## 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

### Citations:

Zbl 1092.11045; Zbl 1120.11010; Zbl 1158.11009
Full Text:

### References:

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