Solutions of 𝑘-Hypergeometric Differential Equations

Abstract

We solve the second-order linear differential equation called the -hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

1. Introduction

In 1769, Euler [1] formed the hypergeometric differential equation of the form which has three regular singular points: , , and . The hypergeometric differential equation is a prototype: 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 variables

The solutions of hypergeometric differential equation include many of the most interesting special functions of mathematical physics. Solutions to the hypergeometric differential equation are built out of the hypergeometric series. The solution of Euler’s hypergeometric differential equation is called hypergeometric function or Gaussian function introduced by Gauss [2].

The equation has two linearly independent solutions at each of the three regular singular points , , and . Kummer [3] derived a set of distinct solutions of hypergeometric differential equation. These include the hypergeometric function of Gauss and all of them could be expressed in terms of Gauss’s function. For more details on Kummer’s solutions, see [4].

Recently, Díaz et al. [5–7] have introduced Pochhammer -symbol. They have introduced -gamma and -beta functions and proved a number of their properties. They have also studied -zeta functions and -hypergeometric functions based on Pochhammer -symbols for factorial functions. In 2010, Kokologiannaki [8] and Krasniqi [9] followed the work of Diaz et al. and obtained some important results for the -beta, -gamma, -hypergeometric, and -zeta functions. Also, Krasniqi [10] gave a limit for the -gamma and -beta functions. In 2012, Mubeen and Habibullah [11, 12] introduced a variant of fractional integrals to be called -fractional integral which was based on -gamma function and gave its application. They also introduced an integral representation of some generalized confluent -hypergeometric functions and -hypergeometric functions by using the properties of Pochhammer -symbols, -gamma, and -beta functions. Furthermore, in 2013, Mubeen [13] defined a second-order linear -hypergeometric differential equation having one solution in the form of -hypergeometric function .

2. Basic Concepts

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications. The solutions of hypergeometric differential equation include many of the most interesting special functions of mathematical physics. Solutions to the hypergeometric differential equation are built out of the hypergeometric series.

Definition 1. The Pochhammer -symbol is defined as and, for , , where .

Definition 2. The -hypergeometric functions with three parameters , two parameters in the numerator and one parameter in the denominator, are defined by for all , where and .

Definition 3. In mathematics, the method of Frobenius [14], named after Ferdinand George Frobenius, is a method to find an infinite series solution for a second-order ordinary differential equation of the form about the regular singular point . After dividing this equation by , we obtain a differential equation of the form which is not solvable with regular power series methods if either or is not analytic at . The Frobenius method enables us to obtain a power series solution to such a differential equation, provided that and are themselves analytic at or, being analytic elsewhere, both their limits at exist.

3. The Solutions of the -Hypergeometric Differential Equation

In this section, we solve the following ordinary linear second-order -hypergeometric differential equation defined by Mubeen [13] using Frobenius method. We usually use this method for complicated ordinary differential equations. This method is used to find an infinite series solution for a second-order ordinary differential equation about regular singular points of that equation. We prove that this equation has three regular singular points, namely, at and and around , and then we will be able to consider a solution in the form of a series.

As this is a second-order differential equation, we must have two linearly independent solutions. The problem however will be that our assumed solutions may or may not be independent or worse may not be defined (depending on the values of the parameters of the equation). This is why we study the different cases for parameters and modify our assumed solutions accordingly.

3.1. Solution at

Let Then and .

Hence, and are singular points.

Let us start with .

To see if it is regular, we study the following limits: Hence, both limits exist and is a regular singular point.

Therefore, we assume the solution of the form with .

Hence By substituting these into the -hypergeometric differential equation (7), we get In order to simplify this equation, we need all powers of to be the same, equal to , the smallest power. Hence, we switch the indices as follows: Thus isolating the first terms of the sums starting from , we get Now, from the linear independence of all powers of , that is, of the functions , , , and so forth, the coefficients of vanish for all . Hence, from the first term, we have which is the indicial equation.

Since , we have Hence, the solutions of the above indicial equation are given below: Also, from the rest of the terms, we have Hence, we get the following recurrence relation: for .

Let us now simplify this relation by giving in terms of instead of .

From the recurrence relation, we have the following: for .

Hence, our assumed solution takes the form

3.2. Analysis of the Solutions in terms of the Difference “” of the Two Roots

Now, we study the solutions corresponding to the different cases for the expression (this reduces to studying the nature of the parameter whether it is an integer or not).

