On the p,q-Humbert Functions from the View Point of the Generating Function Method

The main object of the present paper is to construct new p,q-analogy definitions of various families of p,q-Humbert functions using the generating function method as a starting point. This study shows a class of several results of p,q-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.


Introduction
In the last quarter of 20th century, q-calculus appeared as a connection between mathematics and physics. We have also a generalization of q-calculus with one more parameter, we can say it is a two-parameter quantum calculus. Generally, it is called ðp, qÞ-calculus. The theory of ðp, qÞ-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 ðp, qÞ-calculus, see, for example, [1][2][3][4][5][6]. Burban and Klimyk [3], Duran et al. [7][8][9][10], Jagannathan [11], Jagannathan and Srinivasa [12], Sahai and Yadav [13] have earlier investigated some properties of the two parameter quantum calculus. Sadjang [14][15][16] introduced the two (ðp, qÞ-analogues of the Laplace transform, two ðp, qÞ-Taylor formulas for polynomials, ðp, qÞ-Appell polynomials and developed some their properties. Mursaleen et al. [17,18] investigated the ðp, qÞ-analogues of Bernstein operators and approximation properties of ðp, qÞ-Bernstein operators that are a generalization of q-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 J m,n ðxÞ in [20,21]. In [22], Srivastava and Shehata have earlier studied the q-Humbert functions. The motivation of these generalizations q-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 ðp, qÞ-Humbert functions for 0 < jqj < jpj ≤ 1 for p, q in ℂ. We mainly use the ðp, qÞ-calculus in the theory of special functions. This work is organized as follows. More precisely, we define the numerous (known or new) ðp, qÞ-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 ðp, qÞ-Humbert functions are discussed. In Section 4, the conclusion and perspectives are given to illustrate the main results.
The q-number ½α q and q-factorial ½n q ! are defined as follows: (see [22]) In [22], the q-Humbert functions is defined by The ðp, qÞ-number (bibasic number or twin-basic number) is denoted by ½α p,q and is defined by the following notation [15] α For p, q, α ∈ ℂ and 0 < jqj < jpj ≤ 1 for p, q, α ∈ ℂ, the ðp, qÞ-number and ðp, qÞ-factorial are given as follows: (see [11,12,15]) The ðp, qÞ-number ½n p,q is a natural generalization of the q-number in (3) such that The ðp, qÞ-number satisfies the following addition properties The ðp, qÞ-factorial is denoted by ½n p,q ! and is defined by (see [6,11,12]) where As in the q-case, there are many definitions of the ðp, qÞexponential function. The following two ðp, qÞ-analogues of exponential function will be frequently used throughout this paper: The ðp, qÞ-exponential function is defined by (see [12,16]) The ðp, qÞ-complementary exponential function is defined by It is easy to see that (see [15,16]) Let f be a function defined on a subset of real or complex plane. The ðp, qÞ-derivative operator of the function f is defined as follows (see [15,24,25]) and ðD p,q f Þð0Þ = f ′ ð0Þ, provided that f is differentiable at 0, which satisfies the following relations (see [14,16]) Journal of Function Spaces The ðp, qÞ-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 ðp, qÞ-Humbert functions. In particular, we will present some particular cases of functions which are belonging to the family of ðp, qÞ-Humbert functions which are introduced as the third ðp, qÞ-Humbert functions.

Definitions of New ðp, qÞ-Analogue of the ðp, qÞ-Humbert Functions and Some Basic Properties
Here we apply the notion of ðp, qÞ-analogue of the generating function to obtain explicit formulas for generalized ðp, qÞ-Humbert functions and give some interesting significant properties for these functions.
Definition 1. Let us define the product of symmetric ðp, qÞexponential functions as the generating function of the ðp, qÞ-Humbert functions of the first kind as follows: Remark 2. Note that in eq. (19), if we put p = 1, then ðp, qÞ-Humbert functions reduces to the q-Humbert functions defined in [22].
From (19) and using (11), we have Replace r by m + k and i by n + k to get Explicitly, we get the explicit expression of ðp, qÞ-Humbert functions J ð1Þ m,n ðx | p, qÞ as the following power series By (9), the series expansions of the ðp, qÞ-Humbert functions J ð1Þ m,n ðx | p, qÞ are given as Proof. From the definition of ðp, qÞ-Humbert functions J ð1Þ m,n ðx | p, qÞ, we have Replacing s = k − m, we obtain The equation (27) can be proved in a like manner.

