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

Belbachir, Hacène; Souddi, Nassira

doi:10.2298/fil1901211b

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…To evaluate (18), repeat the process from the last section using the saddle-point method to expand f l (w) around the saddle-point w = z −1 instead of f (w). It follows from Lemmas 1 and 2 and Theorem 1 of [14] that (18) may be expanded as the infinite sum…”

Section: Enlarged Region Of Validitymentioning

confidence: 99%

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Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

2023

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In this paper, the uniform approximations of the Apostol–Frobenius–Genocchi polynomials of order α in terms of the hyperbolic functions are derived through the saddle-point method. Moreover, motivated by the works of Corcino et al., an approximation with enlarged region of validity for these polynomials is also obtained. It is found out that the methods are also applicable for the case of the higher order generalized Apostol-type Frobenius–Genocchi polynomials and Apostol–Frobenius-type poly-Genocchi polynomials with parameters a, b, and c. These methods demonstrate the techniques of computing the symmetries of the defining equation of these polynomials. Graphs are illustrated to show the accuracy of the exact values and corresponding approximations of these polynomials with respect to some specific values of its parameters.

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Section: Enlarged Region Of Validitymentioning

confidence: 99%

“…When a = 1, b = e m , c = e, and λ = 1 in (36), this results in the generalized Frobenius-Genocchi polynomials of order α with parameter m, denoted by G α n (z; u; m), defined by Belbachir and Souddi [18] by means of the following generating function:…”

Section: Generalized Apostol-type Frobenius-genocchi Polynomialsmentioning

confidence: 99%

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

2023

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“…Many other examples of D-algebraic EGFs, whose OGFs can be shown to satisfy τ -equations, and are thus (strongly) D-transcendental, can be found in various references in [26,27,24,22,17,13,33,49,25,46,52,37,55,15,40,2,19,32,35,45,34,11,43], although these ones almost never mention explicitly the corresponding τ -equations.…”

Section: Various Other Sequencesmentioning

confidence: 99%

Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

Bostan¹,

Vizio²,

Raschel³

2020

Preprint

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We show that Klazar's results on the differential transcendence of the ordinary generating function of the Bell numbers over the field C({t}) of meromorphic functions at 0 is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result of differential transcendence over C({t}), that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.

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

Cited by 2 publications

References 9 publications

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

Asymptotic Approximations of Higher-Order Apostol–Frobenius–Genocchi Polynomials with Enlarged Region of Validity

Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

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