 t_3 t_4 q^{n-1}}
\atop {q^{\frac{1}{2}} t_1 t_3 z, \,\, t_1 t_2, \,\, t_1 t_4 }}; q, q 
\right ] .\eqno (5.1.21)  $$
 The Ramanujan-type biorthogonality relation (5.1.20) was derived by a 
different method in 
\cite{ataki94}, but it can be also directly  evaluated in exactly the same
way as  the integral (5.1.17) (see section 5.3), namely, by using the 
definition 
(5.1.21), the representation (5.1.19) and the Pfaff-Saalsch\"{u}tz (5.3.7) and 
Chu-Vandermonde (5.3.11) $q$-sums. The normalization in (5.1.21) is chosen in 
such a way that for $ t_3 = t_4 = 0 $ these rational functions and the 
biorthogonality relation (5.1.20) reduce to the biorthogonal polynomials 
(5.1.12) 
and the relation (5.1.17), respectively. Consequently, for $ t_j = 0 , \,\, 
1 \leq j \leq 4 $, the rational functions (5.1.21) coincide with the 
Rogers-Szeg\H{o} polynomials (5.1.1), satisfying the Ramanujan-type 
orthogonality relation (5.1.5).
 
\vspace{24pt} 
%%\centerline 
\noindent{\large \bf 5.2 The Stieltjes-Wigert ladder $(q > 1)$ }
\vspace{12pt}

The Stieltjes-Wigert polynomials \cite{gasper90}
$$ S_n (z; q) = \sum _{k=0}^n \left[ {n}\atop{k} \right ]_q \, 
q^{k^2}\, z^k = {}_1\phi \,_1 (q^{-n};\, 0; \, q,\, z q^{n+1}) \eqno (5.2.1) $$
represent the lower level in another ladder of \, ${}_4 \phi \,_3$ 
biorthogonal rational functions \cite{salam94}, corresponding to the values 
$q > 1$ of the parameter $q$.
They are related to the Rogers-Szeg\H{o} polynomials (5.1.1) by the Fourier
transformation \cite{atana94}
$$ S_n (\alpha\, e^{2\kappa x}; \, q)\, e^{-\frac{x^2}{2}} = 
\frac{1}{\sqrt {2 \pi}} \, \int \limits_{-\infty}^\infty \, H_n (\alpha \, 
e^{-2 i \kappa y}; \,q)\, e^{ixy - \frac{y^2}{2}}\, d y\, , \eqno (5.2.2) $$
where $\alpha $ is an arbitrary complex number. It is interesting to compare
(5.2.2) with the relation
$$ S_n (x; q) = \frac{1}{\pi} \, \int \limits_0^\pi \, H_n (x \, 
e^{i \varphi}; q)\,
\vartheta _3 ( \varphi , q) \, d \varphi \, , \eqno (5.2.3) $$
derived by L.Carlitz in Ref. \cite{carl58}.

 From (5.1.5) and (5.2.2) follows
that the Stieltjes-Wigert polynomials (5.2.1) satisfy the Ramanujan-type
orthogonality relation of the form  \cite{atana94}
$$ \int \limits_{-\infty}^\infty \, S_m (-q ^{-\frac{1}{2}} \, e^{2 \kappa x};
\, q) \, S_n ( - q^{-\frac{1}{2}}\, e^{2 \kappa x} ; \, q) \, e^{-x^2} 
\, d x = \sqrt \pi \, \frac{(q; q)_m}{q^m}\, \delta _{mn} \, .\eqno (5.2.4) $$
This can also be readily verified by a direct evaluation of (5.2.4) with the 
aid of the explicit representation (5.2.1), the  formula ( {\it cf.} (5.1.6))
$$ \int\limits _{-\infty}^{\infty} \, e^{2 x y - x^2} \, d x = \sqrt {\pi} \,
e^{y^2} \, , \eqno (5.2.5) $$
and the identity (4.1.7).

