Document Type : Research Paper

Authors

• Ugur Duran
• Mehmet Acikgoz

Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, 27310, Turkey

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 (${}_p$. 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