- Volume 56 Issue 1
The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.
generating functions;special numbers and polynomials;Euler numbers;Frobenius-Euler numbers;Apostol-Bernoulli numbers;Apostol-Euler numbers;q-Apostol type Frobenius-Euler numbers and polynomials;
- M. Alkan and Y. Simsek, Generating function for q-Eulerian polynomials and their decomposition and applications, Fixed Point Theory Appl. 2013 (2013), 72, 14 pp. https://doi.org/10.1186/1687-1812-2013-14
- T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. https://doi.org/10.2140/pjm.1951.1.161
- A. Bayad and T. Kim, Identities for Apostol-type Frobenius-Euler polynomials resulting from the study of a nonlinear operator, Russ. J. Math. Phys. 23 (2016), no. 2, 164-171. https://doi.org/10.1134/S1061920816020023
- I. N. Cangul, A. S. Cevik, and Y. Simsek, Generalization of q-Apostol-type Eulerian numbers and polynomials, and their interpolation functions, Adv. Stud. Contemp. Math. 25 (2015), no. 2, 211-220.
- R. Dere and Y. Simsek, Normalized polynomials and their multiplication formulas, Adv. Difference Equ. 2013 (2013), no. 31, 10 pp. https://doi.org/10.1186/1687-1847-2013-10
- L.-C. Jang, On a q-analogue of the p-adic generalized twisted L-functions and p-adic q-integrals, J. Korean Math. Soc. 44 (2007), no. 1, 1-10. https://doi.org/10.4134/JKMS.2007.44.1.001
- M.-S. Kim and J.-W. Son, Some remarks on a q-analogue of Bernoulli numbers, J. Korean Math. Soc. 39 (2002), no. 2, 221-236. https://doi.org/10.4134/JKMS.2002.39.2.221
- T. Kim, Sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math. (Kyung-shang) 9 (2004), no. 1, 15-18.
- T. Kim, A note on exploring the sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math. (Kyungshang) 11 (2005), no. 1, 137-140.
- T. Kim, L.-C. Jang, and C.-S. Ryoo, Note on q-extensions of Euler numbers and polynomials of higher order, J. Inequal. Appl. 2008 (2008), Art. ID 371295, 9 pp.
- T. Kim, S.-H. Rim, and Y. Simsek, A note on the alternating sums of powers of consecutive q-integers, Adv. Stud. Contemp. Math. (Kyungshang) 13 (2006), no. 2, 159-164.
- T. Kim, S.-H. Rim, Y. Simsek, and D. Kim, On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions, J. Korean Math. Soc. 45 (2008), no. 2, 435-453. https://doi.org/10.4134/JKMS.2008.45.2.435
- T. Kim, C. S. Ryoo, L. C. Jang, and S. H. Rim, Exploring the sums of powers of consecutive q-integers, Internat. J. of Math. Ed. Sci. Tech. 36 (2005), no. 8, 947-956. https://doi.org/10.1080/00207390500138165
- H. Ozden, I. N. Cangul, and Y. Simsek, Multivariate interpolation functions of higherorder q-Euler numbers and their applications, Abstr. Appl. Anal. 2008 (2008), Art. ID 390857, 16 pp.
J. Satoh, q-analogue of Riemann's
$\zeta$-function and q-Euler numbers, J. Number Theory 31 (1989), no. 3, 346-362. https://doi.org/10.1016/0022-314X(89)90078-4
- J. Satoh, A construction of q-analogue of Dedekind sums, Nagoya Math. J. 127 (1992), 129-143. https://doi.org/10.1017/S002776300000413X
- Y. Simsek, On q-analogue of the twisted L-functions and q-twisted Bernoulli numbers, J. Korean Math. Soc. 40 (2003), no. 6, 963-975. https://doi.org/10.4134/JKMS.2003.40.6.963
- Y. Simsek, q-analogue of twisted l-series and q-twisted Euler numbers, J. Number Theory 110 (2005), no. 2, 267-278. https://doi.org/10.1016/j.jnt.2004.07.003
- Y. Simsek, Multiple interpolation functions of higher order (h, q)-Bernoulli numbers, AIP Conference Proceedings 1048 (2008), no. 486.
- Y. Simsek, Complete sum of products of (h, q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl. 16 (2010), no. 11, 1331-1348. https://doi.org/10.1080/10236190902813967
- Y. Simsek, Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms 1 (2012), 395-403. https://doi.org/10.3390/axioms1030395
- Y. Simsek, Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 2013 (2013), 87, 28 pp. https://doi.org/10.1186/1687-1812-2013-28
- Y. Simsek, Identities associated with generalized Stirling type numbers and Eulerian type polynomials, Math. Comput. Appl. 18 (2013), no. 3, 251-263.
- Y. Simsek, On generating functions for the special polynomials, Filomat 31 (2017), no. 1, 9-16. https://doi.org/10.2298/FIL1701009S
- Y. Simsek, A. Bayad, and V. Lokesha, q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions, and q-Stirling numbers, Math. Methods Appl. Sci. 35 (2012), no. 8, 877-884. https://doi.org/10.1002/mma.1580
- Y. Simsek, D. Kim, T. Kim, and S.-H. Rim, A note on the sums of powers of consecutive q-integers, J. Appl. Funct. Differ. Equ. 1 (2006), no. 1, 81-88.
- Y. Simsek, T. Kim, D. W. Park, Y. S. Ro, L. C. Jang, and S. H. Rim, An explicit formula for the multiple Frobenius-Euler numbers and polynomials, JP J. Algebra Number Theory Appl. 4 (2004), no. 3, 519-529.
- Y. Simsek, O. Yurekli, and V. Kurt, On interpolation functions of the twisted generalized Frobenius-Euler numbers, Adv. Stud. Contemp. Math. (Kyungshang) 15 (2007), no. 2, 187-194.
- H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, 2001.
- H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Inc., Amsterdam, 2012.
- H. M. Srivastava, M. Garg, and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russ. J. Math. Phys. 17 (2010), no. 2, 251-261. https://doi.org/10.1134/S1061920810020093
- H. M. Srivastava, T. Kim, and Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241-268.
- H. Tsumura, On a p-adic interpolation of the generalized Euler numbers and its applications, Tokyo J. Math. 10 (1987), no. 2, 281-293. https://doi.org/10.3836/tjm/1270134514
- H. Tsumura, A note on q-analogues of the Dirichlet series and q-Bernoulli numbers, J. Number Theory 39 (1991), no. 3, 251-256. https://doi.org/10.1016/0022-314X(91)90048-G
Supported by : University of Akdeniz