Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials

Bayad, Abdelmejid; Navas, Luis M.

doi:10.1016/j.amc.2015.05.128

Cited by 2 publications

(5 citation statements)

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“…We have

t θ (x, y, λ, t) = B (- x, - y, λ, t),

where

double-struckB false(x, y, λ, t false)

is the generating function for the two‐dimensional Apostol‐Bernoulli polynomials introduced by Bayad and Navas . More precisely, we have the following expansion

B (x, y, λ, t) = \sum_{n = 0}^{\infty} {scriptB}_{n} (x, y; λ) \frac{t^{n}}{n!},

which converges for

false| t + \log λ false| < 2 π

.…”

Section: The Multidimensional Appell Polynomials and The Multidimensimentioning

confidence: 99%

“…When n =1, we recover the two‐dimensional Apostol‐Bernoulli polynomials defined in Bayad and Navas by the following generating function

t θ_{λ} (t) e^{x t + y t^{m}} = t \frac{e^{x t + y t^{m}}}{λ e^{t} - 1} = \sum_{l = 0}^{\infty} {scriptB}_{l} (x, y; λ) \frac{t^{l}}{l!} .

Note that…”

Section: The Multidimensional Appell Polynomials and The Multidimensimentioning

confidence: 99%

“…where B(x, , , t) is the generating function for the two-dimensional Apostol-Bernoulli polynomials introduced by Bayad and Navas. 3 More precisely, we have the following expansion…”

Section: The Multidimensional Appell Polynomialsmentioning

confidence: 99%

“…For instance, the special values of these functions are related to multidimensional Appell polynomials. Bayad and Navas considered the special case of two‐dimensional Bernoulli‐Euler‐Apostol polynomials. Navas et al established that a class of some classical polynomials can be expressed as the special values of some analytic functions.…”

Section: Introductionmentioning

confidence: 99%

“…One of the goals of this paper is to unify the approaches developed by Simsek, Bayad‐Navas, Varona et al, and Bayad‐Simsek on the special values of the twisted Barnes zeta functions. This is done by introducing the notion of theta functions .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the multidimensional zeta functions associated with theta functions, and the multidimensional Appell polynomials

Bayad

Hajli

2019

Math Methods in App Sciences

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We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well‐known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.

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“…We have

t θ (x, y, λ, t) = B (- x, - y, λ, t),

where

double-struckB false(x, y, λ, t false)

is the generating function for the two‐dimensional Apostol‐Bernoulli polynomials introduced by Bayad and Navas . More precisely, we have the following expansion

B (x, y, λ, t) = \sum_{n = 0}^{\infty} {scriptB}_{n} (x, y; λ) \frac{t^{n}}{n!},

which converges for

false| t + \log λ false| < 2 π

.…”

Section: The Multidimensional Appell Polynomials and The Multidimensimentioning

confidence: 99%

“…When n =1, we recover the two‐dimensional Apostol‐Bernoulli polynomials defined in Bayad and Navas by the following generating function

t θ_{λ} (t) e^{x t + y t^{m}} = t \frac{e^{x t + y t^{m}}}{λ e^{t} - 1} = \sum_{l = 0}^{\infty} {scriptB}_{l} (x, y; λ) \frac{t^{l}}{l!} .

Note that…”

Section: The Multidimensional Appell Polynomials and The Multidimensimentioning

confidence: 99%

“…where B(x, , , t) is the generating function for the two-dimensional Apostol-Bernoulli polynomials introduced by Bayad and Navas. 3 More precisely, we have the following expansion…”

Section: The Multidimensional Appell Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%