Evaluations of some Euler-type series via powers of the arcsin function

Jiaye Lin; Jiaye Lin

doi:10.3934/math.2025372

AIMS Mathematics

2025, Volume 10, Issue 4: 8116-8130. doi: 10.3934/math.2025372

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Evaluations of some Euler-type series via powers of the arcsin function

Jiaye Lin ^,

School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 12 December 2024 Revised: 26 March 2025 Accepted: 01 April 2025 Published: 09 April 2025
MSC : 05A19, 40A25

In this paper, by using special integrals and integer powers of the $ \arcsin $ function, we obtained the recurrences or explicit expressions of some parametric Euler-type series involving multiple harmonic sums and multiple $ t $-harmonic sums. According to our results, these Euler-type series are all expressible in terms of $ \pi $, $ \ln(2) $, and zeta values. In particular, by specifying the parameters, we presented as examples the evaluations of some special series, including some known ones in the literature and some new ones.
- harmonic numbers,
- infinite series identities,
- Euler-type series,
- inverse sine expansions
Citation: Jiaye Lin. Evaluations of some Euler-type series via powers of the arcsin function[J]. AIMS Mathematics, 2025, 10(4): 8116-8130. doi: 10.3934/math.2025372

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Abstract

In this paper, by using special integrals and integer powers of the $ \arcsin $ function, we obtained the recurrences or explicit expressions of some parametric Euler-type series involving multiple harmonic sums and multiple $ t $-harmonic sums. According to our results, these Euler-type series are all expressible in terms of $ \pi $, $ \ln(2) $, and zeta values. In particular, by specifying the parameters, we presented as examples the evaluations of some special series, including some known ones in the literature and some new ones.

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