On extended Hurwitz-Lerch zeta function

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• 作者：Min-Jie Luo^a ; ^{mathwinnie@live.com} ; Rakesh Kumar Parmar^b ; ^{rakeshparmar27@gmail.com} ; Ravinder Krishna Raina^c ; ¹ ; ^{rkraina_7@hotmail.com}
• 关键词：Abel's summation formula ; Complete monotonicity ; Extended beta function ; Extended Hurwitz&ndash ; Lerch zeta function ; Generating function ; Lindelö ; f&ndash ; Wirtinger expansion
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：448
• 期：2
• 页码：1281-1304
• 全文大小：486 K
• 卷排序：448

文摘

This paper investigates an extended form of a beta function Bp,q(x,y)Bp,q(x,y). We first study the convergence problem of the function Bp,q(x,y)Bp,q(x,y) and consider the completely monotonic and log-convex properties of this function. As a result, we obtain a pair of Laguerre type inequalities. Next, we provide a new double integral representation for the function Bp,q(x,y)Bp,q(x,y). Subsequently, we consider the convergence problem of the extended Hurwitz–Lerch zeta function Φλ,μ;ν(z,s,a;p,q)Φλ,μ;ν(z,s,a;p,q) defined by its series representation. Upon using the series manipulation techniques, we obtain two series identities. We also find various integral representations for the function Φλ,μ;ν(z,s,a;p,q)Φλ,μ;ν(z,s,a;p,q). Lastly, we apply Fourier analysis to the function zaΦμ;ν(z,s,a;p,q)zaΦμ;ν(z,s,a;p,q) and obtain a Lindelöf–Wirtinger type expansion. Some interesting and promising results are also illustrated.