Modified degenerate Carlitz's q-bernoulli polynomials and numbers with weight (α,β)

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 (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.