Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

[1]  Taekyun Kim,et al.  Degenerate r-Stirling Numbers and r-Bell Polynomials , 2018 .

[2]  Leonard Carlitz,et al.  A degenerate Staudt-Clausen theorem , 1956 .

[3]  Abdelfattah Mustafa,et al.  New results on higher-order Daehee and Bernoulli numbers and polynomials , 2015, 1503.00104.

[4]  Chang Liu,et al.  Application of Probabilistic Method on Daehee Sequences , 2018 .

[5]  Taekyun Kim.  $q$-Volkenborn Integration and Its Applications , 2005 .

[6]  Sums of powers of consecutive q-integers , 2005, math/0501531.

[7]  Taekyun Kim,et al.  Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind , 2018, Science China Mathematics.

[8]  D. Kim,et al.  Note on the sums of powers of consecutive $q$-integers , 2005, math/0503209.

[9]  Leonard Carlitz,et al.  $q$-Bernoulli and Eulerian numbers , 1954 .

[10]  Taekyun Kim,et al.  Some identities involving special numbers and moments of random variables , 2019, Rocky Mountain Journal of Mathematics.

[11]  W. H. Schikhof.  Ultrametric Calculus: An Introduction to p-Adic Analysis , 1984 .

[12]  T. Kim.  A note on the alternating sums of powers of consecutive q-integers , 2005, math/0508233.

[13]  Taekyun Kim,et al.  Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤp , 2009 .

[14]  Taekyun Kim.  A note on degenerate stirling polynomials of the second kind , 2017 .

[15]  L. Jang,et al.  Exploring the sums of powers of consecutive q-integers , 2005 .

[16]  Taekyun Kim,et al.  Degenerate Laplace transform and degenerate gamma function , 2017, 1701.06881.