Some Explicit Formulas for the Frobenius-Euler Polynomials of Higher Order

Two Parametric Kinds of Eulerian-Type Polynomials Associated with Euler’s Formula

“…(m) n (λ) denotes the so-called Apostol-Euler numbers of order m (cf. [14,[21][22][23][24]). The λ-Stirling numbers of the second kind are defined by…”

“…where H (m) n (u) denotes the so-called Frobenius-Euler numbers of order m (cf. [4,5,10,12,13,17,19,20,[22][23][24]). By using (6) and (7), we have…”

Some explicit formulas for the generalized Frobenius-Euler polynomials of higher order

“…For m = 1, we get the generalized Bernoulli polynomials B n (x, α) and the generalized Euler polynomials E n (x, α), see for instance [12,16] and references therein.…”

“…For x = 0, we get the generalized Bernoulli numbers B We have the following relation H For m = 1, we have the Frobenius-Euler polynomials of order α, H n (x, α|λ), see for instance [16,17] and references therein. We have E n (x, α| − 1) .…”

“…We will also consider the case where α be a variable at the end of each of the following sections. Recently, Boutiche et al [16] introduce the generalized Stirling polynomials of the second kind as…”