As in the previous section for the Rogers-Szeg\H{o} ladder we could have 
started with the orthogonality relation (5.2.4) and the generating function 
\cite{koekoek94}
$$\sum _{n=0}^\infty \, \frac{t^n}{(q; q)_n} \, S_n (z; q) = e_q (t) \, 
{}_0\phi \,_1 (0; q, q z t) \eqno (5.2.6) $$
 for the Stieltjes-Wigert polynomials (5.2.1). But, for the approach under 
consideration, it is important that generating functions are expressed in 
terms of the $q$-exponential functions (5.1.10) and (5.1.14), because 
biorthogonality relations at every next level follow from their properties 
(5.1.15) and (5.3.5).
Fortunately,  there exist along with (5.2.4) another orthogonality relation
({\it  cf.} Ref.\cite{carl58})
$$ \int \limits_{-\infty}^{\infty} \, S_m (- q^{\frac{1}{2} - m}\,
 e^{2 \kappa x};
\, q) \, S_n (- q^{\frac{1}{2} - n}\, e^{-2 \kappa x}; \, q) \,
e^{-x^2}\, d x = $$
$$ = \sqrt {\pi} \, (- 1)^n \, q^{-\frac{n(n-1)}{2}} \,
(q; \, q)_n \, \delta _{mn} \,,  \eqno (5.2.7)  $$
since the Stieltjes and Hamburger moment problems \cite{shohat70}
for the Stieltjes-Wigert polynomials (5.2.1) are indeterminate.
This is a direct consequence of (5.1.5) and the identity
$$ H_n (z; q^{-1} ) = S_n ( q^{-n} z; q) \, , \eqno (5.2.8) $$
which follows from the definitions (5.1.1) and (5.2.1) upon using the 
transformation property of the $q$-binomial coefficients \cite{gasper90}
$$\left [ {{n}\atop {k}} \right ] _ {q^{-1}} = q^{k(k-n)}\, 
\left [ {{n}\atop {k}} \right ] _ q \, . \eqno (5.2.9) $$

In analogy with the $q^{-1}$-Hermite polynomials $ h_n (x \vert q) = 
i^{-n} \, H_n ( i x \vert q^{-1} ) $ \cite{askey89} it is convenient to 
consider Stieltjes-Wigert polynomials of the form
$$ s_n (z; \, q) = i^{-n}\, H_n (z; q^{-1}) = i^{-n}\, S_n (q^{-n} z; q) \, .
\eqno (5.2.10) $$ 
To distinguish them from the standard Stieltjes-Wigert polynomials (5.2.1), 
they 
will be denoted by the small letter $s$. From (5.2.7) it is evident that
$$\displaystyle{ \int \limits_{-\infty}^{\infty}} s_m (-q^{\frac{1}{2}} \, 
e^{2 \kappa x};\, q)\, 
s_n (-q^{\frac{1}{2}} \, e^{-2 \kappa x};\, q) \, e^{-x^2} \, d x = 
\sqrt {\pi} \, q^{-\frac{n(n-1)}{2}}\,(q; q)_n \,\delta _{mn}. 
\eqno (5.2.11) $$
A generating function for the polynomials (5.2.10) is \cite{carl58}
$$ \sum_{n=0}^\infty \, \frac{(- i t )^n}{(q; q)_n}\, q^{\frac{n(n-1)}{2}}\,
s_n (z; q) = E_q(-t)\, E_q (-z t) \, . \eqno (5.2.12) $$
This can be easily checked by using the well-known Cauchy multiplication 
rule \, 
$$
 \sum_{n=0} ^{\infty} \, \sum _{k=0}^n \, a_k  \, b_{n-k} = 
\sum _{n=0}^\infty \, a_n \, \sum _{k=0}^\infty \, b_k 
$$
for the product of 
two sequences $\{a_n\}$ and $\{b_k\}$, together with the definition of the 
$q$-exponential function (5.1.14). 
Combining (5.2.12) with (5.2.11) leads to the integral 
$$\int\limits _{-\infty}^\infty \, E_q ( q^{\frac{1}{2}} \,t \, 
e^{2 \kappa x}) \,
E_q ( q^{\frac{1}{2}} \,\tau \, e^{- 2 \kappa x}) \,e^{-x^2}\, d x =
\sqrt {\pi} \, e_q(t)\, e_q (\tau)\, E_q (- t \tau ) \, , \eqno (5.2.13) $$
which is a particular case of Ramanujan's identity (5.1.10) with the vanishing
parameter $m$.

