Abstract
In this paper, we study the Carlitz’s degenerate Bernoulli numbers and polynomials, and give some formulae and identities related to those numbers and polynomials.
Similar content being viewed by others
References
Açikgöz M, Erdal D, Araci S (2010) A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials. Adv Differ Equ 2010:951764
Bayad A, Kim T (2011) Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russ J Math Phys 18(2):133–143
Carlitz L (1956) A degenerate Staudt–Clausen theorem. Arch Math (Basel) 7:28–33
Carlitz L (1979) Degenerate stirling. Bernoulli and Eulerian numbers. Utilitas Math 15:51–88
Dere R, Simsek Y (2012) Applications of umbral algebra to some special polynomials. Adv Stud Contemp Math 22(3):433–438
Ding D, Yang J (2010) Some identities related to the Apostol–Euler and Apostol–Bernoulli polynomials. Adv Stud Contemp Math (Kyungshang) 20(1):7–21
He Y (2013) A convolution formula for Bernoulli polynomials. Ars Comb 108:97–104
Kim T (1994) An analogue of Bernoulli numbers and their congruences. Rep Fac Sci Eng Saga Univ Math 22(2):21–26
Kim T (2008) q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys 15(1):51–57
Kim T (2015) Barnes’ type multiple degenerate Bernoulli and Euler polynomials. Appl Math Comput 258:556–564
Kim T, Adiga C (2004) Sums of products of generalized Bernoulli numbers. Int Math J 5(1):1–7
Kim DS, Kim T, Dolgy DV, Komatsu T (2015) Barnes-type degenerate Bernoulli polynomials. Adv Stud Contemp Math 24(1):121–146
Kudo A (2000) A congruence of generalized Bernoulli number for the character of the first kind. Adv Stud Contemp Math (Pusan) 2:1–8
Lim D, Do Y (2015) Some identities of Barnes-type special polynomials. Adv Differ Equ 205:42
Luo Q-M, Qi F (2003) Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials. Adv Stud Contemp Math (Kyungshang) 7(1):11–18
Ozden H (2011) p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials. Appl Math Comput 218(3):970–973
Shiratani K (1972) Kummer’s congruence for generalized Bernoulli numbers and its application. Mem Fac Sci Kyushu Univ Ser A 26:119–138
Simsek Y (2008) Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation function. Adv Stud Contemp Math 16(2):251–278
Wang NL (2014) Some identities involving generalized Bernoulli numbers. J Inn Mong Norm Univ Nat Sci 43(4):403–407
Washington LC (1997) Introduction to cyclotomic fields. In: Graduate texts in mathematics, 2nd edn, vol. 83, Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, T., Kim, D.S. & Kwon, HI. Some Identities of Carlitz Degenerate Bernoulli Numbers and Polynomials. Iran J Sci Technol Trans Sci 41, 749–753 (2017). https://doi.org/10.1007/s40995-017-0286-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0286-x