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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）， 70–80. Available from http：//www.cscanada.net/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（t→u） = （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
[1] Chaudhry， M. A.， & Zubair， S. M. （1994）. Generalized incomplete gamma functions with applications. J. Comput. Appl. Math.， 55， 303–324.
[2] Chaudhry， M. A.， Qadir， A.， Rafique， M.， & Zubair， S. M. （1997）. Extension of Euler’s beta function. J. Comput. Appl. Math.， 78（1）， 19–32.
[3] Chaudhry， M. A.， Qadir， A.， Srivastava， H. M.， & Paris， R. B. （2004）. Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput.，159（2）， 589–602.
[4] Gradshteyn， I. S.， & Ryzhik， I. M. （2000）. Table of integrals， series， and products （6th ed.）. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. San Diego， CA： Academic Press， Inc.
[5] Gupta， A. K.， & Nagar， D. K. （2000）. Matrix variate distributions. Boca Raton： Chapman & Hall/CRC.
[6] Iranmanesh， A.， Arashi， M.， Nagar， D. K.， & Tabatabaey， S. M. M. （2013）. On inverted matrix variate gamma distribution. Comm. Stat. Theor. Meth.， 42（1）， 28–41.
[7] Johnson， N. L.， Kotz， S.， & Balakrishnan， N. （1994）. Continuous univariate distributions-2 （2nd ed.）. New York： John Wiley & Sons.
[8] Luke， Y. L. （1969）. The special functions and their approximations （Vol. 1）. New York： Academic Press.
[9] Miller， A. R. （1998）. Remarks on a generalized beta function. J. Comput. Appl. Math.， 100（1）， 23–32.
[10] Nagar， D. K.， Rold′an-Correa， A.， & Gupta， A. K. （2013）. Extended matrix variate gamma and beta functions. J. Multivariate Anal.， 122， 53–69.
[11] Nagar， D. K.， & Rold′an-Correa， A. （2013）. Extended matrix variate beta distributions. Progr. Appl. Math.， 6（1）， 40–53.
[12] Whittaker， E. T. （1903）. An expression of certain known functions as generalized hypergeometric functions. Bull. Amer. Math. Soc.， 10（3）， 125–134.
[13] Whittaker， E. T.， & Watson， G. N. （1996）. A course of modern analysis. Reprint of the 4th ed. （1927）. Cambridge Mathematical Library. Cambridge： Cambridge University Press.
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）， 70–80. Available from http：//www.cscanada.net/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（t→u） = （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
[1] Chaudhry， M. A.， & Zubair， S. M. （1994）. Generalized incomplete gamma functions with applications. J. Comput. Appl. Math.， 55， 303–324.
[2] Chaudhry， M. A.， Qadir， A.， Rafique， M.， & Zubair， S. M. （1997）. Extension of Euler’s beta function. J. Comput. Appl. Math.， 78（1）， 19–32.
[3] Chaudhry， M. A.， Qadir， A.， Srivastava， H. M.， & Paris， R. B. （2004）. Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput.，159（2）， 589–602.
[4] Gradshteyn， I. S.， & Ryzhik， I. M. （2000）. Table of integrals， series， and products （6th ed.）. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. San Diego， CA： Academic Press， Inc.
[5] Gupta， A. K.， & Nagar， D. K. （2000）. Matrix variate distributions. Boca Raton： Chapman & Hall/CRC.
[6] Iranmanesh， A.， Arashi， M.， Nagar， D. K.， & Tabatabaey， S. M. M. （2013）. On inverted matrix variate gamma distribution. Comm. Stat. Theor. Meth.， 42（1）， 28–41.
[7] Johnson， N. L.， Kotz， S.， & Balakrishnan， N. （1994）. Continuous univariate distributions-2 （2nd ed.）. New York： John Wiley & Sons.
[8] Luke， Y. L. （1969）. The special functions and their approximations （Vol. 1）. New York： Academic Press.
[9] Miller， A. R. （1998）. Remarks on a generalized beta function. J. Comput. Appl. Math.， 100（1）， 23–32.
[10] Nagar， D. K.， Rold′an-Correa， A.， & Gupta， A. K. （2013）. Extended matrix variate gamma and beta functions. J. Multivariate Anal.， 122， 53–69.
[11] Nagar， D. K.， & Rold′an-Correa， A. （2013）. Extended matrix variate beta distributions. Progr. Appl. Math.， 6（1）， 40–53.
[12] Whittaker， E. T. （1903）. An expression of certain known functions as generalized hypergeometric functions. Bull. Amer. Math. Soc.， 10（3）， 125–134.
[13] Whittaker， E. T.， & Watson， G. N. （1996）. A course of modern analysis. Reprint of the 4th ed. （1927）. Cambridge Mathematical Library. Cambridge： Cambridge University Press.