Further study on the conformable fractional Gauss hypergeometric function

1.   Introduction

Many research efforts has been devoted to generalize the classical analysis of special functions to either the fractional calculus or the higher dimensional setting ^[1,2,3]. Fractional calculus has recently attracted considerable attention. It is defined as a generalization of differentiation and integration to an arbitrary order. It has become a fascinating branch of applied mathematics, which has recently stimulated mathematicians and physicists. Indeed, it represents a powerful tool to study a myriad of problems from different fields of science, such as statistical mechanics, control theory, signal and image processing, thermodynamics and quantum mechanics (see ^{[4,5,6,7,8,9]}).

There is a significant value in exploring conformable derivatives, which is obvious from the enormous number of their meaningful and successful applications in many fields of science in the last few years. We mention, but not limited to, some efforts which proved the vital role of conformable derivatives. For instance, the authors in ^[10] studied the dynamics of the traveling wave with fractional conformable nonlinear evaluation equations arising in nonlinear wave mechanics. In ^[11], the authors formulated the exact solutions of the time-fractional Dodd-Bullough-Mikhailov equation, Sinh-Gordon equation, and Liouville equation by utilizing simplest equation method in the conformable fractional derivative sense. Recently, in ^[12], Rabha et al. introduced different vitalization of the growth of COVID-19 by using controller terms based on the concept of conformable calculus. For more on the conformable and the fractal derivative and their applications to real world problems, we refer the reader to see ^{[13,14,15,16,17,18,19,20,21]} and the references therein. Some growing progress in the fractional differential equations describing real life phenomena has been discussed in ^[22,23,24].

Over the last four decades, several interesting and useful extensions of many of the familiar special functions, such as the Gamma, Beta, and Gauss hypergeometric functions have been considered by various authors ^{[25,26,27,28,29]}. Functions of hypergeometric type constitute an important class of special functions. The hypergeometric function $_{2}F_{1}(\mu, \nu; c;x)$ plays a significant role in mathematical analysis and its applications. This function allows one to solve many interesting mathematical and physical problems, such as conformal mapping of triangular domains bounded by line segments or circular arcs and various problems of quantum mechanics. Most of the functions that occur in the analysis are classified as special cases of the hypergeometric functions. Gauss first introduced and studied hypergeometric series, paying particular attention to the cases when a series converges to an elementary function which leads to the study of the hypergeometric series. Eventually, elementary functions and several other important functions in mathematics can be expressed in terms of hypergeometric functions. Hypergeometric functions can also be described as the solutions of special second-order linear differential equations, which are the hypergeometric differential equations. Riemann was the first to exploit this idea and introduced a special symbol to classify hypergeometric functions by singularities and exponents of differential equations. The hypergeometric function is a solution of the following Euler's hypergeometric differential equation

which has three regular singular points $0, 1,$ and $\infty$ and the parameters $\mu, \nu$ and $c$. The generalization of this equation to three arbitrary regular singular points is given by Riemann's differential equation. Any second order differential equation with three regular singular points can be converted to the hypergeometric differential equation by changing of variables.

The solution of the hypergeometric differential equation includes many of the most interesting special functions of mathematical physics and engineering, for instance, the Jacobi, Gegenbaure, Legendre, and Laguerre polynomials can be expressed in terms of the Gauss hypergeometric functions and other related hypergeometric functions. Every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric differential equation by means of a suitable change of variable. Recently, as a conformable fractional derivative introduced in ^[30], the authors in ^[31] used the new concept of fractional regular singular points with the technique of fractional power series to solve the CFGHDE about $x = 0$. They also introduced the form of the conformable fractional derivative and the integral representation of the fractional Gaussian function. Besides, the solution of the fractional $k-$hypergeometric differential equation was introduced in ^[32]. As the Gauss hypergeometric differential equation appears in many problems of physics, engineering, applied science, as well as finance and many other important problems, it largely motivates us to conduct the present study.

Motivated by the above discussion, we intend to continue the work of Abu Hammad et al. ^[31] by finding the solutions of the Gauss hypergeometric differential equation via conformable calculus about the fractional regular singular points $x = 1$ and $x = \infty$. Afterward, we give a wide study on the CFGHF includes deriving various generating functions involving expansions and generalizations of the CFGHF and some of the transformation formulas and differential forms. In order to deduce several of contiguous relations, we define a conformable fractional operator. Furthermore, integral representations and Laplace transform of CFGHF in the context of conformable calculus are established. As an application, we give general solutions of a class of conformable fractional differential equations (CFDE), which can be written in terms of the CFGHF.

