Modied 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 polyno- mials and numbers. We rstly introduce the modied Carlitz's q-Bernoulli polynomials and numbers with weight (p. We then dene the modied degenerate Carlitz's q-Bernoulli polynomials and num- bers with weight (α;β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modied Carlitz's q-Bernoulli polynomials with weight (α;β), the modied degenerate Carlitz's q-Bernoulli polynomials with weight (α;β), the Stirling numbers of the rst kind and second kind.