Combinatorial Sums and Identities associated with Functional Equations of Generating Functions for Fubini Type Polynomials

Year 2023, Volume: 36 Issue: 2, 807 - 817, 01.06.2023

Neslıhan Kılar

https://doi.org/10.35378/gujs.989270

Cited By: 3

Abstract

Using generating functions with their functional equations method, a great number of novel combinatorial sums, formulas, and recurrence relation including Fubini type polynomials and numbers, Stirling type numbers, and Apostol type polynomials are given. Applying Riemann integral to this generating function with their functional equations, some identities involving Cauchy and Stirling numbers are obtained. Moreover, some interpretations about the results are given. 

Keywords

Apostol type polynomials and numbers, Fubini type polynomials and numbers, Special polynomials and numbers, Generating functions, Combinatorial sums

References

• [1] Belbachir, H., Rahmani, M., Sury, B., “Sums involving moments of reciprocals of binomial coefficients”, Journal of Integer Sequences, 14(6): Article 11.6.6, (2011).
• [2] Chang, C. H., Ha, C.W., “A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials”, Journal of Mathematical Analysis and Applications, 315: 758-767, (2006).
• [3] Comtet, L., “Advanced combinatorics”, D. Reidel Publication Company, Dordrecht-Holland/Boston-U.S.A., (1974).
• [4] Good, I.J., “The number of ordering of 𝑛 candidates when ties are permitted”, The Fibonacci Quarterly, 13(1): 11-18, (1975).
• [5] Kargin, L., “On Cauchy numbers and their generalizations”, Gazi University Journal of Science, 33(2): 456-474, (2020).
• [6] Kargin, L, Cekim, B., “Higher order generalized geometric polynomials”, Turkish Journal of Mathematics, 42: 887-903, (2018).
• [7] Kızılateş, C., “New families of Horadam numbers associated with finite operators and their applications”, Mathematical Methods in the Applied Sciences, 44(18): 14371-14381, (2021). DOI: https://doi.org/10.1002/mma.7702
• [8] Kilar, N., “Fubini type numbers and their generating functions”, Akdeniz University; MSc Thesis in Mathematics, Antalya, (2017).
• [9] Kilar, N., “Formulas and combinatorial sums including special numbers on 𝑝-adic integrals”, Montes Taurus Journal of Pure and Applied Mathematics, 1(1): 129-139, (2019).
• [10] Kilar, N., Simsek, Y., “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, (2017).
• [11] Kilar, N., Simsek, Y., Some relationships between Fubini type polynomials and other special numbers and polynomials, AIP Conference Proceedings, 2116, 100017: 100017-1–100017-4, (2019). DOI: https://doi.org/10.1063/1.5114093
• [12] Kilar, N., Simsek, Y., “Identities and relations for Fubini type numbers and polynomials via generating functions and 𝑝-adic integral approach”, Publications de l’Institut Mathématique. Nouvelle Série, 106(120): 113-123, (2019).
• [13] Kilar, N., Simsek, Y., “Formulas and relations of special numbers and polynomials arising from functional equations of generating functions”, Montes Taurus Journal of Pure and Applied Mathematics, 3(1): 106-123, (2021).
• [14] Kim, T., Kim, D.S., Jang G.W., Kwon, J., Symmetric identities for Fubini polynomials, Symmetry, 10(6), 219: 1-7, (2018).
• [15] Kucukoglu, I., “Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials”, Communications de la Faculté des Sciences de l’Université d’Ankara. Séries A1. Mathematics and Statistics, 70(1): 426-442, (2021).
• [16] Lu, D.Q., Srivastava, H.M., “Some series identities involving the generalized Apostol type and related polynomials”, Computers & Mathematics with Applications, 62: 3591-3602, (2011).
• [17] Luo, Q.M., Srivastava, H. M., “Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials”, Computers & Mathematics with Applications, 51: 631-642, (2006).
• [18] Roman, S., “The Umbral Calculus”, Dover Publ. Inc., New York, (2005).
• [19] Simsek, Y., “Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications”, Fixed Point Theory and Applications, 87(2013): 1-28, (2013).
• [20] Simsek, Y., “Combinatorial sums and binomial identities associated with the Beta-type polynomials”, Hacettepe Journal of Mathematics and Statistics, 47(5): 1144-1155, (2018).
• [21] Simsek, Y., “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, (2019).
• [22] Srivastava, H.M., “Some generalizations and basic (or 𝑞-) extensions of the Bernoulli, Euler and Genocchi polynomials”, Applied Mathematics & Information Sciences, 5(3): 390-444, (2011).
• [23] Srivastava, H.M., Choi, J., “Zeta and 𝑞-zeta functions and associated series and integrals”, Elsevier Science Publishers, Amsterdam, (2012).
• [24] Srivastava, H.M., Kızılateş, C., “A parametric kind of the Fubini-type polynomials”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113: 3253-3267, (2019).