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Analytical and asymptotic representations for two sequence related to Gauss’ lemniscate functions

Han Xue-Feng (School of Mathematics and Informatics, (Henan Polytechnic University, Jiaozuo City, Henan Province, People’s Republic of China), hanxuefeng8110@sohu.com
Chen Chao-Ping (School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, People’s Republic of China), chenchaoping@sohu.com
Srivastava H.M. (Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada + Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan, Republic of China + Department of Mathematics and Informatics, Azerbaijan University, Baku, Azerbaijan + Section of Mathematics, International Telematic University Uninettuno, Rome, Italy), harimsri@math.uvic.ca

Let the sequences Gn and gn be defined by Gn := ∫10 dt/(1−t2n)1/n (n ≧ 2) and gn := ∫∞0 dt/(1 + t2n)1/n (n ≧ 1). In this paper, we first derive analytical representations for these two sequences Gn and gn in terms of the gamma function. By using the obtained analytical representations, we then deduce asymptotic expansions for Gn and gn.

Keywords: Gamma and Beta functions, Lemniscate functions, Asymptotic expansions, Zeta functions, Bell polynomials