NEW GENERATING FUNCTIONS FOR THE KONHAUSER MATRIX POLYNOMIALS

Varma et. al. [Ars Combin. 100 (2011) 193-204] introduced the concept of the Konhauser matrix polynomials. In this paper, we obtain some generating functions for these matrix polynomials. Finally, we focus on some special cases.


Introduction
In 2011, Varma et.al. de…ned the pair of the Konhauser matrix polynomials as follows: ( 1) r n r 1 (A + (kr + 1)I)( x) kr ; (1.1) where A is a matrix in C N N satisfying the condition Re( ) > 1 for every 2 (A); (1.3) is a complex number with Re( ) > 0 and k 2 Z + (see [5]): Here, for any matrix A in C N N ; Pochhammer symbol is de…ned by for the polynomials Y and for the polynomials Z and 1 F k is de…ned as for A 1 ; :::; A k and B are matrices in C N N satisfying condition A i + sI is invertible for s 2 N, i = 1; :::; k;see [5].In the present paper, we obtain multilinear and multilateral generating functions for the pair of the Konhauser matrix polynomials.Some special cases are also given.

multilinear and multilateral generating matrix functions
In this section, we give theorems which derive several substantially more general families of bilinear, bilateral generating functions for the Konhauser matrix polynomials de…ned by (1.1) and (1.2).Using the similar method considered in [1,2,3,4], we obtain the main theorems.
Theorem 2.1.Corresponding to a non-vanishing function (y 1 ; :::; y s ) of complex variables y 1 ; :::; y s (s 2 N) and of complex order , let Upon changing the order of summation in (2.4), if we replace n by n + pl; we can write  (x 1) t where C and D are matrices in C N N satisfying the spectral conditions Re(z) > 1 for each eigenvalue z 2 (C); and Re( ) > 1 for each eigenvalue 2 (D), CD = DC and F 4 (A; B; C; D; x; y) is de…ned by where C + nI and D + nI are invertible for every integer n 0 in p x + p y < 1: Then we obtain the following example which provides a class of bilateral generating functions for the Jacobi matrix polynomials and the Konhauser matrix polynomials Y (y + 1) 2 ; where q (y 1) 2 + q (y+1) 2 < 1: , where the Gegenbauer matrix polynomials C D n (y) are de…ned by means of the generating function in [6]: where D is a matrix in C N N satisfying the spectral condition where A and B are matrices in C N N such that A satis…es conditions in (1:6): + k (y; k 2 ) ( ; 2 N 0 ) for s = 1 in Theorem 2.2, then we can give the following example which provides a class of bilateral generating functions for the pair of the Konhauser matrix polynomials.
shown to yield various classes of multilateral and multilinear generating functions for the Konhauser matrix polynomials de…ned by (1.1) and (1.2).

(
A) n = A(A + I):::(A + (n 1)I); n 1 ; (A) 0 = I: They show that the Konhauser matrix polynomials Z (A; ) n (x; k) and Y (A; ) n (x; k) are biorthogonal with respect to the weight matrix function x A e x : Furthermore they derive the following generating matrix functions: (A;) n (x; k); where A and B are matrices in C N N satisfying the conditions Re( ) > 1 for every 2 (A) AB = BA: The proof is similar to Theorem 2.1.Now, we obtain some special cases for generating functions.Firstly, if we set + k (y ) = P (C;D) + k (y) ( ; 2 N 0 ) for s = 1 in Theorem 2.1, where the Jacobi matrix polynomials P (C;D) n (y) are de…ned by means of the generating function in [7]: t n = F 4 I + D; I + C; I + C; I + D;