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 B x (a, b) for negative integer values of a and b. Further we prove that the function ∂ m+n ∂a m ∂b n B x (a, b) exists for m, n = 0, 1, 2, . . . and all a and b.

Some results on the beta function and the incomplete beta function

“…We first consider the integral For further results on the Beta function and the incomplete Beta function, see [3][4][5][6].…”

Some Notes of the Fractional Integral and Derivatives and Its Applications

“…We use neutrix limit [9][10][11][12][13][14][15][16] to defne the frational derivatives for (1)- (6).…”

On defining the q-beta function for negative integers

“…Fisher at al. used neutrices to define the gamma and beta function for all real values [4,5].Özçaḡ et al applied the neutrix limit to extend the definition of the incomplete beta function and its partial derivatives to negative integers, [11,12]. It was shown that the neutrix limit of the q-gamma function and its partial derivatives exist for all values, [13].…”