The integrand in (5.2.13) represents a weight function for the biorthogonal 
polynomials
$$\widetilde p_n (z; t, \tau ; q) = i^{-n} \, p_n (z;\, t, \, \tau; \, q^{-1} )  = i^{-n} \, (q^{1/2}\, \tau; \, q^{-1} )_n \, {}_2 \phi \,_1 (q^{-n}, \, 
q^{\frac{1}{2}} t^{-1} ; \, \tau \, q^{\frac{3}{2} - n} ; \, q, \, q t z  )
\, ,   \eqno (5.2.14)$$
where $p_n (z; \, t, \tau; \, q) $ are defined by (5.1.12). Indeed, it is not
difficult to verify by a direct evaluation (actually repeating the same line 
of reasoning {\it mutatis mutandis} as for the case of $ 0 < q <1 $, 
considered in the section 5.1), that the following biorthogonality relation 
holds
$$\int\limits_{-\infty}^\infty \, \widetilde p_m ( - e^{2 \kappa x}; \, t, \, 
\tau; 
\, q)\,\widetilde p_n ( - e^{-2 \kappa x}; \, \tau , \, t; \, q)\,  
E_q (q t e^{2 \kappa x}) \,E_q (q \tau e^{-2 \kappa x}) \, e^{-x^2} \, d x =$$

$$\sqrt {\pi} \, q^{-\frac{n(n-1)}{2}} \, (q; q)_n \, E_q (-q^{1-n}\, t \tau)
\, e_q (q^{\frac{1}{2}} t) \, e_q (q^{\frac{1}{2}} \tau )\, \delta _{mn} \, .
  \eqno (5.2.15) $$
Observe that for $t = \tau =0$ the polynomials (5.2.14) reduce to the 
Stieltjes-Wigert polynomials (5.2.10), {\it i.e.},
$$ \widetilde p_n (z; 0, 0; q) = s_n (q^{\frac{1}{2}} z; q) \, . 
\eqno (5.2.16) $$
This follows from (5.2.14), (5.2.1) and (5.2.10) upon using the limit relation
$$ \lim_{a \to \infty} {}_2 \phi \,_1 (q^{-n}, a; c; q, z/a) = {}_1 \phi \,_1
(q^{-n}; c; q, z) \, . \eqno (5.2.17) $$

