zbMATH — the first resource for mathematics

On the \(q\)-extension of Euler and Genocchi numbers. (English) Zbl 1112.11012
The author considers a new \(q\)-extension of ordinary Euler numbers and polynomials, and also a new \(q\)-extension of Genocchi numbers and polynomials. They are defined by using the generating functions as follows: \[ \begin{aligned} \sum^\infty_{n=0} E_{n,q}{t^n\over n!} &= [2]_q e^{{t\over 1-q}} \sum^\infty_{j=0} {(-1)^j\over 1+ q^{j+1}}\Biggl({1\over 1-q}\Biggr)^j {t^j\over j!},\\ \sum^\infty_{n=0} E_{n,q}(x){t^n\over n!} &= [2]_q e^{{t\over 1-q}}\sum^\infty_{j=0} {(-1)^j q^{jx}\over 1+ q^{j+1}} \Biggl({1\over 1-q}\Biggr)^j{t^j\over j!},\\ \sum^\infty_{n=0} G_{n,q}{t^n\over n!} &= [2]_q t\sum^\infty_{n=0} (-1)^n q^n e^{[n]_q t}\text{ and }\\ \sum^\infty_{n=0} G_{n,q}(x){t^n\over n!} &= [2]_q t\sum^\infty_{n=0} (-1)^n q^{n+x} e^{[n+ x]_q t},\end{aligned} \] where \([n]_q= 1+ q+\cdots+ q^{n-1}\) for a positive integer \(n\). He obtains several identities for these \(q\)-extensions including the \(q\)-analogues of the formulae \[ E_m(x)= \sum^m_{k=0} {m\choose k}{G_{k+1}\over k+1} x^{m-k}\quad\text{and}\quad (n^m- n)G_m= \sum^{m-1}_{k-1} {m\choose k} n^k G_k Z_{m-k}(n- 1) \] for ordinary Euler polynomials \(E_m(x)\) and Genocchi numbers \(G_m\), where \(Z_m(n)= 1^m- 2^m+\cdots+ (-1)^{n+1} n^m\).
Reviewer: Kaori Ota (Tokyo)

11B68 Bernoulli and Euler numbers and polynomials
Full Text: DOI
[1] Andrews, G.E., q-analogs of the binomial coefficient congruences of babbage, Wolstenholme and glaisher, Discrete math., 204, 15-25, (1999) · Zbl 0937.05014
[2] Carlitz, L., q-Bernoulli numbers and polynomials, Duke math. J., 15, 987-1000, (1948) · Zbl 0032.00304
[3] Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. amer. math. soc., 76, 332-350, (1954) · Zbl 0058.01204
[4] Howard, F.T., Applications of a recurrence for the Bernoulli numbers, J. number theory, 52, 157-172, (1995) · Zbl 0844.11019
[5] Kim, T., A note on q-volkenborn integration, Proc. jangjeon math. soc., 8, 1, 13-17, (2005) · Zbl 1174.11408
[6] Kim, T., Sums of powers of consecutive q-integers, (), 15-18 · Zbl 1069.11009
[7] Kim, T., Analytic continuation of multiple q-zeta functions and their values at negative integers, Russian J. math. phys., 11, 71-76, (2004) · Zbl 1115.11068
[8] Kim, T.; Jang, L.C.; Pak, H.K., A note on q-Euler and Genocchi numbers, Proc. Japan acad. ser. A math. sci., 77, 139-141, (2001) · Zbl 0997.11017
[9] Kim, T., On p-adic q-L-functions and sums of powers, Discrete math., 252, 179-187, (2002) · Zbl 1007.11073
[10] Kim, T., On the alternating sums of powers of consecutive integers, Proc. jangjeon math. soc., 8, 2, 175-178, (2005) · Zbl 1157.11305
[11] Sagan, B.E.; Zhang, P., Arithmetic properties of generalized Euler numbers, South Asian bull. math., 21, 73-78, (1997) · Zbl 0899.11008
[12] Schlosser, M., q-analogues of the sums of consecutive integers, squares, cubes, quarts and quints, Electron. J. combin., 11, 1, (2004), Research paper 71, p. 12 · Zbl 1064.33014
[13] Shiratani, K.; Yamamoto, Y., On a p-adic interpolation function for the Euler numbers and its derivatives, Mem. fac. sci. kyushu univ., 39, 113-125, (1985) · Zbl 0574.12017
[14] Simsek, Y., Theorem on the twisted L-functions and twisted Bernoulli numbers, (), 205-218 · Zbl 1178.11058
[15] Srivastava, H.M.; Kim, T.; Simsek, Y., q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russian J. math. phys., 12, 101-106, (2005) · Zbl 1200.11018
[16] Srivastava, H.M.; Choi, J., Series associated with the zeta and related functions, (2001), Kluwer Academic Publishers · Zbl 1014.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.