Extended symmetric Pascal matrices via hypergeometric functions

Abstract

In this paper, we give a matrix representation of the hypergeometric functions of the type 2F 1(a,b;c;x) . As a result, we obtain a connection between the hypergeometric functions, the Legendre polynomials and the Delannoy numbers. Moreover, it is shown that each entry of P n ( x, y) P n ( x, y) T can be represented by the hypergeometric functions where [P n(x,y)] ij=x i−jy i+j−2 i−1 j−1 is the extended generalized Pascal matrix which is defined by Zhang and Liu [Linear Algebra Appl. 271 (1998) 169].