A generating function for the polynomials (5.2.14)
$$ \sum _{n=0}^{\infty} \frac{(-i \alpha )^n}{(q; q)_n } \,
 q^{\frac{n(n-1)}{2}} \, \widetilde p_n (z; t, \tau ; q) = 
\frac{e_q (q^{\frac{1}{2}} \alpha \tau)\, e_q (\alpha t z)}
{e_q (\alpha) \, e_q (q^{\frac{1}{2}} \alpha z) } \eqno (5.2.18) $$
can be obtained either by transforming $q$ into $q^{-1}$  in (5.1.18), or by a 
direct evaluation of the left member of (5.2.18) with the aid of the 
$q$-binomial
theorem (4.1.20). The biorthogonality relation (5.2.15) and the generating 
functtion (5.2.18)  imply the integral representation
$$ \int\limits _{-\infty}^{\infty} \, e_q (- q^{\frac{1}{2}} 
t_1 t_3 e^{2 \kappa x} )
\, e_q (-q^{\frac{1}{2}} t_2 t_4 e^{-2 \kappa x}) \, \prod _{j=1}^{4}\, E_q 
( q t_j e^{(-)^{j+1} 2 \kappa x} )\, e^{-x^2} \, d x =   $$
$$= \sqrt \pi \, E_q (-q t_1 t_2  ) \, E_q (-q t_2 t_3) \, 
E_q (- q t_3 t_4 )\, E_q (-q t_1 t_4)e_q (q t_1 t_2 t_3 t_4 ) \,
\prod _{k=1}^{4} \, e_q ( q^{\frac{1}{2}} t_k) \, . \eqno (5.2.19)$$
The integrand in (5.2.19) represents a weight function for the biorthogonality 
relation \cite {ataki94}
$$ \int\limits_{-\infty}^{\infty} dx\, e^{-x^2} \, 
\widetilde r_m ( - e^{2 \kappa x}; t_1, t_2, t_3, t_4; q) \, 
\widetilde r_n ( - e^{-2 \kappa x}; t_2, t_1, t_4, t_3; q)\,
e_q (-q^{\frac{1}{2}} t_1 t_3 e^{2 \kappa x}) \times $$
$$ e_q (-q^{\frac{1}{2}} t_2 t_4 e^{ - 2 \kappa x}) 
 \prod _{j=1}^{4} \,
 E_q ( q t_j e^{(-)^{j+1} 2 \kappa x})  = 
 \frac{\sqrt \pi \, (q; q)_n}{1 - T q^{1-2n}} \, q^{- \frac{n(n-1)}{2}} \, 
\times \eqno (5.2.20) $$
$$ E_q (-q^{1-n} t_1 t_2  ) \, E_q (-q^{1-n} t_3 t_4) \, E_q (- q t_1 t_4 )\, 
E_q (-q t_2 t_3) \, e_q (q^{2-n} T ) \, \prod _{k=1}^{4} \,
e_q ( q^{\frac{1}{2}} t_j) \, , $$
where $ T \equiv t_1 t_2 t_3 t_4 \neq q^k $ , $k$ is an arbitrary integer  
number, and $q^{-1}$-rational functions $ \widetilde r_m (z)$ and
$\widetilde r_n (z) $ of Al-Salam and Ismail are defined as ({\it cf.} 
(5.1.21))
$$ 
 \widetilde r_n (z; t_1, t_2, t_3, t_4; q) = i^{-n} \, r_n (z; t_1, t_2, t_3, 
t_4 ; q^{-1} ) = $$

$$ (i q)^n \, q^{-\frac{n^2}{2}}\, \frac{( \widetilde t_1 \widetilde t_2; q)}
{\widetilde t_2 ^n} \, {}_4 \phi \,_3 \, \left [ {{ q^{-n}, \, \widetilde t_1 
z^{-1} ,\, q^{\frac{1}{2}} \widetilde t_1, \, \widetilde T q^{n-1} } \atop
{q^{\frac{1}{2}} \widetilde t_1 \widetilde t_3 z^{-1}, \,\,    
\widetilde t_1 \, \widetilde t_2 , \,\,\widetilde t_1 \widetilde t_4}} ; 
q, \, q \right ]  \eqno (5.2.21)  $$
with $ \widetilde t_j = t^{-1}_j, \, j = 1, 2, 3, 4,$ and $\widetilde T = 
T^{-1}$. The biorthogonality relation (5.2.20) can be directly evaluated by 
using the definition (5.2.21) and the formulas (5.1.11), (5.3.5), (5.3.7) and 
(5.3.11). 

\vspace{24pt}
%\centerline 
\noindent{\large \bf 5.3 Appendix}
\vspace{12pt}

To evaluate the Ramanujan-type biorthogonality integral
$$ I_{mn} (t, \tau ; q) = \int\limits _{-\infty}^\infty \, p_m ( - e^{2 i 
\kappa x}; \, t , \, \tau ; \, q)\,
p_n ( - e^{- 2 i \kappa x}; \, \tau , \, t ; \, q) \times $$
 
