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On a two-variable \(p\)-adic \(l_{q}\)-function. (English) Zbl 1149.11011

The authors introduce the \(q\)-Euler polynomials \({E}_{n,{q}}^{\ast}(t)\) defined by
\[ \frac{[2]_q}{qe^x+1}e^{xt}=\sum_{n=0}^{\infty}{E}_{n,{q}}^{\ast}(t) \frac{x^n}{n!}. \]
The \(q\)-Euler numbers are \({E}_{n,{q}}^{\ast}:={E}_{n,{q}}^{\ast}(0)\). They also introduce the generalized \(q\)-Euler polynomials \({E}_{n,\chi,q}^{\ast}(t)\) are defined by
\[ \frac{[2]_q\sum_{a=1}^{d}(-1)^aq^a\chi(a)e^{(t+a)x}}{qe^x+1}e^{xt}= \sum_{n=0}^{\infty}{E}_{n,\chi,q}^{\ast}(t)\frac{x^n}{n!}. \]
By using the \(p\)-adic \(q\)-integral they obtain some formulas, e.g.,
\[ \sum_{a=1}^{dn}(-1)^aq^a\chi(a)(t+a)^m= \frac{{E}_{n,\chi,q}^{\ast}(t)+(-1)^{n+1}q^{dn}{E}_{n,\chi,q}^{\ast}(t+dn)}{[2]_q} \]
is a \(q\)-analogue of the classical result
\[ \sum_{k=1}^{m}(-1)^kk^n=\frac{{E}_{n}(0)+(-1)^{m}{E}_n(n+1)}{2}. \]
Furthermore, they also define a two-variable \(p\)-adic \({l}_{q}\)-function by the series expansion:
\[ {l}_{p,q}(s,t,\chi )=\frac{{[2]}_{q}}{{[2]}_{F}}\sum_{\substack{ a=1\\ (p,a)=1}}^{F}{(-1)}^{a}(\chi (a){q}^{a}/{\langle a+pt\rangle }^{s}){\sum }_{m=0}^{\infty }\binom{-s}{m} {(F/\langle a+pt\rangle )}^{m}{E}_{m,{q}^{F}}^{\ast}, \]
where \(\chi\) is the Dirichlet character with conductor \(d=d_{\chi}\) (= odd), \(F\) is a positive integral multiple of \(p\) and \(d\). Then they obtain the following relation
\[ {l}_{p,q}(-n,t,\chi )={E}_{n,{\chi }_{n},q}^{\ast}(pt)-{p}^{n}{\chi }_{n}(p)({[2]}_{q}/{[2]}_{{q}^{p}}){E}_{n,{\chi }_{n},{q}^{p}}^{\ast}(t), \]
where \(n\) is a nonpositive integer. The proof of this original construction is due to Kubota and Leopoldt in 1964, the method in this paper is due to Washington. For related topics see also the papers of the reviewer and H. M. Srivastava [J. Math. Anal. Appl. 308, No. 1, 290–302 (2005; Zbl 1076.33006), Comput. Math. Appl. 51, No. 3–4, 631–642 (2006; Zbl 1099.33011) and Integral Transforms Spec. Funct. 17, No. 6, 451–454 (2006; Zbl 1108.11023)].

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
11M41 Other Dirichlet series and zeta functions
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

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