Title: FORMULAS AND COMBINATORIAL SUMS INCLUDING SPECIAL NUMBERS ON P-ADIC INTEGRALS
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-19-00006; Volume 1 / Issue 1 / Year 2019, Pages 129-139
Document Type: Research Paper
Author(s): Neslihan Kilar a
aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya-Turkey
Received: 2 October 2019, Accepted: 5 November 2019, Available online: 28 November 2019.
Corresponding Author: Neslihan Kilar (Email address: email@example.com)
Full Text: PDF
The main motivation of this work is to give some formulas for the special numbers, which were recently introduced by Kilar and Simsek with the aid of the p-adic integrals methods. These formulas are related to the some well-known families of special numbers and polynomials such as the negative order Bernoulli polynomials, the negative order Euler numbers and polynomials, the Stirling numbers, the array polynomials, the combinatorial numbers including the numbers y1(n,k;λ), the numbers y2(n,k;λ), the numbers y3(n,k;λ;a,b), the central factorial numbers, and combinatorial sums.
Keywords: Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the array polynomials, the numbers y1(n,k;λ), the numbers y2(n,k;λ), the numbers y3(n,k;λ;a,b), the central factorial numbersReferences:
- M. Abramowitz, I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Washington, D.C.: U.S. Dept. of Commerce, National Bureau of Standards, 1972.
- A. Bayad, Y. Simsek, H. M. Srivastava, Some array type polynomials associated with special numbers and polynomials, Appl. Math. Comput. 244, 149-157, 2014.
- P. L. Butzer, K. Schmidt, E. L. Stark, L. Vogt, Central factorial numbers, their main properties and some applications, Numer. Funct. Anal. Optim. 10 (5), 419-488, 1989.
- L. Comtet, Advanced Combinatorics, D. Reidel Publication Company, Dordrecht-Holland,1974.
- G. B. Djordjevic, G.V. Milovanovic, Special classes of polynomials, University of Nis, Faculty of Technology Leskovac, 2014.
- N. Kilar, Y. Simsek, Identities and Relations for Special Numbers and Polynomials: An Approach to Trigonometric Functions, to appear in Filomat, 2020.
- N. Kilar, Y. Simsek, Special numbers arised from trigonometric and hyperbolic functions, Mediterranean International Conference of Pure&Applied Mathematics and Related Areas (MICOPAM 2018), 221-224, 2018.
- T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 19, 288-299, 2002.
- T. Kim, q-Euler numbers and polynomials associated with p-adic q-integral and basic q-zeta function, Trend Math. Information Center Math. Sciences 9, 7-12, 2006.
- D. S. Kim, T. Kim, J. Seo, A Note on Changhee Polynomials and Numbers, Adv. Stud. Theor. Phys. 7, 993-1003, 2013.
- D. S. Kim, T. Kim, Daehee Numbers and Polynomials, Appl. Math. Sci. (Ruse) 7 (120), 5969-5976, 2013.
- D. S. Kim, T. Kim, Some p-adic integrals on ℤp associated with trigonometric functions, Russ. J. Math. Phys. 25 (3), 300-308, 2018.
- Q-M. Luo, H. M. Srivastava, Some generalizations of the Apostol–Bernoulli and Apostol–Euler polynomials, J. Math. Anal. Appl. 308 290–302, 2005.
- Q-M. Luo, H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217, 5702–5728, 2011.
- H. Ozden, Y. Simsek, Modification and unification of the Apostol-type numbers and polynomials and their applications, Appl. Math. Comput. 235, 338–351, 2014.
- S. Roman, The Umbral Calculus, Dover Publications Incorporated, New York, 2005.
- W.H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge, 1984.
- Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (4), 495–508, 2010.
- Y. Simsek, Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 87, 1-28, 2013.
- Y. Simsek, Special numbers on analytic functions, Appl. Math. 5, 1091–1098, 2014.
- Y. Simsek, Analysis of the p-adic q-Volkenborn integrals: an approach to generalized Apostol-type special numbers and polynomials and their applications, Cogent Math. 3, 1–17, 2016.
- Y. Simsek, Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers, Math. Methods Appl. Sci. 40 (7), 2347–2361, 2016.
- Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12, 1–35, 2018.
- Y. Simsek, Explicit Formulas for p-adic Integrals: Approach to p-adic Distributions and Some Families of Special Numbers and Polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1-76, 2019.
- H. M. Srivastava, G-D. Liu, Some identities and congruences involving a certain family of numbers, Russ. J. Math. Phys. 16, 536–542, 2009.
- H. M. Srivastava, Some generalizations and basic (or q−) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci. 5 (3), 390–444, 2011.
- H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.