An extension of the incomplete beta function for negative integers ✩

The incomplete beta function Bx(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 ∂m+n ∂am∂bn Bx(a, b) exists for m,n = 0, 1, 2, . . . and all a and b. © 2007 Elsevier Inc. All rights reserved.