Some identities on the Bernstein and \(q\)-Genocchi polynomials (Q2843800)

Let \(q\in\mathbb C_p\) with \(| 1-q|_p<1\), then the \(q\)-Genocchi polynomial are defined by NEWLINE\[NEWLINE\frac{2t}{qe^t+1}e^{xt}=e^{G_q(x)t}=\sum_{n=0}^\infty G_{n,q}(x)\frac{t^n}{n!},NEWLINE\]NEWLINE replacing \(G_q^n(x)\) by \(G_{n,q}(x)\). For \(x=0\) we have \(G_{n,q}(0)\), the \(n\)th \(q\)-Genocchi number. The author proves some properties of these polynomials, gives an identity involving the \(n\)th Frobenius-Euler number. Using the \(p\)-adic integral on \(\mathbb Z_p\) several further identities are proved.