Extended symmetric Pascal matrices via hypergeometric functions

摘要

In this paper, we give a matrix representation of the hypergeometric functions of the type 2F1(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 Pn(x,y)Pn(x,y)T can be represented by the hypergeometric functions where [Pn(x,y)]ij=xi−jyi+j−2i−1j−1 is the extended generalized Pascal matrix which is defined by Zhang and Liu [Linear Algebra Appl. 271 (1998) 169].