The structure of this paper is formulated as follows. In section 2, we provide some basic concepts and notations which are essential in the sequel. Section 3 is devoted to the solutions of the conformable fractional Gauss hypergeometric differential equation. Various generating functions of CFGHF are established in section 4. In section 5, we present some of transformation formulas and differential forms. We define a differential operator, then use it to establish several of contiguous relations in section 6. Conformable fractional integral representations are derived in section 7. Various recursion formulas are obtained in section 8. In section 9, we establish the fractional Laplace transform of CFGHF and some useful related identities. General solutions of some interesting CFDEs are obtained by means of CFGHF in section 10. We append our study by pointing out some general remarks and conclusions in section 11.

2.   Preliminaries and basic concepts

Various definitions of fractional derivatives are obtained and compared. The most commonly used definitions in literature are due to Riemann-Liouvellie and Caputo. The Riemann-Liouville and Caputo fractional derivatives are non-local operators represented by convolution integrals with weakly singular kernels. Although the non-local fractional derivatives give natural memory and genetic effects in the physical system, the fractional derivatives obtained in this kind of calculus seem very complicated and lose some basic properties of general derivatives, such as product rule and chain rule and others, see for instance ^[33]. Accordingly, Khalil et al. in ^[30] introduced a definition of a local kind derivative called from the authors "conformable fractional derivative". The significance of this definition lies on the fact that their derivative stratifies almost all the well-known properties of the integer-order one. Abdeljawad ^[34] made extensive research on the newly introduced conformable fractional calculus. Also, Martynyuk ^[35] presented a physical interpretation of such conformable derivative. Afterwords, more than hundred published articles appeared relying this derivative see ^[36] and references therein.

In ^[37], Anderson and Ulness made a remark, that since the derivative is local, they suggested the name to be "conformable derivative" instead as introduced "conformable fractional derivative". Some authors ^{[35,38,39,40]} use the name "fractional-like" instead "conformable" derivative. Since we still have the derivative $f^{(\alpha)}(x),$ for $0 < \alpha \le 1,$ we shall keep the original name "conformable fractional derivative".

The authors in ^[36] studied the relationship between the conformable derivatives of different order. They obtained a surprising result that a function has a conformable derivative at a point if and only if it has a first-order derivative at the same point, and that holds for all points except the lower terminal. They also answered the question, "what happens in the lower terminal?" From this point of view, the same authors ^[36] concluded that "not all types of initial value problems involving conformable derivatives are transformed to well studied initial value problems with integer-order derivatives." This phenomenon was supported via an example given in ^[33].

We briefly state the main advantages of using the conformable derivatives as follows:

$(1)$ It simulates all the concepts and properties of an ordinary derivative such as quotient, product, and chain rules while the other fractional definitions fail to satisfy these rules.

$(2)$ Non-differentiable functions can be differentiated in the conformable sense.

$(3)$ It generalizes the well-known transforms such as Laplace and Sumudu transforms and is used as a tool for solving some singular fractional differential equations.

$(4)$ It paves the way for new comparisons and applications.

$(5)$ It can be extended to solve PDEs exactly and numerically as it was given in the literature.

For the sake of clarity and avoiding ambiguity, we recall the definitions of both the conformable derivative and conformable integral introduced in ^[30] as well as some of their properties.

Definition 2.1. Let $f:\Omega \subseteq \left(0, \infty \right) \rightarrow \mathbb{R}$ and $x\in \Omega.$ The conformable fractional derivative of order $\alpha \in (0, 1]$ for $f$ at $x$ is defined as

whenever the limit exists. The function $f$ is called $\alpha -$ conformable fractional differentiable at $x.$ For $x = 0, \; D^{\alpha }f\left(0\right) = \lim\limits_{h\rightarrow 0^{+}}D^{\alpha }f\left(x\right)$ if such a limit exists.

This definition carries very important and natural properties. Let $D^{\alpha }$ denote the conformable fractional derivative (CFD) operator of order $\alpha$. We recall from ^[30,34,41] some of its general properties as follows.

Let $f$ and $g$ be $\alpha -$differentiable. Then, we have

$(1)$ Linearity: $D^{\alpha }\left(af+bg\right) \left(t\right) = aD^{\alpha }f\left(t\right) +bD^{\alpha }g\left(t\right),$ for all$\; a, b\in \mathbb{R}.$

$(2)$ Product rule: $D^{\alpha }\left(fg\right) \left(t\right) = f\left(t\right) D^{\alpha }g\left(t\right) +g\left(t\right) D^{\alpha }f\left(t\right).$

$(3)$ Quotient rule: $D^{\alpha }\left(\frac{f}{g}\right) \left(t\right) = \frac{g\left(t\right) D^{\alpha }f\left(t\right) -f\left(t\right) D^{\alpha }g\left(t\right) }{g^{2}\left(t\right) },$ where $g\left(t\right) \neq 0.$

$(4)$ Chain rule: $D^{\alpha }\left(f\circ g\right) \left(t\right) = D^{\alpha }f\left(g\left(t\right) \right) D^{\alpha }g\left(t\right) g\left(t\right) ^{\alpha -1}.$

Notice that for $\alpha = 1$ in the $\alpha -$ conformable fractional derivative, we get the corresponding classical limit definition of the derivative. Also, a function could be $\alpha -$ conformable differentiable at a point but not differentiable in the ordinary sense. For more details, we refer to ^[30,34,42].

