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
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