Abstract
The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and .
1. Introduction
Special functions, often denoted by series, emerge as a result of solving various problems in classical physics. Typically, these problems demand the flow of electromagnetic, acoustic, or thermal energy. It has developed into a vital resource for the particular and specialized roles played by scientists and engineers these days. These characteristics are critical in a variety of domains, including physical science, mathematics, and engineering. The Gaussian hypergeometric function is among most important special functions. The hypergeometric function and its numerous generalizations have been investigated by many researchers who have provided a notably large number of their formula properties (see, e.g., [1–7]). Over the last decades, many researchers have established the generalizations of the classical gamma and beta functions and provided a number of intriguing and useful properties for the extended functions (see, e.g., [8–16]).
We begin with recalling some definitions of extended beta functions.
Chaudhary and Zubair [2] extended the classical gamma function as follows:where .
Subsequently, in 1997, Chaudhary et al. [17] developed the beta function extensionwhere .
Further, Chaudhary et al. [3] used the extended beta function (2) to present the following extension of the hypergeometric function:where andwhere .
The Pochhammer symbol is defined (for ), in terms of the familiar gamma function , by (see, e.g., [17], p. 2 and p. 5)
Here, let , and denote the sets of complex numbers, nonpositive integers, and positive integers, respectively.
Ozergin et al. [11] described the generalization of the beta function such as
Chand et al. [18] recently described the latest extension of the beta function as follows:where and { and not greater than 2.033 5}.
In this paper, we aim to introduce and investigate extended beta function, extended hypergeometric function, and extended confluent hypergeometric function. To do this, we have divided the paper into three portions. The first part discusses the introductory segment that has been documented by many researchers. In the second part, we gain Mellin transformations, summation relationships, beta distributions, integral representations of hypergeometric and confluent hypergeometric functions, differentiation formulas, and transformation formulas incorporating this new extended beta function. In the last part, we develop a novel property-rich extension of the fractional derivative Riemann–Liouville operator.
2. New Extended Beta Function and Its Properties
In this segment, we propose new generalization of classical beta function as follows:where { and less than 2.0335}.
If and , then equation (8) reduces to classical beta function.
2.1. Main Result
2.1.1. Melline Transform Representation of the New Generalisation of Beta Function
Theorem 1. If and and less than 2.0335, then
Proof. Taking the left hand side and using equation (8), we haveNow, interchanging the order of integration of equation (10) and forming uniform convergence, we haveSubstituting in the above expression, we getUsing result (page no. 821, [5]), we obtain
Theorem 2. For the new generalized beta function, we have the subsequent integral representation of new classical beta function. If , and and less than 2.0335, then
Proof: . Equations (14) and (15) can be easily obtained by substituting and in equation (8). Equation (16) can be obtained by interchanging and ; equation (17) can be obtained by adding equations (15) and (16) with some simplifications. Equation (18) can be achieved by putting in equation (8), and replacing , in equation (18), we get equation (19) and finally for the result given in equation (20), it can be obtained by putting and using in equation (19).
Throughout the paper, we consider the condition , and and less than 2.033 5.
2.1.2. Summation Relation
Theorem 3. The new generalized beta function: if , then we have the subsequent functional relation:
Proof: . Taking the left hand side of equation (21), we getwhich is a required result.
Remark 1. If we choose and , in the above equation (21), then it will convert into the classical beta function.
Theorem 4. If , then we have the subsequent summation relation:
Proof. We start with the left hand side of equation (23) with the help of equation (8),Using the binomial expansion in the above expression, and switching the order of summation and integration, we attain
Theorem 5. If , we have the subsequent summation relation:
Proof: . Considering the binomial series expansion,using the above result in equation (8) and interchanging the summation and integration, then we have
2.1.3. The Beta Distribution of Extended Beta Function
The specification of the new generalized beta function is the beta distribution in statistics. We are now describing new generalized beta distributions
The generalized beta distribution is with parameters and such that and and less than 2.0335 will be applied to a random variable with a probability density function (pdf) assumed by .
