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

Qi, Feng

doi:10.1515/dema-2022-0157

Artykuł - szczegóły

Tytuł artykułu

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

Autorzy

Qi Feng

Wybrane pełne teksty z tego czasopisma

http://www.degruyter.com/view/j/dema

Identyfikatory

DOI

10.1515/dema-2022-0157

Warianty tytułu

Języki publikacji

Abstrakty

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.

Słowa kluczowe

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 representations circular constant partial Bell polynomial closed-form formula Riemann zeta function

moc czynna funkcja cosinus funkcja sinus liczby Stirlinga pierwszego rodzaju tożsamości kombinatoryjne reprezentacja szeregowa formuła zamknięta funkcje zeta

Wydawca

De Gruyter

Czasopismo

Demonstratio Mathematica

Rocznik

2022

Tom

Vol. 55, nr 1

Strony

710--736

Opis fizyczny

Bibliogr. 53 poz.

Twórcy

autor

Qi Feng

qifeng618@gmail.com

Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
Independent Researcher, Dallas, TX 75252-8024, USA

Bibliografia

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Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).

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Bibliografia

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