A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. (English) Zbl 1261.05003
Applying the method of generating function and fermionic \(p\)-adic integral representation on \(\mathbb{Z}_p\), this paper derives some relations between the Frobenius-Euler numbers and polynomials and the Bernstein polynomials. Accordingly, the authors obtain some identities such as
\[
\sum_{l=0}^{n-k}{n-k \choose l} (-1)^l H_{l+k}(-u^{-1})= 1+u^{-1}+u^{-2}H_n(-u^{-1}),\quad k =0,
\]
or
\[
\sum_{l=0}^{n-k}{n-k \choose l} (-1)^l H_{l+k}(-u^{-1})= \sum_{l=0}^k {k \choose l} (-1)^{k+l} (1+u^{-1}+u^{-2}H_{n-l}(-u^{-1})),\quad k > 0.
\]
here \(H_n(u)\) are Frobenius-Euler numbers defined as \(H_n(u):= H_n(u,x)\),
\[
\sum_{n=0}^\infty H_n(u,x) \frac{t^n}{n!} = \frac{1-u}{e^t-u} e^{xt}.
\]
Reviewer: Ping Sun (Shenyang)
MSC:
05A10 | Factorials, binomial coefficients, combinatorial functions |
11B65 | Binomial coefficients; factorials; \(q\)-identities |
28B99 | Set functions, measures and integrals with values in abstract spaces |
11B68 | Bernoulli and Euler numbers and polynomials |
11B73 | Bell and Stirling numbers |