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

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• 作者：Zhongfeng Sun ; ^{zfsun@sdut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{zfsun83@163.com" class="auth_mail" title="E-mail the corresponding author} ; Huizeng Qin ^{qin_hz@163.com" class="auth_mail" title="E-mail the corresponding author} ; Aijuan Li ^{juanzi612@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Incomplete beta function ; Neutrix limit ; Closed form ; Hypergeometric function
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：275
• 期：Complete
• 页码：63-71
• 全文大小：293 K

文摘

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 i1" class="mathmlsrc">i1.gif&_user=111111111&_pii=S0096300315015568&_rdoc=1&_issn=00963003&md5=c0889da6c7728a16880c7a79a5c3cabe">i1.gif"> to all complex values of x and y as complex number z satisfying 0 < |z  | < 1. Moreover, we establish the recursive formula of $\frac{{\partial }^{p+q}}{\partial x^p\partial y^q}B(z;x,y)$ for x≠−q,−q−1,−q−2,…,p,q=0,1,2,…$x\ne -q,-q-1,-q-2,\dots ,p,q=0,1,2,\dots$. In addition, we pay our special attention to the closed forms of $\frac{{\partial }^{p+q}}{\partial x^p\partial y^q}B(z;-n,m)$ for n,m=0,1,2,…,$n,m=0,1,2,\dots ,$ which can be expressed by the elementary function, special constants and Riemann zeta function.