The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications

Sibel Koparal; Neşe Ömür; Merve Güney

doi:10.47000/tjmcs.1902166

The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications

Abstract

In this paper, we define generalized harmonic numbers with parameters $\delta $ and $\gamma $, $H_{l,\kappa }\left( \delta ,\gamma \right) .$ As their implementation, we obtain some results and combinatorial identities by using the partial derivative and generating functions. Additionally, we give graphs of the roots of the equations $H_{l,\kappa }\left( \delta ,\gamma \right)=0 $, $G_{l}\left( \delta ,\gamma \right)=0$ and $G_{l}^{(\kappa )}\left( \delta ,\gamma \right)=0 $ for different parameters. Thus, we have made the motivation and roots of the given equations more understandable. In particular, with this approach, we have improved the understanding of their structural properties and revealed new insights for further research.

Keywords

References

Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, UK, 2000.
Batır, N., Combinatorial identities involving harmonic numbers, Integers, 20(2020), A25.
Cheon, G.-S., El-Mikkawy, M.A., Generalized harmonic numbers identities and a related matrix representation, J. Korean Math. Soc., 44(2)(2007), 467–498.
Chen, Z., Ömürr, N., Koparal S., Khan, W. A. Some identities with multi-generalized q-hyperharmonic numbers of order r, Symmetry, 15(4)(2023), 917.
Choi, J., Certain summation formulas involving harmonic numbers and generalized harmonic numbers, Appl. Comput., 218(3) (2011), 734–740.
Choi, J., Srivastava, H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model., 54(2011), 2220–2234.
Chu, W., Fu, A.M., Dougall-Dixon formula and harmonic number identities, Ramanujan J., 18(2009), 11–31.
Dolgy, D.V., Kim, D.S., Kim, H.K., Kim, T. Degenerate harmonic and hyperharmonic numbers, Proc. Jangjeon Math. Soc., 26(3)(2023), 259–268.

Duran, Ö., Ömür, N., Koparal S., On sums with generalized harmonic, hyperharmonic and special numbers, Miskolc Mathematical Notes, 21(2)(2020), 791–803.
Erd´elyi, A., Magnus,W., Oberhettinger, F., Tricomi, F.G., Higher Transcendental Functions, McGraw-Hill Book Company, New York, Toronto and London, I, 1953.
Euler, L., Meditationes circa singulare serierum genus, Novi Comment. Acad. Sci. Petropolitanae, 20(1776), 140–186.
Gencev, M., Binomial sums involving harmonic numbers, Mathematica Slovaca, 61(2)(2011), 215–226.
Guo, D., Some combinatorial identities concerning harmonic numbers and binomial coefficients, Discrete Mathematics Letters, 8(2022), 41–48.
Guo, D., Chu, W., On sums with generalized harmonic, hyperharmonic and special numbers, Miskolc Mathematical Notes, 21(2)(2020), 791–803.
Guo, D., Chu, W., Summation formulae involving multiple harmonic numbers, Appl. Anal. Discret. Math., 15(2021), 201–212.
Kim, T., Kim, D.S., Lee, H., Kwon, J.,On some summation formulas, Demonstratio Mathematica, 55(2022), 1–7.
Kim, D.S., Kim, H., Kim, T., Some identities on generalized harmonic numbers and generalized harmonic functions, Demonstratio Mathematica, 56(2023), 1–10.
Kim, T., Kim, D.S., Some identities on degenerate harmonic and degenerate higher-order harmonic numbers, Appl. Math. Comput., 486(129045)(2025), 1–10.
Kim, D.S., Kim, T., Some numbers and polynomials related to degenerate harmonic and degenerate hyperharmonic numbers, Appl. Anal. Discrete Math., 19(2)(2025), 284–298.
Kim, T. Kim, D.S., Degenerate hyperharmonic polynomials and numbers, Filomat, 40(4)(2026), 1411–1422.
Kim, T., Kim, D.S., Spivey’s type recurrence relation for degenerate Bell polynomials, Proc. Jangjeon Math. Soc., 28(4)(2025), 393–400.
Kim, D.S., Kim, T., Lee, S.-H., Combinatorial identities involving harmonic and hyperharmonic numbers, Adv. Stud. Contemp. Math. (Kyungshang) 23(3)(2013), 393–413.
Koparal, S., Ömür, N., Elkhiri, L., On generalized hyperharmonic numbers of order r, Hrn ,m (σ), Notes on Number Theory and Discrete Mathematics, 29(4)(2023), 804–812.
Koshy, T., Fibonacci and Lucas Numbers with Applications, Vol. 2, John Wiley & Sons, 2019.
Magnus, W., Oberhettinger, F., Formulas and Theorems for the Special Functions of Mathematical Physics, Translated by John Wermer, Chelsea Publishing Company, New York, 1949.
Ömür, N., Koparal, S., On the matrices with the generalized hyperharmonic numbers of order r, Asian-Eur. J. Math., 11(2018), 1850045.
Ömür, N., Bilgin, G., Some applications of the generalized hyperharmonic numbers of order r, Hrn (α), Adv. Appl. Math. Sci., 17(2018), 617–627.
Spieß, J., Some identities involving harmonic numbers, Math. Comput., 55(1990), 839–863.
Srivastava, H.M., Choi, J., Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science, Publishers, Amsterdam, London and New York, 2012.
Whittaker, E.T., Watson, G.N., A course of modern analysisi, an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the pourth (1927) edition, Cambridge University Press, Cambridge, 1996.
Zhao, J., Multiple q-zeta functions and multiple q-polylogarithms, Ramanujan J., 14(2007), 189–221.

Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Sibel Koparal
0000-0001-9574-9652
Türkiye

Neşe Ömür
0000-0002-3972-9910
Türkiye

Merve Güney ^*
0000-0002-6340-4817
Türkiye

Publication Date

October 13, 2026

Submission Date

March 5, 2026

Acceptance Date

April 13, 2026

Published in Issue

Year 2026 Volume: 18 Number: 3

DOI

https://doi.org/10.47000/tjmcs.1902166

IZ

https://izlik.org/JA57RP77PL

Cite

RIS / Bibtex

APA

Koparal, S., Ömür, N., & Güney, M. (2026). The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications. Turkish Journal of Mathematics and Computer Science, 18(3), 877-892. https://doi.org/10.47000/tjmcs.1902166

AMA

1.Koparal S, Ömür N, Güney M. The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications. TJMCS. 2026;18(3):877-892. doi:10.47000/tjmcs.1902166

Chicago

Koparal, Sibel, Neşe Ömür, and Merve Güney. 2026. “The Generalized Harmonic Numbers With Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications”. Turkish Journal of Mathematics and Computer Science 18 (3): 877-92. https://doi.org/10.47000/tjmcs.1902166.

EndNote

Koparal S, Ömür N, Güney M (October 1, 2026) The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications. Turkish Journal of Mathematics and Computer Science 18 3 877–892.

IEEE

[1]S. Koparal, N. Ömür, and M. Güney, “The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications”, TJMCS, vol. 18, no. 3, pp. 877–892, Oct. 2026, doi: 10.47000/tjmcs.1902166.

ISNAD

Koparal, Sibel - Ömür, Neşe - Güney, Merve. “The Generalized Harmonic Numbers With Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications”. Turkish Journal of Mathematics and Computer Science 18/3 (October 1, 2026): 877-892. https://doi.org/10.47000/tjmcs.1902166.

JAMA

1.Koparal S, Ömür N, Güney M. The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications. TJMCS. 2026;18:877–892.

MLA

Koparal, Sibel, et al. “The Generalized Harmonic Numbers With Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 3, Oct. 2026, pp. 877-92, doi:10.47000/tjmcs.1902166.

Vancouver

1.Sibel Koparal, Neşe Ömür, Merve Güney. The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications. TJMCS. 2026 Oct. 1;18(3):877-92. doi:10.47000/tjmcs.1902166