Any linear homogeneous differential equations of order two with three regular singularities can be reduced to (1.1). The hypergeometric function is known as a solution to the hypergeometric equation (1.1). One of the solutions of the hypergeometric equation is given by the following Gauss hypergeometric series in the form

where $\left(b\right) _{n}$ stands for the usual Pochhammer symbol defined by

Choosing the values of the parameters $\mu, \nu$, and $c$ in an appropriately, one can obtain many elementary and special functions as particular cases of the Gauss hypergeometric series. For instance, the complete elliptic integrals of the first and the second kinds, the Legendre associated functions, ultra-spherical polynomials, and many others are special cases of the function $_{2}F_{1}\left(\mu, \nu; c;x\right).$

Definition 2.2. ^[43] Two hypergeometric functions are said to be contiguous if their parameters $\mu, \nu$, and $c$ differ by integers. The relations made by contiguous functions are said to be contiguous function relations.

Definition 2.3. The point $x = a$ is called an $\alpha -$regular singular point for the equation

if $\lim\limits_{x\rightarrow a}(x^{\alpha }-a)P\left(x\right)$ and $\lim\limits_{x\rightarrow a}(x^{\alpha }-a)^{2}Q\left(x\right)$ exist.

Definition 2.4. ^[34] A series $\sum\limits_{n = 0}^{\infty }a_{n}x^{\alpha n}$ is called a fractional Maclaurin power series.

Remark 2.1. We will use $D^{n\alpha }$ to denote $\underset{n-\text{times}}{\underbrace{ D^{\alpha }D^{\alpha }...D^{\alpha }}}$. If $D^{n\alpha }f$ exists for all $n$ in some interval $[0, \lambda]$ then one can write $f$ in the form of a fractional power series

Definition 2.5. ^[30] Suppose that $f:(0, \infty)\rightarrow \mathbb{R}$ is $\alpha$-differentiable, $\alpha \in (0, 1]$, then the $\alpha$-fractional integral of $f$ is defined by

For the infinite double series, we have the following useful Lemma (see ^[44]), which will be used in the sequel.

Lemma 2.1.

3.   Solutions of the conformable fractional Gauss hypergeometric differential equation

In our current study, we are interested to consider a generalization of the differential equation (1.1) to fractional Gauss hypergeometric differential equation, where the involving derivative is CFD. More precisely, we study the equation in the form

where $\alpha \in (0, 1]$ and $\mu, \nu$ and $c$ are reals such that $c \neq 0, -1, -2, \dots$.

The new concept of fractional regular singular point together with the technique of fractional power series are used to solve the CFGHDE (3.1).

Dividing (3.1) by $x^{\alpha }\left(1-x^{\alpha }\right)$, we get

Comparing (3.2) with (2.2), we have

Clearly $x = 0, \; x = 1$ and $x = \infty$ are $\alpha -$regular singular points for (3.1).

Recently, in ^[31], the authors used the technique of fractional power series to obtain the general solution of (3.1) about $x = 0$ as

where $A$ and $B$ are arbitrary constants and $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ is CFGHF defined by

In this section, we will use a similar technique of ^[31] to solve the Eq (3.1) about the two $\alpha -$regular singular points $x = 1$ and $x = \infty.$

3.1.   Solution of the CFGHE about $x=1$

As $x = 1$ is an $\alpha -$regular singular point of (3.1), therefore, the solution of (3.1) can be obtained in a series of powers of $\left(x^{\alpha }-1\right)$ as follows:

Taking $x^{\alpha } = 1-t^{\alpha },$ this transfers the point $x = 1$ to the point $t = 0$ and therefore, we obtain the series solution of the following transformed CFDE in terms of the series of powers of $t^{\alpha }$:

Putting $c^{\prime } = \mu +\nu +1-c,$ in (3.4), we get

This conformable fractional differential equation is similar to CFGHE (3.1). So, the two linearly independent solutions of (3.5) can be stated in the form

Now, replacing $c^{\prime }$ by $\left(\mu +\nu +1-c\right)$ and $t^{\alpha }$ by $\left(1-x^{\alpha }\right)$ in (3.6), we get

and

Thus, the general solution of Eq (3.1) about $x = 1$ is given by

where $A$ and $B$ are arbitrary constants.

