Certain Finite Sums Pertaining to Leibnitz, Harmonic and Other Special Numbers

Abstract

The present manuscript deals with some certain finite sums and identities pertaining to some special numbers. Using generating functions methods, some relations and identities involving the Apostol type Euler and combinatorial numbers, and also the Fubini type numbers and polynomials, are given. Then, by using some certain classes of special finite sums involving the following rational sum which is defined by Simsek (2021b): y(r,ϑ)=∑_(b=0)^r▒〖(-1)^r/((1+b) ϑ^(b+1) 〖(ϑ-1)〗^(r-b+1) ),〗many new certain finite sums and formulas related to the Leibnitz, Harmonic, Changhee, and Daehee numbers are obtained. Moreover, some applications of these results are presented.

Keywords

• Leibnitz Numbers
• Harmonic Numbers
• Fubini Type Numbers
• Changhee and Daehee Numbers

References

1. Gould, H. W. (1972). Combinatorial identities. Morgantown: Morgantown Printing and Binding Co.
2. Kilar, N. (2023a). Combinatorial sums and identities associated with functional equations of generating functions for Fubini type polynomials. Gazi University Journal of Science, doi:10.35378/gujs.989270
3. Kilar, N. (2023b). Formulas for Fubini type numbers and polynomials of negative higher order. Montes Taurus Journal of Pure and Applied Mathematics, 5(3), 23-36.
4. Kilar, N., & Simsek, Y. (2017). A new family of Fubini numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials. Journal of the Korean Mathematical Society, 54(5), 1605-1621. doi:10.4134/JKMS.j160597
5. Kilar, N., & Simsek, Y. (2019a). Some relationships between Fubini type polynomials and other special numbers and polynomials. AIP Conference Proceedings, 2116, 100017-1-100017-4. doi:10.1063/1.5114093
6. Kilar, N., & Simsek, Y. (2019b). Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach. Publications de l’Institut Mathématique, 106(120), 113-123. doi:10.2298/PIM1920113K
7. Kilar, N., & Simsek, Y. (2021a). Combinatorial sums involving Fubini type numbers and other special numbers and polynomials: Approach trigonometric functions and p-adic integrals. Advanced Studies in Contemporary Mathematics, 31(1), 75-87. doi:10.17777/ascm2021.31.1.75
8. Kilar, N., & Simsek, Y. (2021b). Formulae to Fubini type numbers emerge from application of p-adic integrals. Gazi University Journal of Science Part A: Engineering and Innovation, 8(4), 402-410. doi:10.54287/gujsa.980263

9. Kim, D. S., & Kim, T. (2013). Daehee numbers and polynomials. Applied Mathematical Sciences (Ruse), 7(120), 5969-5976. doi:10.12988/ams.2013.39535
10. Kim, D. S., Kim, T., & Seo, J. (2013). A note on Changhee numbers and polynomials. Advanced Studies in Theoretical Physics, 7, 993-1003. doi:10.12988/astp.2013.39117
11. Kucukoglu, I., & Simsek, Y. (2018). Remarks on recurrence formulas for the Apostol-type numbers and polynomials. Advanced Studies in Contemporary Mathematics, 28(4), 643-657 doi:10.17777/ascm2018.28.4.643
12. Kucukoglu, I., Simsek, Y., & Srivastava, H. M. (2019). A new family of Lerch-type zeta functions interpolating a certain class of higher-order Apostol-type numbers and Apostol-type polynomials. Quaestiones Mathematicae, 42(4), 465-478. doi:10.2989/16073606.2018.1459925
13. Simsek, Y. (2017). Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers. Mathematical Methods in the Applied Sciences, 40(7), 2347-2361. doi:10.1002/mma.4143
14. Simsek, Y. (2018). New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Applicable Analysis and Discrete Mathematics, 12, 1-35. doi:10.2298/AADM1801001S
15. Simsek, Y. (2019). Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials. Montes Taurus Journal of Pure and Applied Mathematics, 1(1), 1-76.
16. Simsek, Y. (2021a). Construction of generalized Leibnitz type numbers and their properties. Advanced Studies in Contemporary Mathematics, 31(3), 311-323. doi:10.17777/ascm2021.31.3.311
17. Simsek, Y. (2021b). Interpolation functions for new classes special numbers and polynomials via applications of p-adic integrals and derivative operator. Montes Taurus Journal of Pure and Applied Mathematics, 3(1), 38-61.
18. Simsek, Y. (2021c). New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, (RACSAM), 115, 1-14. doi:10.1007/s13398-021-01006-6
19. Simsek, Y. (2021d). Miscellaneous formulae for the certain class of combinatorial sums and special numbers. Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques, 46(1), 151-167.
20. Simsek, Y. (2022a). Applications of Apostol-type numbers and polynomials: Approach to techniques of computation algorithms in approximation and interpolation functions. In: N. ¬J.¬ Daras & Th. ¬M. ¬Rassias (Eds.) Approximation and Computation in Science and Engineering, Springer Optimization and Its Applications 180, (pp. 783-860). doi:10.1007/978-3-030-84122-5_40
21. Simsek, Y. (2022b). Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p-adic integrals and special functions. Mathematical Methods in the Applied Sciences, doi:10.1002/mma.8321
22. Simsek, Y. (2022c). Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials. Montes Taurus Journal of Pure and Applied Mathematics, 4(3), 61-79.
23. Srivastava, H. M., & Kızılateş, C. (2019). A parametric kind of the Fubini-type polynomials. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (RACSAM), 113, 3253-3267. doi:10.1007/s13398-019-00687-4