The Modification of Classical Hahn Polynomials of Adiscrete Variable

We consider a modiication of moment functionals for the Hahn classical polynomials of a discrete variable by adding two mass points at the ends of the interval, i.e., in x = 0 and x = N 1. We obtain the resulting orthogonal polynomials and identify them as hypergeometric functions. The corresponding three term recurrence relation and tridiagonal matrices are also studied. x1 Introduction. The study of orthogonal polynomials with respect to a modiication of a linear functional in the linear space of polynomials with real coeecients via the addition of one or two delta Dirac measures has been performed by several authors 5] 6], 15]. A special emphasis is given to the modiications of classical linear functionals (Hermite 15], Laguerre 11], Jacobi 12] and Bessel 10]). Very recently appear some works related to modiications of the discrete classical measures, more concretly the Charlier, Kravchuk and Meixner measures, via addition of one delta Dirac measures at x = 0 1], 2], 3], 4] and 9]. In this paper we study the polynomials orthogonal with respect to the modiication of the weight function of the classical Hahn polynomials via the addition of two diierent masses at the ends of the interval. In fact we nd one expression for the perturbed or generalized monic Hahn polynomials ^ h A;B;;;; n (x) as well as their representation in terms of the 5 F 4 hypergeometric series. We also analyze the relation between tridiagonal matrices of these perturbed (x) and classical ^ h ;; n (x; N) polynomials. The structure of the paper is as follows. In Section 2, we provide the basic properties of the classical orthogonal Hahn polynomials. In Section 3 we deduce expressions of the monic generalized Hahn polynomials in terms of the classical ones ^ h ;; n (x; N) and the rst backward diierence derivatives of the polynomials ^ h 1;; n (x) and ^ h ;;1 n (x). In Section 4 we nd their representation as hypergeometric functions 5 F 4 and in Section 5 we analyze two particular cases: A 6 = 0 B = 0 and A = 0; B 6 = 0. Finally, in Section 6, from the three term recurrence relation (TTRR) of the classical orthogonal polynomials we nd the TTRR which satisfy the perturbed ones and analyze the relation between tridiagonal matrices associated with the perturbed monic