Modified degenerate Carlitz's q-bernoulli polynomials and numbers with weight (\alpha ,\beta )

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 (\alpha ,\beta ) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz's q-Bernoulli polynomials with weight (\alpha ,\beta ), the modified degenerate Carlitz's q-Bernoulli polynomials with weight (\alpha ,\beta ), 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