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Some Identities of Carlitz Degenerate Bernoulli Numbers and Polynomials

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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.

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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

    Article  MATH  MathSciNet  Google Scholar 

  • Carlitz L (1956) A degenerate Staudt–Clausen theorem. Arch Math (Basel) 7:28–33

    Article  MATH  MathSciNet  Google Scholar 

  • Carlitz L (1979) Degenerate stirling. Bernoulli and Eulerian numbers. Utilitas Math 15:51–88

    MATH  MathSciNet  Google Scholar 

  • Dere R, Simsek Y (2012) Applications of umbral algebra to some special polynomials. Adv Stud Contemp Math 22(3):433–438

    MATH  MathSciNet  Google Scholar 

  • 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

    MATH  MathSciNet  Google Scholar 

  • He Y (2013) A convolution formula for Bernoulli polynomials. Ars Comb 108:97–104

    MATH  MathSciNet  Google Scholar 

  • Kim T (1994) An analogue of Bernoulli numbers and their congruences. Rep Fac Sci Eng Saga Univ Math 22(2):21–26

    MATH  MathSciNet  Google Scholar 

  • Kim T (2008) q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys 15(1):51–57

    Article  MATH  MathSciNet  Google Scholar 

  • Kim T (2015) Barnes’ type multiple degenerate Bernoulli and Euler polynomials. Appl Math Comput 258:556–564

    MATH  MathSciNet  Google Scholar 

  • Kim T, Adiga C (2004) Sums of products of generalized Bernoulli numbers. Int Math J 5(1):1–7

    MATH  MathSciNet  Google Scholar 

  • Kim DS, Kim T, Dolgy DV, Komatsu T (2015) Barnes-type degenerate Bernoulli polynomials. Adv Stud Contemp Math 24(1):121–146

    MATH  Google Scholar 

  • Kudo A (2000) A congruence of generalized Bernoulli number for the character of the first kind. Adv Stud Contemp Math (Pusan) 2:1–8

    MATH  MathSciNet  Google Scholar 

  • Lim D, Do Y (2015) Some identities of Barnes-type special polynomials. Adv Differ Equ 205:42

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    MATH  MathSciNet  Google Scholar 

  • Ozden H (2011) p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials. Appl Math Comput 218(3):970–973

    MATH  MathSciNet  Google Scholar 

  • Shiratani K (1972) Kummer’s congruence for generalized Bernoulli numbers and its application. Mem Fac Sci Kyushu Univ Ser A 26:119–138

    MATH  MathSciNet  Google Scholar 

  • 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

    MATH  MathSciNet  Google Scholar 

  • Wang NL (2014) Some identities involving generalized Bernoulli numbers. J Inn Mong Norm Univ Nat Sci 43(4):403–407

    MathSciNet  Google Scholar 

  • Washington LC (1997) Introduction to cyclotomic fields. In: Graduate texts in mathematics, 2nd edn, vol. 83, Springer, New York

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Correspondence to Taekyun Kim.

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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

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  • DOI: https://doi.org/10.1007/s40995-017-0286-x

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