3.2.   Solution of the CFGHE about $x=\infty$

As $x = \infty$ is an $\alpha -$regular singular point of (3.1), thus, the solution of (3.1) can be obtained in a series about $x = \infty$ by putting $x^{\alpha } = \frac{1}{\zeta ^{\alpha }}$ in (3.1). Therefore,

In view of (3.1), we get

Now, to find the solution, we proceed as follows. Let $y = \sum\limits_{n = 0}^{\infty }a_{n}\zeta ^{\alpha \left(s+n\right) }; \; a_{0}\neq 0$ be the series solution of Eq (3.9) about $\zeta = 0.$ Then from the basic properties of the CFD, we get

Thus, owing to (3.9), we have

Therefore,

Then we have

A shift of index yields

Equating the coefficients of $\zeta ^{\alpha s}$ to zero in (3.10), we get the following indicial equation

This Eq (3.11) has two indicial roots $s = s_{1} = \mu$ and $s = s_{2} = \nu$.

Again, equating to zero the coefficient of $\zeta ^{\alpha (s+n+1)}$ in (3.10), yields the recursion relation for $a_{n}$

or

To find the first solution of (3.9), we put $s = \mu$ in (3.13) to get

Note that if $n = 0,$ one can see

and for $n = 1,$ we obtain

Using the Pochhammer symbol, we have

In general, we may write

Letting $a_{0} = A,$ the first solution $y_{1}$ is given by

To find the second solution of (3.9), putting $s = \nu$ in (3.13), we have

from which we get

Thus

Again by Pochhammer symbol yields

and in general

Putting $a_{0} = B,$ the second solution $y_{2}$ is given by

Therefore, the general solution of (3.1) about $x = \infty$ is

where $A$ and $B$ are arbitrary constants.

Remark 3.1. It is worth mentioning that the presented CFDE $(3.9)$ is distinct to that one which was treated in ^[31,32]. In fact $(3.9)$ extended the Gauss hypergeometric differential equation given in ^[45] to the conformable fractional context.

4.   Generating functions

Generating functions provide an important way to transform formal power series into functions and to analyze asymptotic properties of sequences. In what follows, we characterize the CFGHF by means of various generating functions.

Theorem 4.1. For $\alpha \in (0, 1],$ the following generating function holds true

where $\left\vert x^{\alpha }\right\vert < 1,$ and $\left\vert t^{\alpha }\right\vert < 1$.

Proof. For convenience, let $\Im$ denote the left-hand side of (4.1). In view of (3.3), it follows that

Changing the order of summations in (4.2) and make use of identity $\left(\mu \right) _{m}\left(\mu +m\right) _{n} = \left(\mu \right) _{m+n} = \left(\mu \right) _{n}\left(\mu +n\right) _{m},$ yields

Using the equality $\sum\limits_{m = 0}^{\infty }\frac{\left(\mu +n\right) _{m}}{m!}\; t^{\alpha m} = \left(1-t^{\alpha }\right) ^{-\left(\mu +n\right) }; \; \; \left\vert t^{\alpha }\right\vert < 1$ and the definition (3.3) immediately leads to the required result.

Theorem 4.2. For $\alpha \in (0, 1],$ we have the following relation

where $\left\vert x^{\alpha }\right\vert < 1,$ and $\left\vert t^{\alpha }\right\vert < 1$

Proof. For short, set $\Im$ to denote the left-hand side of (4.3). Using (3.3), one gets

Since $\left(-m\right) _{n} = 0$ if $n > m,$ then we may write

Using the congruence relation $\left(-m\right) _{n} = \frac{\left(-1\right) ^{n}m!}{\left(m-n\right)!},$ we get

Using lemma 2.1, Eq (4.5) becomes

Changing the order of summations in (4.6) and make use of identity $\left(\mu \right) _{n+m} = \left(\mu \right) _{n}\left(\mu +n\right) _{m},$ we get

Using the binomial relation $\sum\limits_{m = 0}^{\infty }\; \frac{\left(\mu +n\right) _{m}}{m!}t^{\alpha m} = \left(1-t^{\alpha }\right) ^{-\left(\mu +n\right) }, \; \ \left(\left\vert t^{\alpha }\right\vert < 1\right),$ it follows that

Theorem 4.3. For $\alpha \in (0, 1],$ the following generating relation is valid

where $\left\vert x^{\alpha }\right\vert < 1,$ $\left\vert t^{\alpha }\right\vert < 1,$ and $\left\vert x^{\alpha }+t^{\alpha }\right\vert < 1$.

Proof. Let $\Im$ denote the left-hand side of (4.7), then using (3.3), we obtain

With the help of $\left(\mu \right) _{n+m} = \left(\mu \right) _{m}\left(\mu +m\right) _{n},$ we get

Using lemma 2.1, we have

The binomial theorem immediately gives

as required.