$$ e_q (- t e^{2 i \kappa x})\, e_q ( - \tau e^{-2 i \kappa x)}\,
e^{-x^2}\, d x \,,   \eqno(5.3.1) $$
substitute the explicit expression
$$ p_m (  - e^{2 i \kappa x}; t, \, \tau;
q) = ( q^{-\frac{1}{2}} \tau ; q)_m \, \sum _{j=0}^m \, \frac{(q^{-m}, 
q^{\frac{1}{2}} t ; q)_j}{(q, q^{\frac{3}{2} - m} / \tau ; q)_j} \, 
(-\frac{q}{\tau} \, e^{2 i \kappa x})^j \eqno (5.3.2) $$
into (5.3.1) for both polynomials in it, which are defined by (5.1.12). 
This gives
$$ I_{mn} (t, \tau ; q) = (q^{-\frac{1}{2}} \tau ; q)_m \,
(q^{-\frac{1}{2}} t ; q)_n \, \sum _{j=0}^m \, \frac{(q^{-m}, 
q^{\frac{1}{2}} t ; q)_j}{(q, q^{\frac{3}{2} - m} / \tau ; q)_j} \, 
(-\frac{q}{\tau})^j  \times $$
$$ \sum _{k=0}^n \, \frac{(q^{-n}, q^{\frac{1}{2}} \tau; q)_k}
{(q, q^{\frac{3}{2} -n}/ t; q)_k} \, (-\frac{q}{t} )^k \,
\int\limits _{-\infty}^\infty  e^{2i \kappa (j-k)x -x^2}\,
e_q (-t e^{2i\kappa x})\, e_q (-\tau e^{-2i\kappa x})\, d x \,.\eqno (5.3.3) $$
The integration over the variable $x$ in (5.3.3) is performed by (5.1.9) with 
$ m =  2 i \kappa (j - k), \, \alpha = - t $ and $ \beta = - \tau $,
{\it i.e.},
$$
I_{mn} (t,\tau; q) = \sqrt \pi \, e_q (t  \tau) \, (q^{-\frac{1}{2}} \tau ;q)_m \, (q^{-\frac{1}{2}} t; q)_n \, 
\sum _{j=0}^m \, \frac{(q^{-m}, q^{\frac{1}{2}} t ; q)_j}
{(q, q^{\frac{3}{2} - m} / \tau ; q)_j} \, \left (-\frac{q}{\tau} \right )^j  
\times $$
$$ \sum _{k=0}^n \, \frac{(q^{-n}, q^{\frac{1}{2}} \tau; q)_k}
{(q, q^{\frac{3}{2} -n}/ t; q)_k}\, \left  ( -\frac{q}{t} \right)^k \,
 q^{\frac{1}{2}(j-k)^2}\, E_q (-t \, q^{j -  k + \frac{1}{2}})\,
E_q (- \tau \, q^{k - j + \frac{1}{2}})\,.   \eqno (5.3.4)$$
 The formulas (5.1.11) and
$$ E_q (z) = ( - z; q)_k \, E_q (z q^k)  \eqno(5.3.5) $$
 enable the right side of (5.3.4) to be written as 
$$ I_{mn} (t, \tau; q) = \sqrt \pi \, e_q (t  \tau) \, E_q (- q^{\frac{1}{2}} t)
\, E_q ( - q^{\frac{1}{2}} \tau)\, (q^{-\frac{1}{2}} \tau ;q)_m \,
(q^{-\frac{1}{2}} t; q)_n \times $$   

