Formulas and Finite Sums Covering Beta-type Rational Functions and Euler-Frobenius-type Polynomials

Year 2024, Volume: 11 Issue: 4, 792 - 800, 30.12.2024

Damla Gün , Yılmaz Şimşek

https://doi.org/10.54287/gujsa.1584764

Abstract

The aim of this article is to derive some novel formulas and finite sums covering Stirling type numbers, the Frobenius Euler numbers and polynomials, the beta-type rational functions, and combinatorial numbers with the help of both generating functions and their functional equations, and also some special identities associated with special numbers and polynomials.

Keywords

Special Numbers and Polynomials, Stirling Type Numbers, Generating Functions

References

• Carlitz, L., & Sholander, M. (1963). The product of two Eulerian polynomials. Mathematics Magazine, 36(1), 37–41. https://doi.org/10.2307/2688134
• Gun, D., & Simsek, Y. (2020). Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers. Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matematicas, 114(4), 167. https://doi.org/10.1007/s13398-020-00899-z
• Gun, D., & Simsek, Y. (2023). Modification exponential Euler type splines derived from Apostol-Euler numbers and polynomials of complex order. Applicable Analysis and Discrete Mathematics, 17(1), 197–215. https://doi.org/10.2298/AADM220712011G
• Kucukoglu, I., & Simsek, Y. (2019). Identities and relations on the q-Apostol type Frobenius-Euler numbers and polynomials. Journal of the Korean Mathematical Society, 56(1), 265–284. https://doi.org/10.4134/JKMS.j180185
• Luo, Q.-M. (2006). Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwanese Journal of Mathematics, 10(4), 917–925. https://doi.org/10.11650/twjm/1500403883
• Simsek, Y. (2013a). Identities associated with generalized Stirling type numbers and Eulerian type polynomials. Mathematical and Computational Applications, 18(3), 251-263. https://doi.org/10.3390/mca18030251
• Simsek, Y. (2013b). Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory and Algorithms for Sciences and Engineering, 87. https://doi.org/10.1186/1687-1812-2013-87
• Simsek, Y. (2015). Beta-type polynomials and their generating functions. Applied Mathematics and Computation, 254, 172–182. https://doi.org/10.1016/j.amc.2014.12.118
• Simsek, Y. (2018a). Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals. Turkish Journal of Mathematics, 42(2), 557–577. https://doi.org/10.3906/mat-1703-114
• Simsek, Y. (2018b). Combinatorial sums and binomial identities associated with the Beta-type polynomials. Hacettepe Journal of Mathematics and Statistics, 47(5), 1144-1155. https://doi.org/10.15672/HJMS.2017.505
• Simsek, Y. (2018c). New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Applicable Analysis and Discrete Mathematics, 12(1), 1-35. https://www.jstor.org/stable/90020602
• Simsek, Y. (2023). Generating functions for series involving higher powers of inverse binomial coefficients and their applications. Mathematical Methods in the Applied Sciences, 46(12), 12591-12617. https://doi.org/10.1002/mma.9199
• Srivastava, H. M. (2011). Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Applied Mathematics & Information Sciences. 5(3), 390–444.
• Srivastava, H. M., & Choi, J. (2001). Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers.
• Srivastava, H. M., & Choi, J. (2012). Zeta and q-zeta functions and associated series and integrals. Amsterdam: Elsevier Science Publishers.