The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate.

We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

An identity of L. Carlitz for a bibasic hypergeometric series is used to find some summation formulas for series in which the bases are either q and q2 or q and q3,0 < q < 1 In general, these series are neither balanced nor very-well-poised in the usual sense

It is known that the n-th denominators Qn (α, β, z) of a real J-fraction

where

form an orthogonal polynomial sequence (OPS) with respect to a distribution function ψ(t) on ℝ. We use separate convergence results for continued fractions to prove the asymptotic formula

the convergence being uniform on compact subsets of

We give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

Necessary multiplier conditions for Laguerre expansions are derived and discussed within the framework of weighted Lebesgue spaces.

Explicit solutions to the recurrence relation for associated continuous Hahn polynomials are derived using 3F2 contiguous relations. These solutions are used to obtain a new continued fraction and the associated absolutely continuous measure. An exceptional case is shown to yield entry 33 in Chapter 12 of Ramanujan's second notebook.

Two families of associated Wilson polynomials are introduced. Both families are birth and death process polynomials, satisfying the same recurrence relation but having different initial conditions. Contiguous relations for generalized hypergeometric functions of the type 7F6 are derived and used to find explicit representations for the polynomials and to compute the corresponding continued fractions. The absolutely continuous components of the orthogonality measures of both families are computed. Generating functions are also given.

The second order differential equation of Littlejohn-Shore for Laguerre type orthogonal polynomials is generalized in two ways. First the positive Dirac mass can be situated at any point and secondly the weight can be any classical weight modified by an arbitrary number of Dirac distributions.

A q-analogue of the orthogonality property of the Bessel functions on the zeros is obtained in terms of a q-integral.

Let α and β be any two real numbers and let be the Jacobi polynomial sequences. For any non-zero real number a, is an orthogonal polynomial sequence with respect to convolution if and only if either (i) b = 1, α = 0 and β + 1 is not equal to a negative integer or (ii) b = — 1, β = 0 and α + 1 is not equal to a negative integer.

We examine the convergence and analytic properties of a continued fraction of Ramanujan and its connection to the orthogonal polynomials of Meixner-Pollaczek.

A seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

A new proof of a product formula for Laguerre polynomials, due originally to Watson, is given. Considering the commutative Banach algebra of radial functions on the Heisenberg groups Hn, n ≧ 2, we observe that Watson's formula holds for z = 1,2, 3, …. Then, applying a complex function theory argument, we establish the validity of this formula for other complex values of z, i.e. for Re z > - 1/2.

In his Ph.D. thesis Allaway found all polynomials that can be represented as

and a0bn≠0. We solve the essentially equivalent problem of finding all symmetric polynomials with when are orthogonal with respect to dα(x). The polynomials are the continuous q-ultraspherical polynomials and some of their limiting cases.

We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

A popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form

1.1

Recently, H. M. Srivastava extended certain interesting generating functions of L. Carlitz to the forms:

and

where are general oncand many-parameter sequences of functions. In the present paper some general addition formulas for analogous sequences of functions are derived, and a number of interesting applications of the main results are given.

In this note we obtain a q-analog of a pair of biorthogonal sets of rational functions which have been obtained recently by M. Rahman in connection with the addition theorem for the Hahn polynomials.