MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function-Reference-Cited by-同舟云学术

MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function

Published:2022 Issue:2 Volume:16 Page:427-466
ISSN:1452-8630
Container-title:Applicable Analysis and Discrete Mathematics
language:en
Short-container-title:APPL ANAL DISCRETE M

Author:

Guo Bai-Ni¹^ORCID,Lim Dongkyu²^ORCID,Qi Feng³^ORCID

Affiliation:

1. School of Mathematics and Informatics, Henan Polytechnic University Jiaozuo, Henan, China

2. Department of Mathematics Education Andong National University Andong, Republic of Korea

3. School of Mathematical Sciences Tiangong University Tianjin, China

Abstract

In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific partial Bell polynomials and to derive a series representation of generalized logsine function, and deduce combinatorial identities involving the first kind Stirling numbers.

Publisher

National Library of Serbia

Subject

Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis

Reference61 articles.

1. M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Washington, 1972.

2. E. P. Adams and R. L. Hippisley, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, Smithsonian Institute, Washington, D.C., 1922.

3. B. C. Berndt, Ramanujan’s Notebooks, Part I, With a foreword by S. Chandrasekhar, Springer-Verlag, New York, 1985; available online at https://doi.org/10.1007/978-1-4612-1088-7.

4. J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., Natick, MA, 2004.

5. J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1987.

Cited by 18 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant;Symmetry;2024-09-03

2. New formulas for Maclaurin's series expansions of powers of arctangent and other related topics;Integral Transforms and Special Functions;2024-08

3. Expanding the function $\ln(1+\operatorname{e}^x)$ into power series in terms of the Dirichlet eta function and the Stirling numbers of the second kind;Carpathian Mathematical Publications;2024-06-30

4. A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments;Axioms;2024-05-10

5. Several inequalities for bounding sums of two (hyperbolic) sine cardinal functions;FILOMAT;2024