Some identities on generalized harmonic numbers and generalized harmonic functions

• Autorzy: Kim Dae San , Kim Hyekyung , Kim Taekyun

Identyfikatory

DOI

10.1515/dema-2022-0229

• Języki publikacji: EN

Abstrakty

EN

The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions Fn(x) = B(x+1,n+1),(n=0,1,2,…) using elementary methods. For instance, we show that the Hurwitz zeta function ζ(x+1,r) and r! are expressed in terms of those numbers and functions, for every r=2,3,4,5 .

Słowa kluczowe

EN

generalized harmonic numbers of order α; generalized harmonic function

PL

uogólnione liczby harmoniczne rzędu α; funkcja harmoniczna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220229
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Kim Dae San

• Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

autor

Kim Hyekyung

• Department Of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea

autor

Kim Taekyun

• Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Bibliografia

• [1] S. Araci, A new class of Bernoulli polynomials attached to polyexponential functions and related identities, Adv. Stud. Contemp. Math. (Kyungshang) 31 (2021), no. 2, 195–204, DOI: https://doi.org/10.17777/ascm2021.31.2.195.
• [2] J. Choi, Finite summation formulas involving binomial coefficients, harmonic numbers and generalized harmonic numbers, J. Inequal. Appl. 2013 (2013), 49, DOI: https://doi.org/10.1186/1029-242X-2013-49.
• [3] S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
• [4] J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus, New York, 1996.
• [5] D. S. Kim, T. Kim, and S.-H. Lee, Combinatorial identities involving harmonic and hyperharmonic numbers, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 3, 393–413.
• [6] T. Kim and D. S. Kim, Note on the degenerate gamma function, Russ. J. Math. Phys. 27 (2020), no. 3, 352–358, DOI: https://doi.org/10.1134/S1061920820030061.
• [7] T. Kim and D. S. Kim, On some degenerate differential and degenerate difference operators, Russ. J. Math. Phys. 29 (2022), no. 1, 37–46, DOI: https://doi.org/10.1134/S1061920822010046.
• [8] T. Kim and D. S. Kim, Some identities on degenerate hyperharmonic numbers, Georgian Math. J. 2022, DOI: https://doi.org/10.1515/gmj-2022-2203.
• [9] T. Kim and D. S. Kim, Some relations of two type 2 polynomials and discrete harmonic numbers and polynomials, Symmetry 12 (2020), no. 6, 905, DOI: https://doi.org/10.3390/sym12060905.
• [10] T. Kim, D. S. Kim, H. Lee, and J. Kwon, On some summation formulas, Demonstr. Math. 55 (2022), no. 1, 1–7, DOI: https://doi.org/10.1515/dema-2022-0003.
• [11] J. Choi, Certain summation formulas involving harmonic numbers and generalized harmonic numbers, Appl. Math. Comput. 218 (2011), no. 3, 734–740, DOI: https://doi.org/10.1016/j.mcm.2011.05.032.
• [12] J. Choi and H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model. 54 (2011), 2220–2234, DOI: https://doi.org/10.1016/j.mcm.2011.05.032.
• [13] G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.
• [14] W. Magnus and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics, Translated by John Wermer, Chelsea Publishing Company, New York, N.Y., 1949.
• [15] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions, Reprint of the fourth (1927) edition, Cambridge University Press, Cambridge, 1996.
• [16] T. K. Kim and D. S. Kim, Some identities involving degenerate stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys. 30 (2023), no. 1, 62–75, DOI: https://doi.org/10.1134/S1061920823010041.
• [17] T. Kim and D. S. Kim, Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math. 148 (2023), no. 1 102535, DOI: https://doi.org/10.1016/j.aam.2023.102535.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).