Abstract

The main object of the present paper is to construct new -analogy definitions of various families of -Humbert functions using the generating function method as a starting point. This study shows a class of several results of -Humbert functions with the help of the generating functions such as explicit representations and recurrence relations, especially differential recurrence relations, and prove some of their significant properties of these functions.

1. Introduction

In the last quarter of 20th century, -calculus appeared as a connection between mathematics and physics. We have also a generalization of -calculus with one more parameter, we can say it is a two-parameter quantum calculus. Generally, it is called -calculus. The theory of -calculus or post quantum calculus has recently been applied in many areas of mathematics, physics and engineering, such as biology, mechanics, economics, electrochemistry, probability theory, approximation theory, statistics, number theory, quantum theory, theory of relativity, and statistical mechanics, etc. For more details on this topic -calculus, see, for example, [1–6]. Burban and Klimyk [3], Duran et al. [7–10], Jagannathan [11], Jagannathan and Srinivasa [12], Sahai and Yadav [13] have earlier investigated some properties of the two parameter quantum calculus. Sadjang [14–16] introduced the two (-analogues of the Laplace transform, two -Taylor formulas for polynomials, -Appell polynomials and developed some their properties. Mursaleen et al. [17, 18] investigated the -analogues of Bernstein operators and approximation properties of -Bernstein operators that are a generalization of -Bernstein operators. Khan and Lobiya [19] have nicely discussed a lot of applications in different approximation theory areas, such as per Weirstarass approximation theorems, basic hypergeometric functions, orthogonal polynomials and can be used in differential equations as well as computer-aided geometric designs. Recently, Pasricha and Varma presented and introduced the Humbert function in [20, 21]. In [22], Srivastava and Shehata have earlier studied the -Humbert functions. The motivation of these generalizations -Humbert functions is to provide appropriate application areas of mathematical, physical and engineering such as numerical analysis, approximation theory and computer-aided geometric design (see the recent papers [1, 6, 19, 23] and the references therein).

The main purpose of this paper is to obtain explicit formulas for the various families of -Humbert functions for for in . We mainly use the -calculus in the theory of special functions. This work is organized as follows. More precisely, we define the numerous (known or new) -Humbert functions and discuss some significant properties such as explicit representations, recurrence relations and some new generating functions in Section 2. In Section 3, especially recurrence relations and some interesting differential recurrence relations for the -Humbert functions are discussed. In Section 4, the conclusion and perspectives are given to illustrate the main results.

1.1. Basic Definitions and Miscellaneous Results

To convenience of the reader, we provide a summary of the mathematical notations and some basic definitions of -calculus where for , operations and notations we need to be used in this work. We use the following standard notations: , . The symbols and denote the sets of natural numbers and complex numbers, respectively.

The -number and -factorial are defined as follows: (see [22])

In [22], the -Humbert functions is defined by

The -number (bibasic number or twin-basic number) is denoted by and is defined by the following notation [15]

For and for , the -number and -factorial are given as follows: (see [11, 12, 15])

The -number is a natural generalization of the -number in (3) such that

The -number satisfies the following addition properties

The -factorial is denoted by and is defined by (see [6, 11, 12]) where

As in the -case, there are many definitions of the -exponential function. The following two -analogues of exponential function will be frequently used throughout this paper:

The -exponential function is defined by (see [12, 16])

The -complementary exponential function is defined by

It is easy to see that (see [15, 16])

Let be a function defined on a subset of real or complex plane. The -derivative operator of the function is defined as follows (see [15, 24, 25])

and , provided that is differentiable at , which satisfies the following relations (see [14, 16])

The -derivative operator satisfy the following product rules as follows: (see [11, 12, 14, 15])

Our purpose is to generalize the class of Bessel functions, by using the same approach exposed above and is to define our main problem on the generalized -Humbert functions. In particular, we will present some particular cases of functions which are belonging to the family of -Humbert functions which are introduced as the third -Humbert functions.

2. Definitions of New -Analogue of the -Humbert Functions and Some Basic Properties

Here we apply the notion of -analogue of the generating function to obtain explicit formulas for generalized -Humbert functions and give some interesting significant properties for these functions.

Definition 1. Let us define the product of symmetric -exponential functions as the generating function of the -Humbert functions of the first kind as follows:

Remark 2. Note that in eq. (19), if we put , then -Humbert functions reduces to the -Humbert functions defined in [22].

Remark 3. When , the -Humbert functions reduce to the classical Humbert functions defined in [20, 21].

From (19) and using (11), we have

Replace by and by to get

Explicitly, we get the explicit expression of -Humbert functions as the following power series

By (9), the series expansions of the -Humbert functions are given as or equivalently, we have

Lemma 4. Let and are integers, then the function satisfies the following relations

Proof. From the definition of -Humbert functions , we have

Replacing , we obtain

The equation (27) can be proved in a like manner.

Lemma 5. The function satisfies the following properties where and are integers.

Proof. From the definition of -Humbert functions , we have

Upon setting in the Eq. (31), we get

Upon setting in the Eq. (31), we get

Now, we define that the generating function of -Humbert functions of the second kind.

Definition 6. The generating function of-Humbert functions of the second kind is defined by

From the generating function of the -Humbert functions , we have