If is any real number, thenand for , mean of the distribution isand variance of the distribution is
The moment generating function of the distribution isand cumulative frequency distribution of can be expressed aswherewhich is the new extended incomplete beta function, for , equation (35) convergence, and , where is the incomplete beta function (page no. 960, [5]) demarcated by
It is also given by the serieswhere is the Pochhammer symbol. It is to be noted that the problem of stating in the form of other special functions leftovers open. Apparently, this distribution should be convenient in extending the statistical results for variables that are strictly positive to deal with those variables that can take large negative values arbitrarily.
2.2. Graphical Representation between and
Figure 1 describes the graphical form of equation (33) by using different values of within the range (−2, 2) and some fixed values of . We can see that the graph is decreasing in nature. The different values between and are given in Table 1.
2.2.1. Generalised Gauss Hypergeometric and Confluent Hypergeometric Function
Here, we define the generalized hypergeometric function and confluent hypergeometric function with the help of new generalization of classical beta function given in equation (8).where , then
2.2.2. Integral Representation of Gauss and Confluent Hypergeometric Function by Using New Generalization of Beta Function
Theorem 6. If , , then for the new generalized Gauss hypergeometric function , we have the subsequent integral representation demarcated as
Proof: . Considering the left hand side of equation (41) by means of equations (38) and (8), we yieldChanging the order of integration and summation with some simplifications, we attain the right hand side of equation (41).
Further, putting in (41), we getAgain, if we choose in equation (41), we have
Remark 2. By choosing in equation (41), we have the subsequent relation between new generalization of Gauss hypergeometric function
Theorem 7. If , then new generalization of confluent hypergeometric function is defined as follows:
Proof: . Considering the left hand side of (48), by using equation (39) and equation (8), we yieldChanging the order of integration and summation, we obtain the required result (48). By replacing and separately in equation (48) and simplifying it, we obtain equations (49) and (50).
2.2.3. Differentiation Formulas on New Extension of Beta Function
To represent the new generalization of Gauss hypergeometric function and confluent hypergeometric function, we use the following conditions:
Theorem 8. If , then we have the following differential formula:
Proof: . Taking the derivative of with respect to , we haveReplacing in the above equation (54), then it will reduce toRecursive application of this technique provides us the general form of equation (53).
Theorem 9. For confluent hypergeometric function, we have the following differential formula:
Proof: . Proof is same as in the above theorem.
2.2.4. Transformation Formula on New Extension of Beta Function
Theorem 10. For the new generalization of Gauss hypergeometric function with the condition , we have the subsequent transformation formula:
Proof: . Using the expression,and replacing in equation (41), we obtainHence,Replacing in equation (57), we obtain the result given in equation (58), and again replacing by in equation (57), we get equation (59).
3. New Extension of Riemann–Liouville (R-L) Fractional Derivative Operator
In the past years, different kinds of extensions of R-L fractional operators were performed by many authors (see [19–22]). Here, we announce novel extension of R-L fractional derivative operator using new type of extended beta function and this function is also reducible into classical Riemann–Liouville fractional derivative operator with certain conditions.
The classical Riemann fractional derivative of of order defined demarcated bywhere the path of integration is a line from 0 to in the complex when .
Let be the smallest integer greater than and . Then,
Here, we defined new definition of extended Riemann–Liouville (R-L) fractional derivative of of order which is given bywhere { and less than 2.0335} and for and less than 2.0335, , then
If we put in the above equations (65) and (66), then it will become the classical Riemann–Liouville (R-L) fractional derivative equations (63) and (64).
Theorem 11. Let , for some and . Then, we have
Proof: . We have from equation (65),and putting in the above equation, then we have
Theorem 12. Let and less than 2.0335, , then
Proof: . Taking the left hand side and using equation (65), we haveand putting , then it becomes
3.1. Extended Riemann–Liouville (R-L) Derivative to a Function Analytic at the Origin
Theorem 13. Let function be analytic at the origin with its power series given by for some , then
Proof: . By the definition of extended Riemann–Liouville (R-L) derivative operator and using in it, we obtainChanging the order of summation and integration, we have
Theorem 14. Let function be analytic at the origin with its power series expansion for some . Then, we have
Proof. By using equation (73), in L.H.S. of the above equation, we haveand putting , thenThe series given in equation (78), , is uniformly convergent in disc for and the integral is convergent provided that and and less than 2.033 5. Now, interchanging the order of summation and integration, we get the result given in equation (76).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this article.