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

Zhongfeng Sun, Huizeng Qin and Aijuan Li

Applied Mathematics and Computation, 2016, vol. 275, issue C, 63-71

Abstract: 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,…. 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,…, which can be expressed by the elementary function, special constants and Riemann zeta function.

Keywords: Incomplete beta function; Neutrix limit; Closed form; Hypergeometric function (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:275:y:2016:i:c:p:63-71

DOI: 10.1016/j.amc.2015.11.054

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