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.
Duran,U and Acikgoz,M . (2017). Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $). International Journal of Nonlinear Analysis and Applications, 8(2), 135-144. doi: 10.22075/ijnaa.2017.11767.1588
MLA
Duran,U , and Acikgoz,M . "Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)", International Journal of Nonlinear Analysis and Applications, 8, 2, 2017, 135-144. doi: 10.22075/ijnaa.2017.11767.1588
HARVARD
Duran U, Acikgoz M. (2017). 'Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)', International Journal of Nonlinear Analysis and Applications, 8(2), pp. 135-144. doi: 10.22075/ijnaa.2017.11767.1588
CHICAGO
U Duran and M Acikgoz, "Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $)," International Journal of Nonlinear Analysis and Applications, 8 2 (2017): 135-144, doi: 10.22075/ijnaa.2017.11767.1588
VANCOUVER
Duran U, Acikgoz M. Modified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($\alpha ,\beta $). IJNAA. 2017;8(2):135-144. doi: 10.22075/ijnaa.2017.11767.1588
International Journal of Nonlinear Analysis and Applications