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We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.

Discriminants and their discrete and q-analogs are usually studied in the q-analysis theory. In this paper we propose a unified realization of discriminants using vertex operators coming from infinite dimensional Lie algebras and their quantum deformations as well as Yangian deformations. In this picture all of them appear as matrix coefficients of certain products of vertex operators according to respective cases.

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.

A simple proof is given for Bailey's sum of the very well poised $ _6{\psi _6}$q-extension of the partial fraction decomposition of .

We prove that the function

$\displaystyle {\left( {\frac{b} {a}} \right)^{\mu - v}}\frac{{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}$ is the Laplace transform of an infinitely divisible probability distribution when $ v > \mu \geqslant 0$. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when $ x$

$\displaystyle {\left( {\frac{b} {a}} \right)^{\mu - v}}\frac{{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}$ also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as , the infinite divisibility property holds for both functions.

We derive a family of reproducing kernels for the q-Jacobi polynomials $ \Phi _n^{(\alpha ,\beta )}(x){ = _2}{\Phi _1}({q^{ - n}},{q^{n - 1 + \beta }};{q^\alpha };q,qx)$ satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer's theorem.

We show that certain functions involving quotients of gamma functions are completely monotonic. This leads to inequalities involving gamma functions. We also establish the infinite divisibility of several probability distributions whose Laplace transforms involve quotients of gamma functions.

The expansion of arbitrary power series in various classes of polynomial sets is considered. Some applications are also given.

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.

We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is $ \{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\} $ is a positive parameter. The weight function of the polynomials of the second kind is $ {\{ 1 - ({(1 + c)^2}/4\operatorname{c} )T_k^2(x)\} ^{1/2}}/\vert{U_{k - 1}}(x)\vert$ $ c \geqslant 1$ we pick up discrete masses located at the zeros of $ {U_{k - 1}}(x)$-analogues are also treated. Their continuous spectra are also the above mentioned set. The $ q$-ultraspherical polynomials and another set of -ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.