An Extension of the Incomplete Beta Function For Negative Integers

Abstract

The incomplete beta function B-x(a, b) is defined for a, b > 0 and 0 < x < 1. Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function Bx (a, b) for negative integer values of a and b. Further we prove that the function partial derivative(m+n)/partial derivative a(m)partial derivative b(n)B(x)(a, b) exists for m, n = 0, 1, 2,... and all a and b. (C) 2007 Elsevier Inc. All rights reserved.