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\(q\)-Genocchi numbers and polynomials associated with fermionic \(p\)-adic invariant integrals on \(\mathbb Z_{p}\). (English) Zbl 1149.11010

The authors use the fermionic \(p\)-adic invariant integral
\[ \int_{\mathbb{Z}_p}f(x) \,d\mu_{-q}(x)=\lim_{N\to\infty}\frac{1}{[p^N]_q}\sum_{x=0}^{p^N-1}f(x)(-q)^x \]
to construct \(p\)-adic Genocchi numbers and polynomials of higher order. They derive some formulas, e.g.,
\[ {G}_{n+k,q}^{(k)}(x)= {2}^{k}k! \binom{n+k}{k} \sum_{l=0}^{\infty}\;\sum_{d_0+d_1+\dots+d_k=k-1,d_i\in\mathbb{N}} (-1)^l (l+x)^n, \] where \(G_{n,q}^{(k)}(x)\) are the \(q\)-Genocchi polynomials of order \(k\) defined by \[ \frac{t^k2^k}{(e^t+1)(qe^t+1)\cdots(q^{k-1}e^t)}e^{xt}=\sum _{n=0}^{\infty}G_{n;q}(x)\frac{t^n}{n!}. \]

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
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References:

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