# A note on the second kind $q$-Apostol Bernoulli numbers, polynomials, and Zeta function

Volume 12, Issue 1, pp 56--64 Publication Date: October 10, 2018       Article History
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### Authors

C. K. An - Department of Mathematics, Hannam University, Daejeon 306-791, Korea. H. Y. Lee - Department of Mathematics, Hannam University, Daejeon 306-791, Korea. Y. R. Kim - Department of Mathematics, Hannam University, Daejeon 306-791, Korea.

### Abstract

In this paper we consider a new type of the $q$-Apostol Bernoulli numbers and polynomials. Firstly, we define the $q$-Apostol Bernoulli numbers and polynomials by making use of their generating function. Also, we observe many properties, i.e., the recurrence formula, the difference equation, the differential relation.

### Keywords

• The second kind $q$-Apostol Bernoulli polynomials
• the second kind $q$-Apostol Bernoulli numbers
• zeta function

•  05A30
•  11M35
•  11B83

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