5.   Transmutation formulas and differential forms

5.1.   Transmutation formulas

Theorem 5.1. For $\left\vert x^{\alpha}\right\vert < 1$ and $\left\vert \frac{x^{\alpha }}{ 1-x^{\alpha }}\right\vert < 1$, the following identity holds true

Proof. Consider

In view of the expansion $\left(1-x^{\alpha }\right) ^{-\mu } = \sum\limits_{n = 0}^{\infty}\frac{\left(\mu\right)_{n}}{n!} x^{\alpha n}; \left\vert x^{\alpha }\right\vert < 1,$ we may write

Using the identity $\left(\mu +k\right) _{n} = \left(\mu \right) _{k+n},$ it is obvious

In virtue of lemma 2.1 and using the fact $\left(-n\right) _{k} = \frac{\left(-1\right) ^{k}n!}{\left(n-k\right)!}$, one easily gets

Since $\left(-n\right) _{k} = 0$ if $k > n,$ then (5.3) becomes

Since the inner sum on the right of (5.4) is a terminating hypergeometric series, then

Due to $_{2}F_{1}\left(-n, c-\nu; c;1\right) = \frac{\left(\nu \right) _{n}}{\left(c\right) _{n}}$, the proof is therefore completed.

Theorem 5.2. For $\left\vert x^{\alpha}\right\vert < 1$, the following identity is true

Proof. By using assertion of theorem 5.1 and assuming that $y^{\alpha } = \frac{-x^{\alpha }}{1-x^{\alpha }},$ it follows that

From the assumption, we have $x^{\alpha } = \frac{ -y^{\alpha }}{1-y^{\alpha }}$ which gives

With a combinations of (5.8) and (5.1), the result follows.

5.2.   Some differential forms

According to the notation $D^{\alpha n}$, and due to the fact $D^{\alpha }x^{p} = px^{p-\alpha }$ with $\alpha \in (0, 1]$ and $\left\vert x^{\alpha }\right\vert < 1,$ we state some interesting conformable fractional differential formulas for $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ as follows

Such formulas can be proved using the series expansions of $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ as given in (3.3). However, we are going to prove the validity of (5.11) and (5.14), while the other formulas can be proved similarly. First, note that

The action of the conformable derivative gives

Knowing that $\left(\mu \right) _{n+k} = \left(\mu \right) _{n}\left(\mu +n\right) _{k},$ it can be seen

as required.

In view of (5.6), we have

Using (5.10), we get

Return to (5.6), we obtain

Remark 5.1. In case of $\mu = -n$ in $(5.16)$, we obtain

6.   A differential operator and contiguous relations of the CFGHF

Following ^[44], define the conformable fractional operator $\theta ^{\alpha }$ in the form

This operator has the particularly pleasant property that $\theta ^{\alpha }x^{n\alpha } = nx^{n\alpha },$ which makes it handy to be used on power series. In this section, relying on definition 2.2, we establish several results concerning contiguous relations for the CFGHF. To achieve that, we have to prove the following lemma.

Lemma 6.1. Let $\alpha \in (0, 1],$ then the CFGHF $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ satisfies the following

Proof. Using (3.3) and (6.1), it follows that

Similarly, we have

Analogously, we obtain

Therefore,

The following result is an immediate consequence of Lemma 6.1.

Theorem 6.1. Let $\alpha \in (0, 1],$ then the CFGHF $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ satisfies the following contiguous relations

and

Proof. Using (6.2) and (6.3) immediately give (6.5) and similarly (6.2) and (6.4) assert (6.6).

Theorem 6.2. Let $\alpha \in (0, 1],$ then the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ satisfies the following contiguous relation

Proof. Consider

A shift of index gives

Since

then Eq (6.8) yields

Hence, we can write

From (6.2), we obtain

which implies together with (6.9) the required relation.

Theorem 6.3. For $\alpha \in (0, 1],$ then the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ satisfies the following contiguous relation

Proof. By operating $\theta ^{\alpha }{}$ $_{2}F_{1}\left(\mu -1, \nu; c;x^{\alpha }\right),$ we obtain

A shift of index yields

But $\frac{\left(\nu +n\right) }{\left(c+n\right) } = 1-\frac{c-\nu }{c+n},$ thus (6.12) becomes

which yield

Now, replacing $\mu$ by $\left(\mu -1\right)$ in (6.2) implies that

From (6.13) and (6.14), the relation (6.10) is verified. Similarly, since $\mu$ and $\nu$ can be interchanged without affecting the hypergeometric series, (6.11) yields.

Observe that from the contiguous relations we just derived in Theorems 6.1, 6.2, and 6.3, we can obtain further relations by performing some suitable eliminations as follows.

From (6.7) and (6.10), we get

A combination of (6.7) and (6.11) gives

Inserting (6.5) in (6.15) implies

Moreover, from (6.15) and (6.16), we get

Use (6.6) and (6.16) to obtain

By interchanging $\mu$ and $\nu$ in (6.15), we have

We append this section by driving the CFGHE. The conformable fractional operator (6.1) can be employed to derive a conformable fractional differential equation characterized by (3.3).

