Abstract
The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz’s q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers.
Highlights
There are various ways of studying special polynomials and numbers, including generating functions, combinatorial methods, umbral calculus techniques, matrix theory, probability theory, p-adic analysis, differential equations, and so on.In [1], it was shown that odd integer power sums can be represented in terms of some values of the type 2 Bernoulli polynomials.In addition, some identities of symmetry, involving the type 2 Bernoulli polynomials, odd integer power sums, the type 2 Euler polynomials, and alternating odd integer power sums, were obtained by introducing appropriate quotients of bosonic and fermionic p-adic integrals on Z p
In [1], it was shown that the moments of two random variables, constructed from random variables with Laplace distributions, are closely connected with the type 2 Bernoulli numbers and the type 2
In recent years, studying degenerate versions of various special polynomials and numbers, which began with the paper by Carlitz in [2], has attracted the interest of many mathematicians
Summary
There are various ways of studying special polynomials and numbers, including generating functions, combinatorial methods, umbral calculus techniques, matrix theory, probability theory, p-adic analysis, differential equations, and so on. (q-Euler) polynomials were introduced by virtue of the bosonic (fermionic) p-adic q-integrals It was noted, among other things, that the odd q-integer (alternating odd q-integer) power sums are expressed in terms of the type 2 q-Bernoulli (q-Euler) polynomials. In this short paper, we would like to introduce the type 2 degenerate q-Bernoulli polynomials and the corresponding numbers by making use of the bosonic p-adic q-integrals, as a degenerate version of and as a q-analogue of the type 2 Bernoulli polynomials, and derive several basic results for them. We study type 2 degenerate q-Bernoulli polynomials and investigate some identities and properties for these polynomials
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