Geodesic
Formulas Involving Sums of Powers, Special Numbers and Polynomials Arising From $p$-Adic Integrals, Trigonometric and Generating Functions
Publications de l'Institut Mathématique, _N_S_108 (2020) no. 122, p. 103

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DOI

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and $p$-adic integrals involving the fermionic $p$-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.
DOI : 10.2298/PIM2022103K
Classification : 05A15, 11B68, 11B73, 11S80, 26A42, 33B10
Keywords: Bernoulli numbers and polynomials, Euler numbers and polynomials, Fubini numbers, Tangent numbers, $p$-adic integrals, Riemann integral, Trigonometric functions, Generating functions 
Neslihan Kilar; Yilmaz Simsek. Formulas Involving Sums of Powers, Special Numbers and Polynomials Arising From $p$-Adic Integrals, Trigonometric and Generating Functions. Publications de l'Institut Mathématique, _N_S_108 (2020) no. 122, p. 103 . doi: 10.2298/PIM2022103K
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     author = {Neslihan Kilar and Yilmaz Simsek},
     title = {Formulas {Involving} {Sums} of {Powers,} {Special} {Numbers} and {Polynomials} {Arising} {From} $p${-Adic} {Integrals,} {Trigonometric} and {Generating} {Functions}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {103 },
     year = {2020},
     volume = {_N_S_108},
     number = {122},
     doi = {10.2298/PIM2022103K},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM2022103K/}
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