Relation (3.3) with the operator $\theta ^{\alpha }$ defined by (6.1) gives

A shift of index yields

This shows $y = \; _{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ is a solution of the following CFDE

Owing to $\theta ^{\alpha }y = \frac{1}{\alpha }x^{\alpha }D^{\alpha }y$ and $\theta ^{\alpha }\theta ^{\alpha }y = \frac{1}{\alpha ^{2}}x^{2\alpha }D^{\alpha }D^{\alpha }y+\frac{1}{\alpha }x^{\alpha }D^{\alpha },$ then Eq (6.21) can be written in the form

which coincide with (3.1).

7.   Conformable fractional integral of the CFGHF

Taking into account the $\alpha$-integral given in Definition 2.5, we provide some forms of fractional integral related to the $\alpha$-Gauss hypergeometric function. Thus according to Definition 2.5, it follows that

In this regard, we state the following important result given in ^[34].

Lemma 7.1. Suppose that $f:\left[0, \infty \right) \rightarrow \mathbb{R}$ is $\alpha$-differentiable for $\alpha \in (0, 1],$ then for all $x > 0$ one can write:

With the aid of (7.1) and (7.2), the following result can be deduced.

Theorem 7.1. For $\alpha \in (0, 1]$, then the conformable fractional integral $I_{\alpha }$ of CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ can be written as

Proof. Relation (5.9), gives

Acting by the conformable fractional integral on both sides we obtain

Using (7.2), we have

Therefore, it follows that

as required.

Theorem 7.2. For $\alpha \in (0, 1]$, then the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ has a conformable fractional integral representation in the form

where $d_{\alpha }t = t^{\alpha -1}dt$

Proof. In view of theorem 7.1, we obtain

Hence,

as required.

Now, following ^[44], we state the following result.

Theorem 7.3. For $\alpha \in (0, 1]$ and $c > \nu > 0,$ the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ has an integral representation

Proof. From the definition of CFGHF (3.3), we have

Using the integral form of beta function, we get

By using the identity $\sum\limits_{n = 0}^{\infty }\frac{\left(\mu \right) _{n} }{n!}\; t^{n} = \left(1-t\right) ^{-\left(\mu \right) }, \; \left\vert t\right\vert < 1$, it follows that

as required.

8.   Recursion formulas for formulas for the CFGHF

Employing the assertion in theorem 7.3, and owing to the results given in ^[46], we state the following recursion formulas.

Theorem 8.1. Let $\alpha \in (0, 1].$ The following recursion formulas hold for the CFGHF

where $\left\vert x^{\alpha }\right\vert < 1, \; n\in \mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}.$

Proof. By means of (7.4), we have

In virtue of conformable derivative, we may write

Again using (7.4), we have

Thus,

or

Applying this last identity (8.3), we get

Again apply (8.3) recursively n-times, we obtain

Using (5.9), we have

Furthermore, the assertion of theorem 7.3 gives

Relying on the integral representation (7.4), we have

Therefore,

or

Applying relation (8.5) recursively, we obtain

Repeating the recurrence relation (8.5) $n$-times and dappling the derivative formula (5.9), we have

The relation (8.2) follows directly from (8.6) by replacing $\mu$ by $\left(\mu +n\right)$ where $n\in \mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}.$

Theorem 8.2. For $\alpha \in (0, 1].$ The following recursion formulas hold true for the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$

and

where $\left\vert x^{\alpha }\right\vert < 1, \; n\in \mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}$.

Proof. Employing (8.1) of theorem 8.1 with $n = 1,$ we obtain

with $n = 2,$ we have

Making use of (8.9) and (8.10) implies

Using (8.9) and (8.11) with $n = 3,$ it follows that

Relation (8.12) can be written in the form

In general, we may write that

In order to prove (8.8), we note from (8.2) of theorem 8.1 $\left(\text{with }n = 1\right)$ that

Similarly, $\left(\text{with } n = 2\right)$ yields

Inserting (8.13) in (8.14), we get

Using the Pochhammer symbol, we may write (8.15) as

Thus, in general, we may write

just as required in (8.8).

Theorem 8.3. For $\alpha \in (0, 1],$ the following recursion formulas hold true for the CFGHF, $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$

Proof. In view of (7.4), we have

Using the binomial theorem, we obtain

Using the definition of the Pochhammer symbol, we may write (8.17) as

Applying (7.4), we obtain

just as required in theorem 8.3.

9.   Fractional Laplace transform of the CFGHF

In ^[34], Abdeljawad defined the fractional Laplace transform in the conformable sense as follows:

Definition 9.1. ^[34] Let $\alpha \in (0, 1]$ and $f:[0, \infty)\rightarrow \mathbb{R}$ be real valued function. Then the fractional Laplace transform of order $\alpha$ is defined by

Remark 9.1. If $\alpha = 1,$ then $(9.1)$ is the classical definition of the Laplace transform of integer order.

