George Gasper

Some systems of multivariable orthogonal q-Racah polynomials

In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.

Positivity and Special Functions

Publisher Summary This chapter focuses on positivity and special functions. The solutions of many problems depend on the determination of when a specific function is positive or nonnegative. In many cases, a series or integral representation for the function can be derived in terms of more elementary functions. Sometimes the problem can be reduced to a simpler one involving fewer parameters or it can be transformed into another problem that is easier to handle. Various hypergeometric series representations for the Hahn polynomials are used to write a sum of products of gamma functions. The chapter also describes absolutely monotonic and completely monotonic functions.

Positive Sums of the Classical Orthogonal Polynomials

An expansion as a sum of squares of Jacobi polynomials

Positive Integrals of Bessel Functions

P_n^{(\alpha ,\beta )} (x)

Projection formulas for orthogonal polynomials of a discrete variable

is used to prove that if

An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas

0 \leqq \lambda \leqq \alpha + \beta

Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients

and

Summation Formulas for Basic Hypergeometric Series

\beta \geqq - {1 / 2}

q-EXTENSIONS OF BARNES', CAUCHY'S, AND EULER'S BETA INTEGRALS

, then \[( * )\qquad \sum_{k = 0}^n {\frac{{(\lambda + 1)_{n - k} }}{{(n - k)!}}} \frac{{(\lambda + 1)_k }}{{k!}}\frac{{P_k^{(\alpha ,\beta )} }}(x){{P_k^{(\beta ,\alpha )}(i) }} \geqq 0,\quad - 1 \leqq x < \infty ,\] and the only cases of equality occur when

Positivity of the poisson kernel for the continuous q Jacobi polynomials and some quadratic transformation formulas for basic hypergeometric series

x = - 1