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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 \(H_n(u):= H_n(u,x)\), \[ \sum_{n=0}^\infty H_n(u,x) \frac{t^n}{n!} = \frac{1-u}{e^t-u} e^{xt}. \]

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
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Full Text: arXiv