Relation of Some Known Functions in terms of Generalized Meijer -Functions

Abstract

The aim of this paper is to prove some identities in the form of generalized Meijer -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer -function and solve an integral involving the product of modified Bessel functions.

1. Introduction

The elementary functions such as trigonometric functions, exponential functions, logarithmic functions, hypergeometric functions, Bessel functions, Mittag-Leffler functions, and even binomial expansion are all special functions, which are widely applicable in many fields, especially in the field of sciences. The hypergeometric functions were firstly defined in series form by [1], also many properties were presented. Due to the wide range of applications, these hypergeometric functions have gained attention of the researchers. Almost all the elementary functions can be expressed in the form of hypergeometric functions.

The exponential functions, sine and cosine functions, and binomial expansion in the form of hypergeometric functions are given in [2, 3] as, respectively,where and

The Bessel function is defined [4, 5] as

Zhu et al. [6] estimated some weighted Simpson-like type integral inequalities, used them in some estimation type results to obtain the first-order differentiable functions, and applied them in some known special functions, modified Bessel function, -digamma function, etc. Sarivastava et al. [7] worked on Mittag-Leffler type functions and estimated the Faber polynomial coefficient of biclose-to-convex functions connected with the Borel distribution of the Mittag-Leffler type. They considered the Fekete-Szegö type inequalities for biclose-to-convex function and also presented several results and related consequences. The researchers [8–10] also worked on inequalities involving special functions and proved various applications.

The concept of -symbol was introduced by Diaz [11, 12]. The -theory gave boost to the field of special functions. The researchers [13–15] started to work on this particular -symbol and proved many properties and identities.

Diaz defined Pochhammer -symbol, gamma -function, and beta -function, for , respectively, as

The integral form of gamma -function is given as

Mubeen and Rehman [16] gave the -form of some elementary functions.

The Meijer -functions which are considered to be the general functions are the particular special functions which have gained the attention of many researchers [17–24]. Many special functions can be obtained by considering the specific variation of parameters of Meijer -function. Most of the special functions are -functions or can be expressed in the form of product of -functions with elementary functions [25–29].

Meijer -function is defined [30] aswhere is well known gamma function and .

The definitions of Meijer -function in the form of hypergeometric function are given [31], respectively, asfor or , andfor or and .

The relation of Meijer -functions with elementary functions as exponential, logarithmic, cosine, and Bessel functions is given by Santosh [32] aswhere and are the Bessel and modified Bessel functions, respectively.

Roach [33] gave the relation of hypergeometric functions with Meijer -function asand proved some results related to Meijer -function. Dyda et al. [34] worked on fractional Laplace operators and Meijer -functions and discussed some transformation properties of Meijer -function given aswhere and . Pishkoo and Darus [35] gave series representations of three basic univalent -functions by using Mellin-Barnes type contour integral representation.

In our current findings, we first define the generalized form of Meijer -function in the integral and hypergeometric forms and obtain some known special functions such as the Bessel function, exponential function, sine function, cosine function, sine and cosine hyperbolic functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansions, and logarithmic function by using the definitions of generalized Meijer -functions for different choice of parameters. We define the generalized form of modified Bessel functions of the first and second kind and prove results for the product of modified Bessel functions in the form of generalized Meijer -function.

2. Generalized Meijer -Functions and Their Relation with Some Known Functions

In this section, we define the generalized form of (7) and (8) in the form of hypergeometric -function and (6) in integral form for . The relations of some known functions with generalized Meijer -function are also considered.

Definition 1. For or and , the generalized Meijer -function can be defined as

Definition 2. For or and , the generalized Meijer -function can be defined as

Definition 3. The integral form of generalized Meijer -function for is given aswhere is the well-known gamma -function and .
Now, we prove some known special functions by considering different choices of parameters of generalized Meijer -functions defined in (12) and (13), for . Different identities are proved as follows.

Proposition 1. Let , , , , then (12) gives the generalized Meijer -function in the form of Bessel function as

Proof. By choosing , , , in (12), we havewhich gives the desired result.

Proposition 2. Let , , , , , then (12) gives

Proof. By choosing , , , , in (12), we haveSince , so

Proposition 3. Let , , , then (12) gives the relation of Meijer -function with binomial series as

Proof. By choosing , , in (12), we have

Proposition 4. Let , , in (12), then the generalized Meijer -function can be expressed in series form as

Proof. By choosing , , in (12), we have

Corollary 1. By taking same choices as in the above case in (13), we can obtain

Proposition 5. Let , , , then (12) gives the relation of generalized Meijer -function with sine function as

Proof. By choosing , , in (12), we have

Proposition 6. Let , , , then (12) gives the relation of generalized Meijer -function with cosine function as

Proof. By choosing , , in (12), we have

Proposition 7. Let , , in (12), then we have

Proof. By choosing , , in (12), we have

Corollary 2. By taking the same choice of parameters as in the previous proposition and , then (13) gives

Proof. By choosing , , in (13), we obtain

Proposition 8. Let , , , then (12) gives the relation of generalized Meijer -function with the product of exponential and sine hyperbolic function as

Proof. By choosing , , in (12), we have

Corollary 3. For , , , , (13) gives

Proof. By choosing , , , in (13), we can obtain

Proposition 9. Let , , in (13), then

Proof. By choosing , , in (13), we obtain

Proposition 10. Let , , , , , , , then (13) gives the relation of generalized Meijer -function with logarithmic function as

Proof. By choosing , , , , , , in (13), we have

Proposition 11. Let , , , in (13), then we havewhere Si is the sine integral.

Proof. By choosing , , , in (13), we have

3. Product of Modified Bessel Functions in terms of Generalized Meijer -Function and Their Relations

In this section, we first define the modified Bessel function of the first and second kind in the form of generalized Meijer -function and then prove the relation of the product of modified Bessel functions with the generalized Meijer -function by taking the specific choice of parameter of the Bessel functions. At the end, we solve an integral involving the product of modified Bessel functions in the integrand.

For complex argument , the modified Bessel functions of the first and second kind are defined, respectively, aswhere is an integer.

Theorem 1. Let in (43) and (44), then the product of modified Bessel functions in terms of generalized Meijer -function is given as

Proof. We first consider the left hand side asSo, from (46) and (47), we haveBy working on (48), we obtainNow, by considering the right hand side and applying (12), we haveFrom (49) and (50), we obtain the required result.

Theorem 2. Let be the variable of integration, , and be the argument of the modified Bessel functions of the first and second kind, then we have

Proof. From Theorem 3.1, we can writeNow, multiplying by and taking integral, we haveAlso, by using (12) on the right hand side, we haveHence, from (53) and (54), we obtained the desired result.

4. Conclusions

In this research work, we gave some functions as sine, cosine, exponential, logarithmic functions, etc., in terms of the generalized form of Meijer -function. We proved the theorems, which gave the relation of the product of Bessel functions of the first and second kind with the generalized Meijer -function, and solved the integral involving modified Bessel functions as an integrand, respectively.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and they read and approved the final manuscript for publication.