ON THE RITT ORDER AND TYPE OF A CERTAIN CLASS OF FUNCTIONS DEFINED BY BE -DIRICHLETIAN ELEMENTS

We introduce the notions of Ritt order and type to functions defined by the series ∞ ∑ n=1 fn ( σ +iτ0 ) exp (−sλn), s = σ +iτ, (σ ,τ)∈ R×R (∗) indexed by τ0 on R, where ( λn )∞ 1 is a D-sequence and (fn) ∞ 1 is a sequence of entire functions of bounded index with at most a finite number of zeros. By definition, the series are BE -Dirichletian elements. The notions of order and type of functions, defined by BDirichletian elements, are considered in [3, 4]. In this paper, using a technique similar to that used by M. Blambert and M. Berland [6], we prove the same properties of Ritt order and type for these functions.


Preliminary lemmas
Definition 1.1 (B.Lepson [10]).An entire function f is said to be of bounded index if there exists a nonnegative integer ν such that max f (k) for all j and for all s.The least such integer ν is called the index of f .
Theorem A (F. Gross [8]).An entire function with at most a finite number of zeros is of bounded index if and only if it is of the form P (s)exp(αs), where P (s) is polynomial and α is a complex constant.
Theorem B (S. M. Shah [16]).Let f (s)= P (s)exp(αs), where α is any complex number and P (s) is a polynomial of degree less than n.Then f is of bounded index and the index ν ≤ p, where p is any integer such that p ≥ n − 1 and Let λ n ∞ 1 be a D-sequence (that is a positive strictly increasing unbounded sequence) and (f n ) ∞  1 be a sequence of entire functions f n of bounded index ν n with at most a finite number of zeros from Theorem A. As a result of the two theorems, we have ∀s ∈ C, ∀n ∈ N\{0} f n (s) = P n (s) exp α n s , (1.3) where P n (s) is a polynomial of degree m n and α n is a complex constant, that is, a n,j s j with a n,mn = 0 and s ∈ C. (1.4) Let us suppose that ∃k ∈]0,λ 1 [, ∀n ∈ N\{0} α n ∈ d (0,k) , (1.5) where d (0,k) is the closed disc centered at 0 and of radius k.
Consider the space of elements indexed by τ 0 on R. By definition, {f τ 0 } is the BE -Dirichletian element.Let ) Consider the associated Dirichletian element whose coefficients are strictly positive and denote, by σ f A c , the abscissa of convergence of {f A }.
Let us state three lemmas due to M. Blambert and M. Berland [6] which we use later.These demonstrations are obvious because this sequence (1.10) where τ 1 is any arbitrary real number.

Main theorems.
Let us define the following quantities.For each σ on C, where (2.5) The quantities defined above are finite.

Remark. The function σ M(σ
and Therefore, ∃n 1 ∈ N\{0} such that  are the Ritt-order and the lower Ritt-order of function f τ 0 defined by BE -Dirichletian element {f τ 0 }.Also, M(σ ; f A ) is defined in a similar manner with f A in the place of f τ 0 .It is trivial that (2.21) Proof.(1) We get the inequalities, ∀τ 0 ∈ R, τ 0 is any arbitrary real number.Consider the closed interval and then (M.Blambert and M. Berland [6])
we have (2.100) Remark.The notions of Ritt-type of order of functions, defined by B−Dirichletian elements, are considered in [3] with the same result of this theorem. (2.102) Proof.(1) We have the inequality, ∀τ 0 ∈ R, Suppose that the inequality is false.Then Under the conditions stated in Theorem 2.4, R. K. Srivastava [17] proved that which implies that ∃n ∈ N\{0}, ∀n ≥ n , where λ is a constant lying in [1, ∞[ and Let us consider ϕ n , the function defined by and indexed by n ≥ n 2 .Choose This takes the maximum value at (2.114) As ∀n ∈ N\{0}, (2.119) Thus, we get the contradiction that which proves, under the stated conditions, that it is impossible to find a τ 0 of R such (2.121) As a result of this theorem, we have an expression for (2.122)

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: are the Ritt-type and the lower Ritt-type of order of f τ 0 .