Extension of the partial derivatives of the incomplete beta function for complex values

In this paper, by using the hypergeometric function and the neutrix limit, we extend the definition of the partial derivatives of the incomplete beta function ∂p+q∂xp∂yqB(z;x,y)(p,q=0,1,2,…) to all complex values of x and y as complex number z satisfying 0 < |z  | < 1. Moreover, we establish the recursive formula of ∂p+q∂xp∂yqB(z;x,y) for x≠−q,−q−1,−q−2,…,p,q=0,1,2,…x≠−q,−q−1,−q−2,…,p,q=0,1,2,…. In addition, we pay our special attention to the closed forms of ∂p+q∂xp∂yqB(z;−n,m) for n,m=0,1,2,…,n,m=0,1,2,…, which can be expressed by the elementary function, special constants and Riemann zeta function.