Also, the author in ^[34] gave the following interesting results.

Lemma 9.1. ^[34] Let $\alpha \in (0, 1]$ and $f:[0, \infty)\rightarrow \mathbb{R}$ be real valued function such that

$L_{\alpha }\left[f\left(t\right) \right] = F_{\alpha }\left(s\right)$ exist. Then $F_{\alpha }\left(s\right) = L\left[f\left(\alpha t\right) ^{\frac{1}{\alpha }}\right],$ where $L\left[f\left(t\right) \right] = \int\limits_{0}^{\infty }e^{-st}f\left(t\right) dt.$

Lemma 9.2. ^[34] The following the conformable fractional Laplace transform of certain functions:

(1) $L_{\alpha }\left[1\right] = \frac{1}{s}, \; \ s > 0,$

(2) $L_{\alpha }\left[t^{p}\right] = \alpha ^{\frac{p}{ \alpha }}\frac{\Gamma \left(1+\frac{p}{\alpha }\right) }{s^{1+\frac{p}{ \alpha }}}, \; \ s > 0,$

(3) $L_{\alpha }\left[e^{k\frac{t^{\alpha }}{\alpha }}\right] = \frac{ 1}{s-k}.$

Owing to the definition of CFGHF and applying the conformable fractional Laplace transform operator of an arbitrary order $\gamma \in (0, 1]$, we have

Using (2) of lemma 9.2, we obtain

Remark 9.2. If $\gamma = \alpha$ in $(9.3)$ we have

Theorem 9.1. Let $\alpha \in (0, 1]$ and $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ be a conformable fractional hypergeometric function, then

Proof. Using (3.3) and (9.1), one can see

But

Using (3) of lemma 9.2, we have

Since $\left(-n\right) _{k} = 0$ if $k > n,$ then we can write

Using $\frac{\left(s\right) _{k}}{s\left(s+1\right) _{k}} = \frac{1}{s+k},$ (9.7) becomes

Substituting (9.8) into (9.6), we have

as required.

Theorem 9.2. Let $\alpha \in (0, 1]$ and $_{2}F_{1}\left(\mu, \nu; c;x^{\alpha }\right)$ be a conformable fractional hypergeometric function, then

Proof. First, we see that

Using (2) of lemma 9.2, it follows that

But $\left(n+2\right) _{2k} = \left(\frac{n+2}{2}\right) _{k}.\left(\frac{n+3}{2}\right) _{k}$ and $\left(2\right) _{2k} = \left(1\right) _{k}.\left(\frac{3}{2}\right) _{k} = \left(\frac{3}{2}\right) _{k}k!$. Therefore,

10.   Applications

The general solution of a wide class of conformable fractional differential equations of mathematical physics can be written in terms of the CFGHF after using a suitable change of independent variable. This technique will be illustrated through the following interesting discussion.

Abul-Ez et al. ^[33] gave the hypergeometric representation of the conformable fractional Legendre polynomials $P_{\alpha n}\left(x\right)$, as

This formula can be easily obtained through the CFGHE as follows.

Note that, the conformable fractional Legendre polynomials $P_{\alpha n}\left(x\right)$ satisfy the conformable fractional differential equation

With the help of $t^{\alpha } = \frac{1-x^{\alpha }}{2},$ we get

Using (10.1), we obtain

Comparing the last Eq (10.2) with the CFGHE (3.1), we obtain the parameters $\mu,$ $\nu$ and $c$, such that

Hence, we may write the conformable fractional Legendre polynomials as

Example 10.1. Consider the following conformable fractional differential equation

Then the general solution of $(10.3)$ can be easily deduced as follows.

Let $t^{\alpha } = \left(1-e^{x^{\alpha }}\right)$, then we have

and

Now, in view of $(10.3)$, it can be easily seen that,

Simplifying $(10.4)$, we get

Comparing $(10.5)$ with the CFGHE $(3.1)$, we obtain $\mu +\nu = 0, \; \mu \nu = -1 {\ and }~ c = \frac{-1}{2}$. Thus, $\mu = 1$ and $\gamma = -1$. Therefore, the general solution of the CFDE $(10.3)$ can be given in the form

where $A$ and $B$ are arbitrary constants.

Example 10.2. Consider the class of conformable fractional differential equation which contains two arbitrary regular $\alpha$-singular points $x = \lambda _{1}$ and $x = \lambda _{2}$:

where $\lambda _{i}(i = 1, 2, \dots, 5) \in \mathbb{R}$ and $\lambda _{1}\neq \lambda _{2}$.

