Mizan Rahman

On classical orthogonal polynomials

Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.

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.

Projection Formulas, a Reproducing Kernel and a Generating Function for q-Wilson Polynomials

A projection formula for the q-Wilson polynomials

On a general q -Fourier transformation with nonsymmetric kernels

p_n (x;a,b,c,d)

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

is obtained which is then used to construct a reproducing kernel. Using Askey and Wilson’s q-analogue of the beta integral an integral representation is obtained for a very well-poised

Product and addition formulas for the continuous q -ultraspherical polynomials

{}_8 \phi _7

Diagonalizaton of certain integral operators II

as a q-analogue of Euler’s integral formula for a

The Pearson equation and the beta integrals

{}_2 F_1

Barnes and Ramanujan-type integrals on the q -linear lattice

. As an application of these results a generating function is obtained for the continuous q-Jacobi polynomials introduced by Askey and Wilson.

Quadratic q-exponentials and connection coefficient problems

Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.