$$\sum _{j=0}^m \, \frac{(q^{-m}, 
q^{\frac{1}{2}}/ \tau ; q)_j}{(q, q^{\frac{3}{2} - m}/ \tau; q)_j} \, 
q^j  \,
 \sum _{k=0}^n \, \frac{(q^{-n},\, q^{\frac{1}{2}} \tau , \,
q^{\frac{1}{2}-j}  / t ;\, q)_k}{(q,\, q^{\frac{1}{2} -j} \tau,\, 
q^{\frac{3}{2} -n}/ t;\, q)_k}\, q^k \, .   \eqno (5.3.6) $$
The sum with respect to the index $k$ in (5.3.6) is a particular case of the 
Pfaff-Saalsch\"utz $q$-sum
$$ {}_3\phi \,_2 \left[ {{q^{-n}, \, a, \, b}\atop {c,\, a b c^{-1} q^{1-n} }}; 
q, \, q \right ] = \frac{( c/a, \, c/b; \, q)_n}{(c, \, c/ {ab}; \, q)_n } 
\eqno (5.3.7)$$
with $ a = q^{\frac{1}{2}} \tau, \, b = q^{\frac{1}{2} - j} /t$ and 
$c = \tau q^{\frac{1}{2} -j} $, therefore it equals to 
$$ \frac{(q^{-j}, \, t \tau ; \, q)_n}{(q^{\frac{1}{2} -j}  \tau, 
q^{-\frac{1}{2}} t; \, q)_n} \, . $$
Consequently, 
$$I_{mn} (t, \tau; q) = \sqrt \pi \,(q^{-\frac{1}{2}} \tau; q)_m \, 
e_q (t \tau q^n) \, E_q (-q^{\frac{1}{2}} t)\, E_q (-q^{\frac{1}{2}} \tau )
\times $$
$$
\sum_{j=0}^m \frac{(q^{-m},\, q^{1/2} / \tau ; \, q)_j}
{(q, q^{\frac{3}{2}-m}/ \tau ; q)_j }\, q^j \, \frac{(q^{-j}; \, q)_n}
{(q^{\frac{1}{2} -j} \, \tau ; q)_n} \, .  \eqno (5.3.8)$$
Due to the factor $(q^{-j}; q)_n $ a sum over the index $j$ in $(5.3.8)$ is 
equal to zero for $n > m $, while for $n \leq m$ it starts with $j = n$. 
Therefore it can be written as
$$ \sum_{j=0}^m \frac{(q^{-m},\, q^{1/2} / \tau ; \, q)_j}{(q, 
q^{\frac{3}{2}-m}/ \tau ; q)_j }\, q^j \, \frac{(q^{-j}; \, q)_n}
{(q^{\frac{1}{2} -j} \, \tau ; q)_n}  = $$

$$\frac{(q^{-m}; q)_n}{(q^{\frac{3}{2} -m} / \tau ; q)_n }\,
\left ( \frac{q^{\frac{1}{2}}}{\tau}\right ) ^n \, \sum_{l=0}^N\,
\frac{(q^{-N}, q^{\frac{1}{2}} / \tau ; q)_l}{( q, \, q^{\frac{3}{2} - N}
/ \tau ; q)_l } \, q^ l \,,  \eqno (5.3.9) $$
 where $N=m-n$ and we have used the formulas (5.1.14) and ( see, for example, 
\cite{gasper90})
$$ (a; q)_{n+k} = (a; q)_n \, (a q^n; \, q)_k \, .    \eqno (5.3.10) $$
The remaining sum over $l$ is calculated by the Chu-Vandermonde formula 
$$ {}_2 \phi \,_1 ( a, q^{-n}; c; q, q) = \frac{(c / a ; q)_n}{(c; q)_n} \, 
a^n \eqno (5.3.11) $$
and gives 
$$ \frac{(q^{1-N}; q)_N}{(q^{\frac{3}{2} - N}/ \tau ; q)_N }\,
 \left ( \frac{q^{\frac{1}{2}}}{\tau} \right ) ^N = 
\delta _{N0} = \delta _{mn} \, , $$
because of the evident relation $( q^{1-N}; q) _N = \delta _{N0}$, valid for
 any nonnegative integer number $N$. Hence, the entire sum over $j$ in 
$(5.3.8)$ is equal to
 $$ \frac{(q^{-n}; q)_n}{( q^{\frac{3}{2}-n} / \tau ; q)_n} \, 
\left ( \frac{q^{\frac{1}{2}}}{\tau} \right )^n \, \delta _{mn} = 
\frac{(q; q)_n}{q^n \, ( q^{-\frac{1}{2}} \tau ; q)_n } \, \delta _{mn} \, , 
\eqno (5.3.12) $$
 where we have again used (5.1.14) with $a = 1$ and $ b = q^{\frac{3}{2}} /
\tau $. Substituting (5.3.12) into (5.3.8), we obtain the following expression 
for the biorthogonality integral (5.3.1)
$$ I_{mn} (t, \tau; q) = \sqrt \pi \, \frac{(q; q)_n}{q^n} \, e_q ( q^n t 
\tau ) \, E_q ( - q^{\frac{1}{2}} t) \, E_q ( - q^{\frac{1}{2}} \tau )\, 
\delta _{mn} \,  $$  
to complete the proof of (5.1.17).