Taking $t^{\alpha } = (x^{\alpha }-\lambda _{1})/(\lambda _{2}-\lambda _{1})$. Then

and

Substituting in $(10.6)$ the obtained equation becomes

Now, we can write $(10.7)$ as the CFGHE such that

Hence, the general solution of $(10.6)$ can be obtained about the regular singular points $t = 0$ and $t = 1$, which means that we can find the general solutions about $x = \lambda _{1}$ and $x = \lambda _{2}$.

As a special case, putting $\lambda _{1} = 1$, $\lambda _{2} = -1$, $\lambda _{3} = 0$, $\lambda _{4} = \alpha$, and $\lambda _{5} = -\alpha ^{2}n^{2}$ in Eq. $(10.6)$, we obtain the conformable fractional Chebyshev differential equation

From $(10.7)$, it follows that $c = 1/2$, $\mu +\nu = 0$, and $\mu \nu = -n^{2}$. Hence, $\mu = -n$, $\nu = n$, and $c = 1/2$. The general solution of $(10.9)$ about $x = 1$ is

and the general solution of $(10.9)$ about $x = -1$ is

The strategy used in the preceding examples can be easily applied to solve some famous differential equations such as, Fibonacci, and Lucas differential equations in the framework of fractional calculus. Handled by Fibonacci, and Lucas differential equations have advantages due to their own importance in applications. The Fibonacci polynomial is a polynomial sequence, which can be considered as a generalization circular for the Fibonacci numbers. It is used in many applications, e.g., biology, statistics, physics, and computer science ^[47]. The Fibonacci and Lucas sequences of both polynomials and numbers are of great importance in a variety of topics, such as number theory, combinatorics, and numerical analysis. For these studies, we refer to ^{[47,48,49,50]}. Table 1 provides briefly the general solutions of such famous differential equations.

11.   Conclusions

The Gaussian hypergeometric function $_{2}F_{1}$ has been studied extensively from its mathematical point of view ^[51]. This occurs, naturally, due to its many applications on a large variety of physical and mathematical problems. For example, in quantum mechanics, the investigation of the Schrödinger equation for some systems involving Pöschl-Teller, Wood-Saxon, and Hulthén potentials leads to solutions expressed in terms of the hypergeometric functions ^[52]. Another significant case is related to the angular momentum theory since the eigenfunctions of the angular momentum operators are written in terms of $_{2}F_{1}$ functions ^[53]. One essential tool related to such problems is then provided by the derivatives of the $_{2}F_{1}$ function with respect to the parameters $\mu$, $\nu$, and $c$ since they allow one, for example, to write a Taylor expansion around given values $\mu _{0}, \nu _{0}$, or $c_{0}$. As a result, the importance of the Gaussian hypergeometric differential equation motivates one to provide a detailed study on the CFGHF. The solutions of the CFGHE are given to improve and generalize those given in ^[31]. Besides, many interesting properties and useful formulas of CFGHF are presented. Finally, supported examples show that a class of conformable fractional differential equations of mathematical physics can be solved through the CFGHF.

Interestingly, the obtained results of the current work have treated various famous aspects such as generating functions, differential forms, contiguous relations, and recursion formulas. Moreover, they have been generalized and developed in the context of the fractional setting. These aspects play essential roles in themselves and their diverse applications. In fact, most of the special functions of mathematical physics and engineering, for instance, the Jacobi and Laguerre polynomials, can be expressed in terms of the Gauss hypergeometric function and other related hypergeometric functions. Therefore, the numerous generating functions involving extensions and generalizations of the Gauss hypergeometric function can play essential roles in the theory of special functions of applied mathematics and mathematical physics, see ^[54].

The derivatives of any order of the GHF $_{2}F_{1}(\mu, \nu; c; x)$ with respect to the parameters $\mu, \nu$, and $c$, which can be expressed in terms of generalizations of multivariable Kampe de Fériet functions, have many applications (see the work of ^[55]). We may recall that the contiguous function relation applications range from the evaluation of hypergeometric series to the derivation of the summation and transformation formulas for such series; these can be used to evaluate the contiguous functions to a hypergeometric function, see ^[43]. Furthermore, using some contiguous function relations for the classical Gaussian hypergeometric series $_{2}F_{1}$, several new recursion formulas for the Appell functions $F_{2}$ with essential applications have been the subject of some research work, see for example ^[46] and reference therein. In conclusion, it is rather interesting to consider a broad generalization of the Gaussian hypergeometric function in future work either in the framework of fractional calculus or in a higher-dimensional setting. Our concluded results can be used for a wide variety of cases.

Acknowledgements

This work was funded by the Academy of Scientific Research and Technology, Egypt, under Science UP grant No. (6479). The authors, therefore, acknowledge with thanks the Academy of Scientific Research and Technology for financial support.

Conflict of interest

The authors declare no conflict of interest.