Abdelmejid Bayad

Some array type polynomials associated with special numbers and polynomials

Abstract The main objective in this paper is first to establish new identities for the λ -Stirling type numbers of the second kind, the λ -array type polynomials, the Apostol–Bernoulli polynomials and the Apostol–Bernoulli numbers. We then construct a λ -delta operator and investigate various generating functions for the λ -Bell type numbers and for some new polynomials associated with the λ -array type polynomials. We also derive several other identities and relations for these polynomials and numbers.

Reduction and duality of the generalized Hurwitz-Lerch zetas

In this paper, by means of integral representation, we introduce the generalized Hurwitz-Lerch zeta functions of arbitrary complex order. For these functions, we establish the reduction formula and its associated dual formula. We then investigate analytic continuations to the whole complex plane and special values. By means of these reduction and dual formulas, we obtain nice and useful formulas for the Bernoulli-Nörlund and Apostol-Euler-Nörlund polynomials.

Note sur une forme de Jacobi méromorphe

Resume Soit L un reseau complexe. Nous etudions les proprietes d’une fonction DL(z; ϕ), periodique de periodes L en la seconde variable, et analytique en la premiere variable, normalisee par lim ⁡ z → 0 z D L ( z ; ϕ ) = 1 cette fonction est liee par un facteur exponentiel simple a la forme Fτ = θ′ (0)θ(u + v)/(θ(u)θ(v)) analytique en τ ∈ ℋ (= demi-plan superieur) et u, v ∈ ℂ, introduite dans [7], § 3, ou le couple (u, v) est proportionnel a (z, ϕ) et θ designe le produit triple de Jacobi. Notre resultat principal est que DL verifie aussi une relation de distribution additive simple, de nature arithmetique. De facon plus precise, si Λ est un reseau tel que L ⊂ Λ et [Λ : L] = l, on a: ∑ t D L ( l z ; ϕ + t ) = D Λ ( z ; ϕ ) ou t parcourt un systeme complet de representants dans ℂ de Λ/L. On retrouve des resultats connus, lorsque ϕ est un point de torsion de ℂ/L.

Higher dimensional Dedekind sums in finite fields

Abstract We introduce Dedekind sums of a new type defined over finite fields. These are similar to the higher dimensional Dedekind sums of Zagier. The main result is the reciprocity law for them.

A differential approach to a polynomial equivalence problem

A new efficient algorithm for solving the linear variant of the isomorphism of polynomials with one secret problem (J. Patarin, 1996) is presented. This paper shows that partial knowledge of a matrix solution allows to recover it entirely by solving a suitable linear system.

CERTAIN COMBINATORIC CONVOLUTION SUMS AND THEIR RELATIONS TO BERNOULLI AND EULER POLYNOMIALS

In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as ap- plications, we show five identities concerning the third and fourth-order convolution sums of divisor functions expressed by their divisor functions and linear combination of Bernoulli or Euler polynomials.

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

We define the two-dimensional (2D) Apostol-Bernoulli and the 2D Apostol-Euler polynomials respectively via the generating functions t e x t + y t m λ e t - 1 = ? n = 0 ∞ B n ( x , y ; λ ) t n n ! , 2 e x t + y t m λ e t + 1 = ? n = 0 ∞ E n ( x , y ; λ ) t n n ! . As parametrized polynomial families they are essentially the same. We study their basic algebraic properties, generalizing some well-known formulas and relations for Apostol-Bernoulli and Bernoulli polynomials. We determine the Fourier series of x ? λ x B n ( x , y ; λ ) , y ? λ x B n ( x , y ; λ ) and ( x , y ) ? λ x B n ( x , y ; λ ) for (x, y) ? 0, 1) × 0, 1). These contain as a special case the Fourier series of the one-dimensional Apostol-Bernoulli and Apostol-Euler polynomials.

Multiple Dedekind–Rademacher sums in function fields

In the previous paper, we introduced the higher-dimensional Dedekind sums in function fields, and established the reciprocity law. In this paper, we generalize our higher-dimensional Dedekind sums and establish the reciprocity law and the Petersson–Knopp identity.

Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices

We study combinatoric convolution sums of certain divisor functions involving even indices. We express them as a linear combination of divisor functions and Euler polynomials and obtain identities , , and . As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method.

The minimal polynomial of 2 cos(π/q) and Dickson polynomials☆

Abstract The number λ q = 2 cos ( π / q ) , q ∈ N , q ⩾ 3 , appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many partial results about the minimal polynomial of this algebraic number. Here we obtain the general formula and it is Mobius inversion for this minimal polynomial by means of the Dickson polynomials and the Mobius inversion theory. Moreover, we investigate the homogeneous cyclotomic, Chebychev and Dickson polynomials in two variables and we show that our main results in one variable case nicely extend to this situation. In this paper, the deep results concerning these polynomials are proved by elementary arguments.