A series involving Catalan numbers: Proofs and demonstrations

Abstract

Tewodros Amdeberhan obtained his Ph.D. in mathematics from Temple University under the supervision of Doron Zeiberger. He currently teaches at Tulane University and holds a permanent membership at DIMACS, Rutgers University. Xiao Guan was born in Beijing, China. He is currently a graduate student working under the guidance of V. Moll at Tulane University. Inspired by his advisor, he is currently concentrating on problems related to the evaluation of definite integrals. He is also interested in random matrices. Lin Jiu is a doctoral student working under the supervision of V. Moll at Tulane University. He obtained both bachelor and master degrees in Mathematics at Beijing Institute of Technology. His research interests involve Symbolic Computations, Special Functions and Number Theory.

Integral Representations of Catalan Numbers and Their Applications

“…It seems that there are close and similar ideas in [3,4] and that the paper [3] is almost an expanded version of [4]. Great minds think alike!…”

“…In 2016, motivated by the above-mentioned problem posed by Beckwith and Harbor [4], Amdeberhan and his four coauthors [3] also proposed a general problem: find a closed-form formula for the series in (6.2). They obtained the sum…”

“…The authors are thankful to Wei-Dong Jiang (Weihai Vocational University, China), to Chun-Na Zeng (Chongqing Normal University, China), and to Victor H. Moll (Tulane University, USA) for their kind helps in October 2016 to find and to supply the formally published versions of the papers [1,3]. …”

Several Series Identities Involving the Catalan Numbers

“…It seems that there are close and similar ideas in [6,7] and that the paper [6] is almost an expanded version of [7]. Great minds think alike!…”

“…which are special cases of the general formula In 2016, motivated by Problem 11765 in [7] mentioned above, Amdeberhan and his four coauthors [6] proposed a general problem: find a closed-form formula for the series…”