TY - JOUR AU - An, C. K. AU - Lee, H. Y. AU - Kim, Y. R. PY - 2019 TI - A note on the second kind \(q\)-Apostol Bernoulli numbers, polynomials, and Zeta function JO - Journal of Nonlinear Sciences and Applications SP - 56--64 VL - 12 IS - 1 AB - 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. SN - ISSN 2008-1901 UR - http://dx.doi.org/10.22436/jnsa.012.01.06 DO - 10.22436/jnsa.012.01.06 ID - An2019 ER - TY - JOUR TI - On the Lerch zeta function AU - T. M. Apostal JO - Pacific J. Math. PY - 1951 DA - 1951// VL - 1 ID - Apostal 1951 ER - TY - JOUR TI - Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function AU - J. Choi AU - P. J. Anderson AU - H. M. Srivastava JO - Appl. Math. Comput. PY - 2008 DA - 2008// VL - 199 ID - Choi2008 ER - TY - BOOK TI - Advanced combinatorics: the art of finite and infinite expansions AU - L. Comtet PB - D. Reidel Publishing Co. PY - 1974 DA - 1974// CY - Dordrecht/ Boston ID - Comtet 1974 ER - TY - JOUR TI - Note on the Euler q-zeta functions AU - T. Kim JO - J. Number Theory PY - 2009 DA - 2009// VL - 129 ID - Kim2009 ER - TY - JOUR TI - A note on q-Euler and Genocchi numbers AU - T. Kim AU - L.-C. Jang AU - H. K. Pak JO - Proc. Japan Acad. Ser. A Math. Sci. PY - 2001 DA - 2001// VL - 77 ID - Kim2001 ER - TY - JOUR TI - A numerical investigation of the roots of the second kind \(\lambda\)-Bernoulli polynomials AU - H. Y. Lee AU - N. S. Jung AU - C. S. Ryoo JO - Neural Parallel Sci. Comput. PY - 2011 DA - 2011// VL - 19 ID - Lee2011 ER - TY - BOOK TI - Formulas and Theorems for the Special Functions of Mathematical Physics AU - W. Magnus AU - F. Oberhettinger AU - R. P. Soni PB - Springer-verlag PY - 1966 DA - 1966// CY - New York ID - Magnus1966 ER - TY - JOUR TI - Analytic continuation of Euler polynomials and Euler zeta function AU - C. S. Ryoo JO - Discrete Dyn. Nat. Soc. PY - 2014 DA - 2014// VL - 2014 ID - Ryoo2014 ER - TY - JOUR TI - Distribution of the roots of the second kind Bernoulli polynomials AU - C. S. Ryoo JO - J. Comput. Anal. Appl. PY - 2011 DA - 2011// VL - 13 ID - Ryoo2011 ER - TY - BOOK TI - Handbook of number theory II AU - J. Sándor AU - B. Crstici PB - Kluwer Academic Publishers PY - 2004 DA - 2004// CY - Dordrecht ID - Sándor2004 ER - TY - JOUR TI - On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers AU - Y. Simsek JO - J. Korean Math. Soc. PY - 2003 DA - 2003// VL - 40 ID - Simsek2003 ER - TY - BOOK TI - Series associated with the zeta and related functions AU - H. M. Srivastava AU - J. Choi PB - Kluwer Academic Publishers PY - 2001 DA - 2001// CY - Dordrecht ID - Srivastava2001 ER -