Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68228220171201Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)135144279110.22075/ijnaa.2017.11767.1588ENUgur DuranDepartment of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, 27310, Turkey0000-0002-5717-1199Mehmet AcikgozDepartment of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, 27310, TurkeyJournal Article20170627The 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.https://ijnaa.semnan.ac.ir/article_2791_48a0eba5d8560ea93b810f1b3562b4eb.pdf