\vspace{24pt}
%\centerline
{\large \bf Acknowledgments}
\vspace{12pt}

I would like to thank Francisco Marcella\'n for inviting me
to this workshop in honour of Andr\'e Ronveaux. I am very
grateful to Renato \'Alvarez-Nodarse, who has carefully read
the manuscript and helped to correct some misprints.

%\vspace{24pt}
%\centerline
%{\large \bf REFERENCES}
%\vspace{12pt}

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\bibitem {atak95} {\sf N.M. Atakishiyev:} {\em Orthogonality of Askey-Wilson polynomials with respect
          to a measure of Ramanujan type.} {\sc Theor. Math. Phys.}, {\bf 102}
 (1995), 23--28.

\bibitem{ata95} {\sf N.M. Atakishiyev:} {\em Kravchuk functions for the finite oscillator approximation.}
 In {\sc NASA Conference Publication 3286},
Edited by D.Han and K.B.Wolf, Greenbelt, Maryland, 1995, 153--164.

\bibitem{nata95} {\sf N.M. Atakishiyev:} {\em  Ramanujan-type continuous measures for
classical $q$-polynomials.} {\sc Theor. Math. Phys.}
{\bf 105} (1995), 1500--1508.

\bibitem{nat94} {\sf N.M. Atakishiyev:} {\em Note on the transformation $q \to q^{-1}$
for orthogonal $q$-polynomials.} {\sc  Preprint No. 26}, IIMAS-UNAM , M\'exico, 1994.


\bibitem{nat94a} {\sf N.M. Atakishiyev:} {\em Note on the transformation $q\rightarrow q^{-1}$
 for orthogonal q-polynomials.} {\sc Proceedings of the IV Wigner Symposium} 
(Guadalajara, Mexico, August 7 -- 11 , 1995), Edited by N.M.Atakishiyev, T.H.Seligman, and K.B.Wolf,
World Scientific, Singapore,  1996, 397--401.

\bibitem{ataki94} {\sf N.M. Atakishiyev:} {\em A Ramanujan-type measures for the Al-Salam 
and Ismail biorthogonal rational functions.} In {\sc Proceedings of the Workshop 
on Symmetries and Integrability of Difference Equations} (May 22-29, 1994, 
Est\'erel, Qu\'ebec, Canada), Edited by D.Levi, L.Vinet and P.Winternitz,
CRM Proceedings and Lecture Notes, American Mathematical Society, Montr\'eal,
1996. 21--27.

\bibitem{ataf95} {\sf N.M. Atakishiyev and P. Feinsilver:}{\em  Two Ramanujan's integrals
with a complex parameter.} In {\sc Proceedings of the IV Wigner Symposium} 
(Guadalajara, Mexico, August 7 -- 11 , 1995), Edited by N.M.Atakishiyev, 
T.H.Seligman, and K.B.Wolf, World Scientific, Singapore, 1996, 406--412.

\bibitem{afw94} {\sf N.M. Atakishiyev, A. Frank and K.B. Wolf:} {\em A simple difference realization
 of the Heisenberg.} {\sc J. Math. Phys.}, {\bf 35} (1994), 3253--3260.

\bibitem{atanag94} {\sf N.M. Atakishiyev and Sh.M. Nagiyev:} {\em On the wave functions of a 
covariant linear oscillator.} {\sc Theor. Math. Phys.}, {\bf 98} (1994), 162--166. 

\bibitem{atana94} {\sf N.M. Atakishiyev and Sh.M. Nagiyev:} {\em On the Rogers-Szego' polynomials} 
{\sc J. Phys. A: Math. Gen.} {\bf 27}  (1994), L611--L615.  

\bibitem{ars95} {\sf N.M. Atakishiyev, M. Rahman and S.K. Suslov:} 
{\em On classical orthogonal polynomials.} 
{\sc Constructive Approximation} {\bf 11} (1995), 181--226.

\bibitem{atsu88} {\sf N.M. Atakishiyev and S.K. Suslov:} 
{\em About one class of special functions.}{\sc Revista Mexicana 
de Fisica} {\bf 34} (1988), 152--167.

