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.