Case 1 (“” not an integer). Let be not an integer. Then Since therefore, we have Hence Let Then

Case 2 (“” (i.e., )). Let c = k. Then Since we have using , we get Hence, To calculate this derivative, let Then Since therefore Differentiating both sides of the equation with respect to , we get Since therefore For , we get Hence, Let Then

Case 3 (“” an integer and “”). Here, we discuss the further two cases:
(i) “.” Let . Then from the recurrence relation we see that, when (the smaller root), . Thus, we must make the substitution where is the root for which our solution is infinite.
Therefore, we take and our assumed solution in equation (29) takes the new form Then As we can see, all terms before vanish because of the in the numerator.
Starting from this term, however, the in the numerator vanishes. To see this, note that Hence, our assumed solution takes the form Now To calculate this derivative, let Then following the method in the previous case , we get Since therefore At , we get Hence Let Then
(ii) “”. Let . Then, from the recurrence relation we see that, when (the smaller root), . Thus, we must make the substitution where is the root for which our solution is infinite.
Hence we take and our assumed solution takes the new form Then As we can see, all terms before vanish because of the “” in the numerator.
Starting from this term, however, the “” in the numerator vanishes. To see this, note that Hence, our solution takes the form Now To calculate this derivative, let Then following the method in the previous case , we get At , we get Hence Let Then

3.3. Solution at

Let us now study the singular point .

To see if it is regular, we study the following limits: As both limits exist, therefore is a regular singular point.

Now instead of assuming a solution in the form we will try to express the solutions of this case in terms of the solutions for the point . We proceed as follows.

We have the -hypergeometric differential equation of the form Let . Then Hence, -hypergeometric differential equation (7) takes the form Since , the solution of the -hypergeometric differential equation at is the same as the solution for this equation at . But the solution at is identical to the solution we obtained for the point if we replace by .

Hence, to get the solutions, we just make the substitution in the previous results.

Note also that, for , Hence in our case,

3.4. Analysis of the Solutions in terms of the Difference “” of the Two Roots

Let us now find out the solutions. In the following we replace each by .

Case 1 (“” not an integer). Let be not an integer. Then

Case 2 (“”). Let . Then

Case 3 (“” an integer and “”). Here, we discuss further two cases:
(i) “.” Let . Then
(ii) “.” Let . Then

3.5. Solution Around “”

Finally we study the singularity as . Since we cannot study this directly, therefore we let ; then the solution of the equation as is identical to the solution of the modified equation when .

We have the -hypergeometric differential equation Hence the -hypergeometric differential equation (7) takes the new form Let Here, we only study the solutions when . As we can see that this is a singular point because .

Let us now see that is a regular singular point, for this As both limits exist, therefore is a regular singular point.

Thus, we assume the solution of the form with .

Hence, we set the following: By substituting these into the modified -hypergeometric differential equation (88), we get In order to simplify this equation, we need all powers of to be the same, equal to , the smallest power. Hence, we switch the indices as follows: Thus, by isolating the first terms of the sums starting from , we get Now from the linear independence of all powers of , that is, of the functions , , , and so forth, the coefficients of vanish for all . Hence, from the first term, we have which is the indicial equation.

Since , we have Hence, we get two solutions of this indicial equation: Also, from the rest of the terms we have Hence, we get the recurrence relation of the following form: for .

Let us now simplify this relation by giving in terms of instead of .

From the recurrence relation, we have for .

Hence, our assumed solution in equation (91) takes the form

3.6. Analysis of the Solutions in terms of the Difference “” of the Two Roots

Now, we study the solutions corresponding to the different cases for the expression (this reduces to studying the nature of the parameter whether it is an integer or not).

Case 1 (“” not an integer). Let be not an integer. Then Since therefore, we have Hence Let Then

Case 2 (“”). Let . Then Also, we have which can be written as Since we have , therefore At , To calculate this derivative, let and then we get For , we get Hence Let Then

Case 3 (“” an integer and “”). Here, we have further two cases:
(i) “.” Let .
Then, from the recurrence relation we see that, when (the smaller root), . Thus, we must make the substitution where is the root for which our solution is infinite.
Hence we take and our assumed solution in equation (110) takes the new form Then As we can see, all terms before vanish because of the in the numerator.
Starting from this term, however, the in the numerator vanishes. To see this, note that Hence, our solution takes the form Now To calculate this derivative, let and then we get At , we get Replacing by , we get in terms of : Let Let Then
(ii) “.” Let .
Then, from the symmetry of the situation here, we see that

4. Conclusion

In this research work, we derived the -hypergeometric differential equation. Also, we obtained solutions of -hypergeometric differential equation around all its regular singular points by using Frobenius method. So, we conclude that, if , we obtain Euler’s hypergeometric differential equation. Similarly, by letting , we find solutions of hypergeometric differential equation around all its regular singular points.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.