Lemma 5.
The function J ð1Þ m,n ðx | p, qÞ satisfies the following properties where n and m are integers.
Proof. From the definition of ðp, qÞ-Humbert functions J ð1Þ m,n ðx | p, qÞ, we have Upon setting s = k − m in the Eq. (31), we get Journal of Function Spaces Upon setting s = k − n in the Eq. (31), we get Now, we define that the generating function of ðp, qÞ-Humbert functions of the second kind. Definition 6. The generating function F 2 ðx ; u, t | p, qÞ of ðp, qÞ-Humbert functions of the second kind is defined by From the generating function of the ðp, qÞ-Humbert functions J ð2Þ m,n ðx | p, qÞ, we have Now, substituting r by m + k and i by n + k in the last equation, we obtain the following equality Explicitly, we get the explicit expression of ðp, qÞ-Humbert functions J ð2Þ m,n ðx | p, qÞ as the following power series equivalently, we have Proof. If we set that in (19) and using e 1/p,1/q ðxÞ = E p,q ðxÞ, we get and (34), we obtain (39).
Definition 8. The generating function F 3 ðx ; u, t | p, qÞ of the ðp, qÞ -Humbert functions of the third kind J ð3Þ m,n ðx | p, qÞ is given by Using (42), (11) and (12), we have Substituting r by m + k and i by n + k in the last equation, we obtain the following equality Explicitly, we get the explicit expression of ðp, qÞ-Humbert functions J ð3Þ m,n ðx | p, qÞ of the third kind as the following power series or, equivalently, we get Journal of Function Spaces Definition 9. A fourth generating function F 4 ðx ; u, t | p, qÞ of the ðp, qÞ -Humbert functions J ð4Þ m,n ðx | p, qÞ of the fourth kind is defined by Using (47), (11) and (12), we have Replace r by m + k and i by n + k to get Explicitly, we obtain the explicit expressions of ðp, qÞ -Humbert functions J ð4Þ m,n ðx | p, qÞ as

Journal of Function Spaces
Definition 10. The generating function F 5 ðx ; u, t | p, qÞ of the ðp, qÞ -Humbert functions J ð5Þ m,n ðx | p, qÞ of the fifth kind is defined as Using (11) (12) and (51), we have Upon setting r = m + k and i − n + k in the above equation, we get Explicitly, we obtain the explicit representations of ðp, qÞ -Humbert functions J ð5Þ m,n ðx | p, qÞ of the fifth kind as the following power series Journal of Function Spaces Definition 11. The generating function F 6 ðx ; u, t | p, qÞ of the ðp, qÞ -Humbert functions J ð6Þ m,n ðx | p, qÞ of the sixth kind is defined by From (55), (11) and (12), we have Substituting r by m + k and i by n + k in the last equation, we get the following equality Explicitly, we obtain the explicit representations of ðp, qÞ-Humbert functions J ð6Þ m,n ðx | p, qÞ of the sixth kind as the following power series Journal of Function Spaces Definition 12. The generating function F 7 ðx ; u, t | p, qÞ of the ðp, qÞ -Humbert functions J ð7Þ m,n ðx | p, qÞ of the seventh kind is defined by From (11), (12) and ((59), we have Replacing r by m + k and i by n + k in the above equation, we obtain the following equality Explicitly, we get the explicit expressions of ðp, qÞ-Humbert functions J ð7Þ m,n ðx | p, qÞ of the seventh kind as the following power series From (11), (12) and (63), we have Substituting r by m + k and i by n + k in the above equation, we get the following equality Explicitly, we obtain the explicit expressions of ðp, qÞ-Humbert functions J ð6Þ m,n ðx | p, qÞ of the eighth kind as the following power series and Further examples can be discussed, but are omitted for the sake of conciseness.   ð77Þ Taking x = p 2/3 q 1/3 x, u = p −2/3 q 2/3 u and t = p 1/3 q −1/3 t in Eq. (19), then we get the result Using the generating function (19), and taking x = p 1/3 q 2/3 x, u = p −1/3 q 1/3 u and t = p −1/3 q 1/3 t, we have Thus, we obtain the recurrence relation (71). Similarly, the other equations of this theorem can be proved.