\bibitem{ats93} {\em N.M. Atakishiyev and S.K. Suslov:} {\em  Difference hypergeometric
functions.} In {\sc  Progress in Approximation Theory: An International 
Perspective}, Springer Series in Computational Mathematics, Edited by 
A.A.Gonchar and E.B.Saff, {\bf 19}, 1993, 1--35

\bibitem{ataw94} {\sf N.M. Atakishiyev and K.B. Wolf:}{\em Approximation on a finite set of 
points through Kravchuk functions.} {\it Revista Mexicana de Fisica} {\bf 40} (1994), 366--377.

\bibitem{berg94} {\sf C. Berg C. and M.E.H Ismail:} {\em q-Hermite polynomials and classical 
orthogonal polynomials.} {\sc Can. J. Math.}, {\bf 48} (1996), 43--63.  

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Mehler's formula for q-Hermite polynomials.} {\sc Indiana Univ. Math. Journ.} {\bf 29}
 (1980), 577--580.

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theta functions.} {\sc Publicationes Matematicae.} {\bf 5},  
 222--228 (1958).  

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F.G. Tricomi:} {\bf Higher Trans\-cend\-ental  Functions.} Vol. {\bf 2},
 New York: McGraw-Hill, 1953.

\bibitem{gasper90} {\sf G. Gasper and M. Rahman:} {\bf Basic Hypergeometric
Series.} Cambridge: Cambridge University Press, 1990.

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hypergeometric orthogonal polynomials and its $q$-analogue.} {\sc Report 94--05}, 
Delft University of Technology, Delft, 1994.

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(Non-relativistic Theory).} Oxford: Pergamon Press, 1991.
 
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Classical Orthogonal Polynomials of a Discrete Variable.} Berlin, Heidelberg:
Springer Verlag, 1991.
  
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 {\em  Polynomial Solutions of hypergeometric type
 difference Equations and their classification}. {\sc Integral Transforms and Special
 Functions.} {\bf 1}, (1993), 223--249. 
 
\bibitem{pastro85} {\sf P.I. Pastro:}
{\em Orthogonal polynomials and some beta integrals of Ramanujan.}
 {\sc J. Math. Anal. Appl.}, {\bf 112} (1985), 517--540.   

\bibitem{ram88} {\sf S. Ramanujan:} {\bf The Lost Notebook and Other Unpublished
Papers.} New Delhi: Narosa Publishing House, 1988.

\bibitem{rogers95} {\sf L.J. Rogers:} {\em Third memoir on the expansion
of certain infinite products.} {\sc Proc. London Math. Soc.}
 {\bf 26} (1895), 15--32.

\bibitem{shohat70} {\sf J.A. Shohat and J.D. Tamarkin:} {\bf The Problem of Moments.}
Mathematical Surveys, No. 1, Providence, RI: American Mathematical Society, 
1970.

\bibitem {szego26} {\sf G. Szeg\H{o}:} {\em Ein Beitrag zur Theorie der Thetafunktionen.}
{\sc Sitz. Preuss. Akad. Wiss., Phys.-Math. Klasse},  {\bf 19}
 (1926), 242--252; reprinted in {\bf Collected Papers} ( R.Askey, ed.), 
{\bf 1}, Boston, MA: Birkh\H{a}user, 1982, 795--805.

\bibitem{whitta90}  {\sf E.T. Whittaker and G.N. Watson:} {\bf A Course of Modern
Analysis.} $4th$ edition, Cambridge: Cambridge University Press,
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\end{thebibliography}

\begin{flushleft}
{{\bf Natig M. Atakishiyev ${}^{\dagger}$} \\
Instituto de Matem\'aticas, UNAM, \\
Apartado Postal 139--B \\
 62191 Cuernavaca, Morelos, M\'exico
 } \\
{\sf E-Mail}{\small\tt: natig@ce.ifisicam.unam.mx}\\
${}^{\dagger}$ On leave of absence from
Institute of Physics, \\
Azerbaijan Academy of Sciences, \\
Baku 370143, Azerbaijan.
\end{flushleft}


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