Modified Degenerate Carlitz&rsquo;s <em>q</em>-Bernoulli Polynomials and Numbers with Weight (<em>&alpha;</em>, <em>&beta;</em>)

Abstract

The main goal of the present paper is to construct some families of the Carlitz's q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on Z p . We then define the modified degenerate Carlitz's q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz's q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz's q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.Keywords: Carlitz's q-Bernoulli polynomials; Stirling numbers of the first kind; Stirling numbers of the second kind; p-adic q-integral.