Open Phys. 2016; 14:187–191
Research Article
Open Access
P. Agarwal*, M. Chand, İ. Onur Kıymaz, and A. Çetinkaya
A Certain Sequence of Functions Involving the Aleph Function DOI 10.1515/phys-2016-0018 Received December 19, 2015; accepted March 17, 2016
Abstract: Sequences of functions play an important role in approximation theory. In this paper, we aim to establish a (presumably new) sequence of functions involving the Aleph function by using operational techniques. Some generating relations and finite summation formulas of the sequence presented here are also considered. Keywords: Special function; Generating relation; Aleph ℵ− function; Sequence of functions; Finite summation formula. PACS: 02.30.Gp, 02.30.Lt
1 Introduction
and finite summation formulas in terms of the Aleph function, are written in compact and easily computable form in Sections 2 and 3. Finally, some special cases and concluding remarks are discussed in Section 4. Throughout this paper, let C, R, R+ , Z−0 , and N be sets of complex numbers, real numbers, positive real numbers, non-positive integers and positive integers respectively. Also N0 := {0} ∪ N. The Aleph function, which is a general higher transcendental function and was introduced by S¨ udland et al. [26, 27], is defined by means of a Mellin-Barnes type integral in the following manner (see, e.g., [23, 24]) ]︃ [︃ ⃒ (︀ ⃒ a , A )︀ , [︀τ (︀a , A )︀]︀ ⃒ (︀ j j)︀ 1,n [︀ k (︀ jk jk )︀]︀ n+1,p k ;r m,n ℵ [z] = ℵp k ,q k ,τ k ;r z ⃒ ⃒ b j , B j 1,m , τ k b jk , B jk m+1,q ;r k ∫︁ 1 −s := Ω m,n p k ,q k ,τ k ;r ( s ) z ds 2πi L
Recently, interest has developed into study of operational techniques, due to their importance in many field of engineering and mathematical physics. The sequences of functions play an important role in approximation theory. They can be used to show that a solution to a differential equation exists. Therefore, a large body of research into the development of these sequences has been published In the literature, there are numerous sequences of functions, which are widely used in physics and mathematics as well as in engineering. Sequences of functions are also used to solve some differential equations in a rather efficient way. Here, we introduce and investigate further computable extensions of the sequence of functions involving the Aleph function, represented with ℵ, by using operational techniques. The generating relations
*Corresponding Author: P. Agarwal: Department of Mathematics, Anand International College of Engineering, Jaipur 303012, Rajasthan, India, E-mail:
[email protected] M. Chand: Department of Mathematics, Singhania University, Pacheri Bari 333515, India, E-mail:
[email protected] İ. Onur Kıymaz: Department of Mathematics, Ahi˙ Evran University, 40100, Kirşehir, Turkey, E-mail:
[email protected] A. Çetinkaya: Department of Mathematics, Ahi˙ Evran University, 40100, Kirşehir, Turkey, E-mail:
[email protected]
where z ∈ C − {0}, i =
(1)
√
−1 and
Ω m,n p k ,q k ,τ k ;r ( s ) = ∏︀m ∏︀n j=1 Γ(b j + B j s) j=1 Γ(1 − a j − A j s) ∑︀r ∏︀q k ∏︀ k τ Γ(1 − b − B jk s) pj=n+1 Γ(a jk + A jk s) jk k=1 k j=m+1 (2) here Γ denotes the familiar Gamma function; the integration path L = L i𝛾∞ (𝛾 ∈ R) extends from 𝛾 − i∞ to 𝛾 + i∞; the poles of Gamma function Γ(1−a j −A j s) (j = 1, 2, . . . , n) do not coincide with those of Γ(b j + B j s) (j = 1, 2, . . . , m); the parameters p k , q k ∈ N0 satisfy the conditions 0 ≤ n ≤ p k , 1 ≤ m ≤ q k ; τ k > 0 (k = 1, 2, . . . , r); the parameters A j , B j , A jk , B jk > 0 and a j , b j , a jk , b jk ∈ C; the empty product in (2) is (as usual) understood to be unity. The existence conditions for the defining integral (1) are given below φ l > 0, φ l ≥ 0,
© 2016 P. Agarwal et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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