Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

Bai-Ni Guo; Dongkyu Lim; Feng Qi; Bai-Ni Guo; Dongkyu Lim; Feng Qi

doi:10.3934/math.2021438

AIMS Mathematics

2021, Volume 6, Issue 7: 7494-7517. doi: 10.3934/math.2021438

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Research article

Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

Bai-Ni Guo ¹,
Dongkyu Lim ^{2
,
,},
Feng Qi ^{3
,
,}

1.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454003, China
2.
Department of Mathematics Education, Andong National University, Andong 36729, South Korea
3.
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China

Received: 22 February 2021 Accepted: 30 April 2021 Published: 07 May 2021
MSC : Primary: 11B83; Secondary: 11C08, 12E10, 26A39, 33B10, 41A58

In the paper, the authors

1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;

2. find closed-form formulas for the sequence

$ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $

where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;

3. derive series representations of generalized logsine functions.

The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.
- general expression,
- closed-form formula,
- arcsine,
- series expansion,
- power,
- special value,
- second kind Bell polynomials,
- series representation,
- generalized logsine function
Citation: Bai-Ni Guo, Dongkyu Lim, Feng Qi. Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions[J]. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438

Related Papers:

Abstract

In the paper, the authors

1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;

2. find closed-form formulas for the sequence

$ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $

where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;

3. derive series representations of generalized logsine functions.

The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.

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