Asymptotic formulas for the triple Gamma function ¦£₃ by means of its integral representation

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• 作者：Junesang  ; Choi^a ;   ; ^{junesang@mail.dongguk.ac.kr} ; H.M.  ; Srivastava^b ;   ; ^{harimsri@math.uvic.ca}
• 关键词：Gamma ; double ; triple and multiple Gamma functions ; Riemann Zeta function ; Hurwitz (or generalized) Zeta function ; Psi (or Digamma) function ; Bernoulli numbers ; Bohr-Mollerup theorem ; Euler-Mascheroni and Glaisher-Kinkelin constant ; Determinants of the La
• 刊名：Applied Mathematics and Computation
• 出版年：2011
• 出版时间：15 November 2011
• 年：2011
• 卷：218
• 期：6
• 页码：2631-2640
• 全文大小：229 K

文摘

There is an abundant literature on inequalities for the Gamma function ¦£ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function ¦£₂ and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function ¦£₃, which is then used to derive an asymptotic formula for ¦£₃. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function ¦£ and the double Gamma function ¦£₂ are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions ¦£_n (n ?4).