Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi
In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the aut...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-10-01
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| Series: | Demonstratio Mathematica |
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| Online Access: | https://doi.org/10.1515/dema-2022-0157 |
| _version_ | 1828099745279639552 |
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| author | Qi Feng |
| author_facet | Qi Feng |
| author_sort | Qi Feng |
| collection | DOAJ |
| description | In this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function. |
| first_indexed | 2024-04-11T08:18:18Z |
| format | Article |
| id | doaj.art-8f3c6ecbffbb4122b7ae5c4b33a39b17 |
| institution | Directory Open Access Journal |
| issn | 2391-4661 |
| language | English |
| last_indexed | 2024-04-11T08:18:18Z |
| publishDate | 2022-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Demonstratio Mathematica |
| spelling | doaj.art-8f3c6ecbffbb4122b7ae5c4b33a39b172022-12-22T04:35:04ZengDe GruyterDemonstratio Mathematica2391-46612022-10-0155171073610.1515/dema-2022-0157Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of PiQi Feng0Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, ChinaIn this article, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the light of properties of partial Bell polynomials, the author establishes Taylor’s series expansions of real powers of two functions containing squares of inverse (hyperbolic) cosine functions in terms of the Stirling numbers of the first kind, presents a closed-form formula of specific partial Bell polynomials at a sequence of derivatives of a function containing the square of inverse cosine function, derives several combinatorial identities involving the Stirling numbers of the first kind, demonstrates several series representations of the circular constant Pi and its real powers, recovers Maclaurin’s series expansions of positive integer powers of inverse (hyperbolic) sine functions in terms of the Stirling numbers of the first kind, and also deduces other useful, meaningful, and significant conclusions and an application to the Riemann zeta function.https://doi.org/10.1515/dema-2022-0157taylor’s series expansionmaclaurin’s series expansionreal powerinverse cosine functioninverse hyperbolic cosine functioninverse sine functioninverse hyperbolic sine functionstirling number of the first kindcombinatorial identitycompositeseries representationcircular constantpartial bell polynomialclosed-form formulariemann zeta functionprimary 41a58secondary 05a1911b7311b8311c0811s0512d0526a2426c0533b10 |
| spellingShingle | Qi Feng Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi Demonstratio Mathematica taylor’s series expansion maclaurin’s series expansion real power inverse cosine function inverse hyperbolic cosine function inverse sine function inverse hyperbolic sine function stirling number of the first kind combinatorial identity composite series representation circular constant partial bell polynomial closed-form formula riemann zeta function primary 41a58 secondary 05a19 11b73 11b83 11c08 11s05 12d05 26a24 26c05 33b10 |
| title | Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi |
| title_full | Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi |
| title_fullStr | Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi |
| title_full_unstemmed | Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi |
| title_short | Taylor’s series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi |
| title_sort | taylor s series expansions for real powers of two functions containing squares of inverse cosine function closed form formula for specific partial bell polynomials and series representations for real powers of pi |
| topic | taylor’s series expansion maclaurin’s series expansion real power inverse cosine function inverse hyperbolic cosine function inverse sine function inverse hyperbolic sine function stirling number of the first kind combinatorial identity composite series representation circular constant partial bell polynomial closed-form formula riemann zeta function primary 41a58 secondary 05a19 11b73 11b83 11c08 11s05 12d05 26a24 26c05 33b10 |
| url | https://doi.org/10.1515/dema-2022-0157 |
| work_keys_str_mv | AT qifeng taylorsseriesexpansionsforrealpowersoftwofunctionscontainingsquaresofinversecosinefunctionclosedformformulaforspecificpartialbellpolynomialsandseriesrepresentationsforrealpowersofpi |