Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions
<p>In the paper, the authors</p> <p>1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;</p> <p>2. find closed-form formulas for the sequence</p> <p class="disp_formula">$ \begin{equation...
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2021-05-01
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doaj-1e8c9d55aace425d8b809193d03e38332021-05-17T01:43:16ZengAIMS PressAIMS Mathematics2473-69882021-05-01677494751710.3934/math.2021438Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functionsBai-Ni Guo0Dongkyu Lim 1Feng Qi 21. School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454003, China2. Department of Mathematics Education, Andong National University, Andong 36729, South Korea3. School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China<p>In the paper, the authors</p> <p>1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;</p> <p>2. find closed-form formulas for the sequence</p> <p class="disp_formula">$ \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*} $</p> <p>where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;</p> <p>3. derive series representations of generalized logsine functions.</p> <p>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.</p> https://www.aimspress.com/article/doi/10.3934/math.2021438?viewType=HTMLgeneral expressionclosed-form formulaarcsineseries expansionpowerspecial valuesecond kind bell polynomialsseries representationgeneralized logsine function |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Bai-Ni Guo Dongkyu Lim Feng Qi |
| spellingShingle |
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 AIMS Mathematics general expression closed-form formula arcsine series expansion power special value second kind bell polynomials series representation generalized logsine function |
| author_facet |
Bai-Ni Guo Dongkyu Lim Feng Qi |
| author_sort |
Bai-Ni Guo |
| title |
Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions |
| title_short |
Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions |
| title_full |
Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions |
| title_fullStr |
Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions |
| title_full_unstemmed |
Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions |
| title_sort |
series expansions of powers of arcsine, closed forms for special values of bell polynomials, and series representations of generalized logsine functions |
| publisher |
AIMS Press |
| series |
AIMS Mathematics |
| issn |
2473-6988 |
| publishDate |
2021-05-01 |
| description |
<p>In the paper, the authors</p>
<p>1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $;</p>
<p>2. find closed-form formulas for the sequence</p>
<p class="disp_formula">$ \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*} $</p>
<p>where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials;</p>
<p>3. derive series representations of generalized logsine functions.</p>
<p>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.</p>
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| topic |
general expression closed-form formula arcsine series expansion power special value second kind bell polynomials series representation generalized logsine function |
| url |
https://www.aimspress.com/article/doi/10.3934/math.2021438?viewType=HTML |
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