An extension of the incomplete beta function for negative integers

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.