Properties of the Extended Whittaker Function

Abstract： In this article， we define an extended form of the Whittaker function by using extended confluent hypergeometric function of thefirst kind and study several of its properties. We also define the extended confluent hypergeometric function of the second kind and show that this function occurs naturally in statistical distribution theory.

Key words： Beta function； Extended beta function； Extended confluent hypergeometric function； Extended gamma function； Extended Gauss hypergeometric function； Gamma distribution； Gauss hypergeometric function

Nagar， D. K.， Mor′an-V′asquez， R. A.， & Gupta， A. K. （2013）. Properties of the Extended whittaker function. Progress in Applied Mathematics， 6（2）， 7080. Available from http：///index.php/pam/article/view/j.pam.1925252820130602.2807

DOI： 10.3968/j.pam.1925252820130602.2807

1. INTRODUCTION

The classical beta function， denoted by B（a，b）， is defined （see Luke [8]） by the Euler’s integral

In 2004， Chaudhry et al. [3] gave definitions of the extended Gauss hypergeometric function and the extended confluent hypergeometric function of thefirst kind， denoted by Fσ（a，b；c；z） andΦσ（b；c；z）， respectively. These definitions were developed by considering the extended beta function （7） instead of beta function（1） that appears in the general term of the series （4） and （5）. They suggested

In this article， we define the extended form of the Whittaker function and derive several results pertaining to it. We also define the extended confluent hypergeometric function of the second kind.

In Section 2， several known properties of the extended beta， extended Gauss hypergeometric and extended confluent hypergeometric functions have been given. The extended Whittaker function and its properties are given in Section 3. Finally， in Section 4， the extended confluent hypergeometric function of the second kind is defined and an application of this function to statistical distributions is also given.

2. SOME KNOWN DEFINITIONS AND RESULTS

We shall begin by briefly reviewing some of the definitions and basic properties of special function and statistical distributions that will be useful in our later work.

An integral representation of the modified Bessel function of the second kind（Gradshteyn and Ryzhik [4， Eq. 3.471.9]） is given by

If we considerσ= 0 in （20）， then the extended Whittaker function reduces to the classical Whittaker function， i.e.， M0，κ，μ（z） = Mκ，μ（z）.

An integral representation for the extended Whittaker function Mσ，κ，μ（z） can be obtained by replacing extended confluent hypergeometric function in （20） by its integral representation （11）. Thus， we get

In the integral representation of the extended Whittaker function given in （21）， substituting t = （1 + u）？1u， with the Jacobian J（tu） = （1 + u）？2， alternative integral representation is obtained as

If we put z = 0 in （21） and compare the resulting expression with （7）， we obtain an interesting relationship between the extended Whittaker function and extended beta function

Using the inequality exp（x） > 1+xn/n！， x > 0， n > 0， in （21）， anthor interesting inequality is obtained as

In this section， we give an extended form of the Tricomi’s function or confluent hypergeometric function of the second kind and show that this function occurs naturally in statistical distribution theory.

The extended confluent hypergeometric function of the second kind， denoted byΨσ（b；c；z）， may be defined as

Finally， we give the following theorem which gives the density of the ratio of two independent random variables in terms of extended confluent hypergeometric function of the second kind.

REFERENCES

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