Mourad E. H. Ismail

A Generalization of Ultraspherical Polynomials.

Some old polynomials of L. J. Rogers are orthogonal. Their weight function is given. The connection coefficient problem, which Rogers solved by guessing the formula and proving it by induction, is derived in a natural way and some other formulas are obtained. These polynomials generalize zonal spherical harmonics on spheres and include as special cases polynomials that are spherical functions on rank one spaces over reductive p-adic groups. A limiting case contains some Jacobi polynomials studied by Hylleraas that arose in work on the Yukawa potential.

The zeros of basic Bessel functions, the functions Jv + ax(x), and associated orthogonal polynomials

Abstract The modified Lommel polynomials are generalized and their orthogonality relation is obtained. As a byproduct we prove that the non-zero roots of Jv + ax(x) are real, simple and interlace with those of Jv + ax − 1(x). The q-Lommel polynomials are shown to play in the theory of basic Bessel function the role played by the Lommel polynomials in the Bessel function theory. The q-Lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support. This is then used to prove that the prositive zeros of x−νJ(2)v(x; q) are real simple and interlace with the zeros of x−ν − 1J(2)v + 1(x; q), when ν > −1. We also establish the complete monotonicity of −1x −1 2 J v + 1 (2) (i√x; q) J v (2) (i√x; q)

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilsons 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehlers formula.

Ladder operators and differential equations for orthogonal polynomials

Under some integrability conditions we derive raising and lowering differential recurrence relations for polynomials orthogonal with respect to a weight function supported in the real line. We also derive a second-order differential equation satisfied by these polynomials. We discuss the Lie algebra generated by the generalized creation and annihilation operators. From the differential equations, Plancherel - Rotach type asymptotics are derived. Under certain conditions, stated in the text, an Airy function emerges.

Completely monotonic functions involving the gamma and q-gamma functions

We give an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic. Each such function gives rise to an infinitely divisible probability distribution. Other similar results are also obtained for specific combinations of the gamma and q-gamma functions.

Completely monotonic functions associated with the gamma function and its q-analogues

Several functions involving the gamma function Γ(x) and the q-gamma function Γq(x) are proved to be completely monotonic. Some of these results are used to establish the infinite divisibility of a number of probability distributions defined via their moment generating functions.

Special Functions, Stieltjes Transforms and Infinite Divisibility

We establish the complete monotonicity of several quotients of Whittaker (Tricomi) functions and of parabolic cylinder functions. These results are used to show that the F distribution of any positive degrees of freedom (including fractional) is infinitely divisible and self-decomposable. We also prove the infinite divisibility of several related distributions, including the square of a gamma variable. We also prove that

Inequalities and monotonicity properties for gamma and q -gamma functions

x^{{{(\nu - \mu )} /2}} {{I_\mu (\sqrt x )} /{I_\nu (\sqrt x )}}

Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics

is a completely monotonic function of x when

q -Taylor theorems, polynomial expansions, and interpolation of entire functions

\mu > \nu > - 1