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On \(p\)-adic interpolating function for \(q\)-Euler numbers and its derivatives. (English) Zbl 1160.11013

Let \[ [x]_{q}=\frac{1-q^x}{1-q}\text{ and }\lim_{q\to1}[x]_q=x. \] For \[ f\in UD(\mathbb Z_p,\mathbb C_p)=\{f\mid f:\mathbb Z_p\to\mathbb C_p\text{ is uniformly differentiable function}\}, \] the \(p\)-adic invariant \(q\)-integral on \(\mathbb Z_p\) was defined by the author [Russ. J. Math Phys. 9, No. 3, 288–299 (2002; Zbl 1092.11045)] as follows: \[ I_{q}(f)=\int_{\mathbb Z_p}f(x)d\mu_{q}(x)=\lim_{N\to\infty}\frac{1}{[p^N]_q} \sum_{x=0} ^{p^N-1}q^x f(x). \] Recently, many applications of this integral have been given. The \(q\)-deformed \(p\)-adic invariant integral on \(\mathbb Z_p\), in the fermionic sense was defined by the author [T. Kim, J. Nonlinear Math. Phys. 14, No. 1–4, 15–27 (2007; Zbl 1158.11009); J. Math. Anal. Appl. 331, No. 2, 779–792 (2007; Zbl 1120.11010)]: \[ I_{-q}(f)=\int_{\mathbb Z_p}f(x)d\mu_{-q}(x)=\lim_{N\to\infty}\frac{1}{[p^N]_{-q}} \sum_{x=0} ^{p^N-1}(-q)^x f(x). \] By busing \(p\)-adic \(q\)-integral on \(\mathbb Z_p\), the author gives Witt’s formula associated with \(q\)-Euler numbers. He also constructs two variable Dirichlet’s type Euler \(l\)-function, which interpolates generalized \(q\)-Euler polynomials at negative integers. He defines partial Euler \(q\)-zeta functions. He gives some properties of this function. He also constructs \(p\)-adic interpolation function of the \(q\)-Euler numbers which is defined as follows: Let \(F\) (=odd) be a positive integral multiple of \(p\) and \(d=d_\chi\). For \(s\in\mathbb Z_p\) \[ l_{p,q}(s,t|\chi)=[2]_q \sum_{a=1}^{F}\chi(a)H_{p,q}^{(E)}(s,a+pt|F) \] where \[ H_{p,q}^{(E)}(s,a+pt|F)=\frac{(-1)^a}{[2]_{q^F}}\langle a\rangle^{-s} \sum_{j=0} ^{\infty}\left(\begin{matrix} -s\\ j\end{matrix}\right) \left(\frac{[F]_q}{[a]_q}\right)^j E_{j,q^F}. \] This functions interpolate generalized \(q\)-Euler polynomials at negative integers. Finally, the author finds derivative of \(l_{p,q}(s,t|\chi)\) function. He gives many applications related to these functions, \(q\)-Euler polynomials, \(p\)-adic \(q\)-integral and \(p\)-adic measure.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
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