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.
1 Introduction
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 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 +,
where z ∈ℂ − {0},
here Γ denotes the familiar Gamma function; the integration path L = Liγ∞ (γ ∈ ℝ) extends from γ − i∞ to γ+ i∞; the poles of Gamma function Γ(1−aj−Ajs) (j = 1, 2, . . ., n) do not coincide with those of Γ(bj + Bjs) (j = 1, 2, . . ., m); the parameters pk, qk ∈ ℕ0 satisfy the conditions 0 ≤ n ≤ pk, 1 ≤ m ≤ qk; τk > 0 (k = 1, 2, . . ., r); the parameters Aj, Bj, Ajk, Bjk > 0 and aj, bj, ajk, bjk ∈ ℂ; the empty product in (2) is (as usual) understood to be unity. The existence conditions for the defining integral (1) are given below
Setting τk = 1 (k = 1, 2, . . ., r) in (1.1) yields the I-function [25], whose further special case when r = 1 reduces to the familiar Fox’s H-function (see [21, 22]).
For our purpose, we also required some known functions and earlier works. In 1971, Mittal [12] gives the Rodrigues formula for the generalized Lagurre polynomials defined as
where pk (x) is a polynomial in x of degree k and
Mittal [13] also proved the following relation for (7)
where s is constant and Ts ≡ x (s + xD).
In this sequel, in 1979, Srivastava and Singh [19] studied a sequence of functions
by employing the operator θ ≡ xa (s + xD), where a and s are constants.
In this paper, a new sequence of functions
where
The following properties of the differential operator
2 Generating Relations
First generating relation:
Second generating relation:
Third generating relation:
Proof of first generating relation:
From (10), we have
Using operational technique (11) in (20), we get
Replacing t by tx−a, (17) is obtained.
Proof of second generating relation:
From (10), we obtain
Applying operational technique (12) in (22), we have
which yields the desired result.
Proof of third generating relation:
We can write the following equation from (10)
Thus we have
or
Applying operational technique (11), the above equation can then be written as
or
Replacing t by tx−a, this gives result (19).
3 Finite Summation Formulas
First finite summation formula:
Second finite summation formula:
Proof of first finite summation formula:
From (10), we obtain
Using operational techniques (13), (14) and (15), we get
On the other hand, taking α = 0 and replacing n by n − m in (32), we find
or
Thus, using (35) in (33), we have the required result (30).
Proof of second finite summation formula:
From (10), we have
Applying operational technique (11) in (36) we obtain
Using operational technique (16), (37) reduces to
Now equating the coefficients of tn, we get
4 Special Cases
Here we consider some interesting special cases of the results given in Section 2 and 3.
It is further noted that a number of other special cases of our main results, as illustrated in Sections 2 and 3, can also be obtained. In this paper, we have studied a new sequence of functions involving the Aleph function by using operational techniques, and we have established some generating relations and finite summation formulas of the sequence. Moreover, in view of close relationships of the Aleph function with other special functions, it does not seem difficult to construct various known and new sequences.
Competing interests:The authors declare that they have no competing interests.
Author’s contributions:The authors contributed equally to this work. The four authors have contributed to the manuscript; they wrote, read, and approved the
manuscript.
Acknowledgement
This work was supported by the Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: PYO-FEN.4001.14.008.
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