Geodesic
A note on the second kind $q$-Apostol Bernoulli numbers, polynomials, and Zeta function
An, C. K. 1 ; Lee, H. Y. 1 ; Kim, Y. R. 1
1 Department of Mathematics, Hannam University, Daejeon 306-791, Korea
Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 1, p. 56-64.

Voir la notice de l'article provenant de la source International Scientific Research Publications

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.
DOI : 10.22436/jnsa.012.01.06
Classification : 05A30, 11M35, 11B83
Keywords: The second kind \(q\)-Apostol Bernoulli polynomials, the second kind \(q\)-Apostol Bernoulli numbers, zeta function

An, C. K.  1 ; Lee, H. Y.  1 ; Kim, Y. R.  1

1 Department of Mathematics, Hannam University, Daejeon 306-791, Korea 
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An, C. K. ; Lee, H. Y. ; Kim, Y. R. . A note on the second kind \(q\)-Apostol Bernoulli numbers, polynomials, and Zeta function. Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 1, p. 56-64. doi : 10.22436/jnsa.012.01.06. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.01.06/

[1] T. M. Apostal On the Lerch zeta function , Pacific J. Math., Volume 1 (1951), pp. 161-167 | Zbl

[2] Choi, J.; Anderson, P. J.; Srivastava, H. M. Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function, Appl. Math. Comput., Volume 199 (2008), pp. 723-737 | DOI | Zbl

[3] L. Comtet Advanced combinatorics: the art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht/ Boston, 1974

[4] Kim, T. Note on the Euler q-zeta functions, J. Number Theory, Volume 129 (2009), pp. 1798-1804 | DOI

[5] Kim, T.; Jang, L.-C.; Pak, H. K. A note on q-Euler and Genocchi numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 77 (2001), pp. 139-141 | DOI | Zbl

[6] Lee, H. Y.; Jung, N. S.; Ryoo, C. S. A numerical investigation of the roots of the second kind \(\lambda\)-Bernoulli polynomials, Neural Parallel Sci. Comput., Volume 19 (2011), pp. 295-306 | Zbl

[7] Magnus, W.; Oberhettinger, F.; Soni, R. P. Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-verlag, New York, 1966

[8] Ryoo, C. S. Analytic continuation of Euler polynomials and Euler zeta function, Discrete Dyn. Nat. Soc., Volume 2014 (2014), pp. 1-6

[9] Ryoo, C. S. Distribution of the roots of the second kind Bernoulli polynomials, J. Comput. Anal. Appl., Volume 13 (2011), pp. 971-976

[10] Sándor, J.; B. Crstici Handbook of number theory II, Kluwer Academic Publishers, Dordrecht, 2004 | DOI

[11] Simsek, Y. On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers, J. Korean Math. Soc., Volume 40 (2003), pp. 963-975 | DOI

[12] Srivastava, H. M.; Choi, J. Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001

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