%0 Journal Article
%T Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)
%J International Journal of Nonlinear Analysis and Applications
%I Semnan University
%Z 2008-6822
%A Duran, Ugur
%A Acikgoz, Mehmet
%D 2017
%\ 12/01/2017
%V 8
%N 2
%P 135-144
%! Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)
%K Carlitz's $q$-Bernoulli polynomials
%K Stirling numbers of the first kind
%K Stirling numbers of the second kind
%K $p$-adic $q$-integral
%R 10.22075/ijnaa.2017.11767.1588
%X 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.
%U https://ijnaa.semnan.ac.ir/article_2791_48a0eba5d8560ea93b810f1b3562b4eb.pdf