A connection between a generalized Pascal matrix and the hypergeometric function

Abstract

The n × n generalized Pascal matrix P( t) whose elements are related to the hypergeometric function 2F 1(a, b; c; x) is presented and the Cholesky decomposition of P( t) is obtained. As a result, it is shown that 2F 1(−a, −b; 1; x) = ∑ k=0 min(a,b) ( a k )( b k )x k is the solution of the Gauss's hypergeometric differential equation, x(1 − x)y″ + [1 + (a + b − 1)x]y′ − aby = 0 . where a and b are any nonnegative integers. Moreover, a recurrence relation for generating the elements of P( t) is given.