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