Title: Arithmetic of Sheffer sequences arising from Riemann, Volkenborn and Kim integrals
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-21-00078; Volume 4 / Issue 1 / Year 2022, Pages 149-169
Document Type: Research Paper
Author(s): Dae San Kim a , Taekyun Kim b , Lee-Chae Jang c , Hye Kyung Kim d, Jongkyum Kwon e
aDepartment of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
bDepartment of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
cGraduate School of Education, Konkuk University, Seoul 05029, Republic of Korea
dDepartment of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Republic of Korea
eDepartment of Mathematics Education, Gyeongsang National University, Jinju 52828, Republic of Korea
Received: 21 December 2021, Accepted: 5 February 2022, Published: 19 February 2022.
Corresponding Author: Jongkyum Kwon (Email address: mathkjk26@gnu.ac.kr)
Full Text: PDF
Abstract
Let {sn(x)} be any sequence of polynomials with rational coefficients which is Sheffer for some Sheffer pair. Then we consider the Riemann integral from 0 to 1, the Volkenborn integral on ℤp and the Kim integral on ℤp of sn(x + y) with respect to y. They all give rise to some different Sheffer polynomials. The aim of this paper is to derive some properties of those polynomials, especially their convolution identities, and to illustrate our results with some examples.
Keywords: Riemann integral, Volkenborn integral, Kim integral, Umbral calculus
References:- S. Araci, Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus, Appl. Math. Comput. 233, 599–607, 2014.
- R. Dere and Y. Simsek, Applications of umbral algebra to some special polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (3), 433–438, 2012.
- B. Diarra, The p-adic q-distribution, Advances in Ultrametric Analysis: 12th International Conference on p-adic Functional Analysis, 45–62.
- B. S. El-Desouky and A. Mustafa, New results on higher-order Daehee and Bernoulli numbers and polynomials, Adv. Difference Equ. 2016 (32), 2016.
- T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (3), 288–299, 2002.
- T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp, Russ. J. Math. Phys. 16, 93–96, 2009.
- D. S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. 7 (120), 5969–5976, 2013.
- D. S. Kim and T. Kim, Degenerate Sheffer sequences and λ-Sheffer sequences, J. Math. Anal. Appl. 493 (1), 1–21 Paper No. 124521, 2021.
- D. S. Kim, T. Kim, J. Kwon, S.-H. Lee and S. Park, On λ-linear functionals arising from p-adic integrals on ℤp, Adv. Difference Equ. 2021 (479), 2021.
- D. S. Kim, T. Kim, S.-H. Lee and J.-J. Seo, Higher-order Daehee numbers and polynomials, Int. J. Math. Anal. 8 (6), 273–283, 2014.
- D. S. Kim, T. Kim and J. J. Seo, A note on Changhee polynomials and numbers, Adv. Stud. Theor. Phys. 7, 1–10, 2013.
- D. S. Kim, T. Kim, J.-J. Seo and S.-H. Lee, Higher-order Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 8 (8), 365–373, 2014.
- S. Roman, The umbral calculus, Pure and Applied Mathematics, 111. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
- S. Roman, P. De Land, R. Shiflett and H. Shultz, The umbral calculus and the solution to certain recurrence relations, J. Comb. Inf. Syst. Sci. 8 (4), 235–240, 1983.
- W. H. Schikhof, Ultrametric calculus, An introduction to p-adic analysis, Cambridge University Press, Cambridge, 2010.
- Y. Simsek, Identities on the Changhee numbers and Apostol-type Daehee polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 27 (2), 199–212, 2017.