International Bulletin of Mathematical Research Volume XX, Issue X, December 2014 Pages 16-27, ISSN: XXXX-XXXX
γ,K Note on Sequence of Functions involving the Product of Eδ,β (x) Mehar Chand Department of Mathematics Fateh College for Women, Rampura Phul Bathinda-151001, India
[email protected]
Abstract A remarkably large number of operational techniques have drawn the attention of several researchers in the study of sequences of functions and polynomials. In this sequel, here, we aim to introduce a new sequence of functions involving a product of the generalized Mittag-Leffler function by using operational techniques. Some generating relations and finite summation formula of the sequence presented here are also considered.
1
Introduction
The idea of representing the processes of calculus, differentiation, and integration, as operators, is called an operational technique, which is also known as an operational calculus. Many operational techniques involve various special functions have found some significant applications in various sub-fields of applicable mathematical analysis. Several applications of operational techniques can be found in the problems of analysis, in particular differential equations are transformed into algebraic problems, usually the problem of solving a polynomial equations. Since last four decades, a number of workers like Chak[5], Gould and Hopper [11], Chatterjea[8], Singh[27], Srivastava and Singh[29], Mittal[15, 16, 17], Chandal[6, 7], Srivastava[24], Joshi and Parjapat[14], Patil and Thakare[18] and Srivastava and Singh[28] have made deep research of the properties, applications and different extensions of the various operational techniques. d The key element of the operational technique is to consider differentiation as an operator D = dx acting on functions. Linear differential equations can then be recast in the form of an operator valued function F(D) of the operator D acting on an unknown function which equals a known function. Solutions are then obtained by making the inverse operator of F acting on the known function.
Indeed, a remarkably large number of sequences of functions involving a variety of special functions have been developed by many authors (see, for example, [28]; for a very recent work, see also [1, 2, 3, 22, 23, 26]). γ,K Here we aim at presenting a new sequence of functions involving a product of the Eδ,β by using operational techniques. Some generating relations and finite summation formula are also obtained. For our purpose, we begin by recalling some known functions and earlier works. In 1971, by Mittal [15] gave the Rodrigues formula for the generalized Lagurre polynomials defined by
Received: December 2014 Keywords: Special function, generating relations , Mittag-Leffler, Sequence of function, finite summation formula, symbolic representation. AMS Subject Classification: 33E10, 44A45.
γ,K Note on Sequence of Functions involving the Product of Eδ,β (x)
(α)
Tkn (x) =
17
1 −α x exp (pk (x)) Dn xα+n exp (−pk (x)) , n!
(1.1)
where pk (x) is a polynomial in x of degree k. Mittal[16] also proved the following relation for (1.1) given by (α+s−1)
Tkn
(x) =
1 −α−n x exp (pk (x)) Tsn [xα exp (−pk (x))] , n!
(1.2)
where s is a constant and Ts ≡ x (s + xD). (α)
In this sequel, in 1979, Srivastava and Singh [28] studied a sequence of functions Vn
Vn(α) (x; a, k, s) =
(x; a, k, s) defined by
x−α exp {pk (x)} θn [xα exp {−pk (x)}] n!
(1.3)
By using the operator θ ≡ xa (s + xD) , where s is constant, and pk (x) is a polynomial in x of degree k. n o∞ (δ ,β ,γ ,K ,α) Here, a new sequence of function Vn j j j j (x; a, k, s)
n=0
(δj ,βj ,γj ,Kj ,α)
Vn
(x; a, k, s) :=
1 −α x n!
r Y j=1
is introduced as follows:
r Y γ ,K γ ,K Eδjj,βj j −pkj (x) , Eδjj,βj j pkj (x) (Txa,s )n xα
(1.4)
j=1
d , a and s are constants, β ≥ 0, kj is a finite and non-negative integer, pkj (x) dx γ ,K are polynomials in x of degree kj , where j = 1, 2, ..., r and Eδjj,βj j (.) is generalized Mittag-Leffler function. For where Txa,s ≡ xa (s + xD) , D ≡
γ,K the sake of completeness, we recall the Eδ,β (.).
In 1903, the Swedish mathematician Gosta Mittag-Leffler [19] introduced the function Eα (z), defined as Eα (z) =
∞ X n=0
zn Γ(αn + 1)
(1.5)
where z is a complex variable and Γ(.) is a Gamma function, α ≥ 0. The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces for α = 0. For 0 < α < 1 it interpolates 1 between the pure exponential and a hypergeometric function 1−z . Its importance is realized during the last two decades due to its involvement in the problems of physics, chemistry, biology, engineering and applied sciences. Mittag-Leffler function naturally occurs as the solution of fractional order differential equation or fractional order integral equations. The generalization of Eα (z) was studied by Wiman [31] in 1905 and he defined the function as
18
Mehar Chand
Eα,β (z) =
∞ X n=0
zn Γ(αn + β)
(α, β ∈ C; 0, 0)
(1.6)
Which is known as Wiman’s function or generalized Mittag-Leffler function as Eα,1 (z) = Eα (z). The former was introduced by Mittag-Leffler[19] in connection with his method of summation of some divergent series. In his papers [19, 20], he investigated certain properties of this function. The function defined by (1.6) first appeared in the work of Wiman [31]. The function (1.6) is studied, among others, by Wiman [31], Agarwal [4], Humbert [12] and Humbert and Agarwal [13] and others. The main properties of these functions are given in the book by Erdelyi et al. ([10], Section 18.1) and a more comprehensive and a detailed account of Mittag-Leffler functions are presented in Dzherbashyan ([9], Chapter 2). γ In 1971, Prabhakar [21] introduced the function Eα,β (z) in the form of
γ Eα,β (z) =
∞ X n=0
(γ)n zn , Γ(αn + β) n!
(1.7)
where α, β, γ ∈ C; 0, 0), 0 and (λ)n denotes the familiar Pochhammer symbol or the shifted factorial, since (1)n = n! (n ∈ N0 )
(λ)n =
Γ(λ + n) = Γ(λ)
1 (n = 0; λ ∈ C − {0}) λ(λ + 1)...(λ + n − 1) (n ∈ N ; λ ∈ C)
(1.8)
γ Recently generalization of Mittage-Leffler function Eα,β (z) of (1.7) studied by Srivastava and Tomovski [30] is defined as follows:
γ,K Eα,β (z) =
∞ X n=0
(γ)Kn z n , Γ(αn + β) n!
(1.9)
where α, β, γ ∈ C; 0, 0), 0; 0 which, in the special case when
K = q(q ∈ (0, 1) ∪ N)
and
min{
