By employing an addition theorem for the confluent hypergeometric function, Paris R.B.[3], has obtained a Kummer-type transformation for a 2F2 (x) hypergeometric function with general parameters in the form of a sum of 2F2 (-x) functions. Recently, Choi Junesang and Rathie Arjun K.[1], has obtained the same result without using the addition theorem. The aim of this paper is to derive the result of Paris R.B.[3], with change in the general parameters without using the addition theorem in the line of Choi Junesang and Rathie Arjun K.[1]. Corresponding author E.mail:-  pandey1172@gmail.com, pandey.awadhesh1972@gmail.co

Bhardwaj, Ramakant

Pandey, Awadhesh Kumar

Thakur, Nitesh Singh

Wadhwa, Kamal

English

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  58 A Method of Transformation for Generalized Hypergeometric Function 2F2   Awadhesh Kumar Pandey,Ramakant Bhardwaj∗,Kamal Wadhwa∗∗and Nitesh Singh Thakur∗∗∗ Department of Mathematics,Patel Institute of Technology, Bhopal, M.P. *Truba Institute of Technology, Bhopal, M.P. **Govt. Narmda P.G. College, Hoshangabad, M.P.              ***Patel College of Science & Technology, Bhopal, M.P.   Abstract By employing an addition theorem for the confluent hypergeometric function, Paris R.B.[3], has obtained a Kummer-type transformation for a 2F2 (x) hypergeometric function with general parameters in the form of a sum of 2F2 (-x) functions. Recently, Choi Junesang and Rathie Arjun K.[1], has obtained the same result without using the addition theorem. The aim of this paper is to derive the result of Paris R.B.[3], with change in the general parameters without using the addition theorem in the line of Choi Junesang and Rathie Arjun K.[1]. Corresponding author E.mail:-  pandey1172@gmail.com, pandey.awadhesh1972@gmail.com  1. Introduction and results required We start with a Kummer-type transformation for a 2F2 (x) hypergeometric function with general parameters in the form of a sum of 2F2 (-x) functions due to Paris R.B.[3, Eq.(3)]:  2F2( , d ; b, c ; ) =   ∑ !"#$%#$   &!(&)* −#& 2F2(, −  , d ; b, c + n ; )                                ……. (1.1) where ()n =  -./0#-. (n =0,1,2,3, ……. ) is the Pochhammer symbol.  Paris R.B.[3], also considered several interesting special cases of (1.1).This result (1.1) was established with the help of the integral representation for 2F2 [5, Eq.(4.8.3.11)]:  2F2( , d ; b, c ; )=  -1-. -1! .#  2 3.! 44*1 − 3#1!.! 4  1F1(d ; c ; 3)  dt                                   ……. (1.2)  and 2F2( , d ; b, c ; )=  -1-. -1! .#  2 31!.! 44*1 − 3#.! 4  1F1(d ; c ; - 3)  dt                              ……. (1.3)  provided R(b) > 0 and R() > 0, and the addition theorem for the confluent hypergeometric function in the form due to Slater L. J.[4, Eq.(2.3.5)]:  1F1(d ; c ; - 3)=   ∑ !"#$%#$   &!(&)* −#& 1F1(d ; c + n ; −3)                                               ……. (1.4) Paris R.B.[3],  remarked that the special case of (1.1) when c = d reduces to the well-known Kummer's first theorem due to [5]:  1F1( ; b ; ) =  1F1(, −  ; b ; −)                                                                                    ……. (1.5)  Choi Junesang and Rathie Arjun K.[1], has derived the following result: 2F2(6 ,  ; 7, b ; ) =   ∑ !"#8%#8   9!(9)* −#9 2F2(, −  , d ; b, c + r ; )                                ……. (1.6)  The aim of this paper is to derive the result of Paris R.B.[3], with change in the general parameters without using the addition theorem in the line of Choi Junesang and Rathie Arjun K.[1].  2. Main Result 2F2(, ,  ; , b ; +y) = /:  ∑.!1#;.#;   <!(<)* − − =#<                                                              2F2(, − , b ; b,  + ? ; −-y)                                                    ……. (2.1)   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications  59 Proof:- Start with the left-hand side of (2.1) and use (1.2), it becomes  2F2(,,  ; , b ; +y) =  -.-1 -.! 1#  2 31! 44*1 − 3#.!1! 4                                                        1F1(, ;  ; 3+yt)  dt                                                                           ……. (2.2)               which can be written as  2F2(,,  ; , b ; +y) =  -.-1 -.! 1# /:  2 31! 44*1 − 3#.!1! 4 !!:                                                                 1F1(, ;  ; 3+yt)  dt                                                                       ……. (2.3)                         Using equation (1.5) in the integrand of the integral in equation (2.3), we have   2F2(, ,  ; , b ; +y) =  -.-1 -.! 1# /:  2 31! 44*1 − 3#.!1! 4 !4!@#!:4!@#                                                    1F1( − , ;  ; −3-yt)  dt                                                              ……. (2.4)            Now expand !4!@#!:4!@# in equation (2.4) as the Maclaurin series, after a little simplification, we obtain  2F2(, ,  ; , b ; +y)  =  -.-1 -.! 1#   /:  ∑!!:#;<!   (<)*   2 31! 44*1 − 3#.!1/A! 4 !4!@#!:4!@#                                                                 1F1( − , ;  ; −3-yt)  dt                                                             ……. (2.5)            Substituting 1-t = z  in equation (2.5) and simplifying, we have  2F2(, ,  ; , b ; +y)  =  -.-1 -.! 1#   /: . ∑!!:#;<!   (<)*     2 1 − B#1! 44*B.!1/A! 4 !4!@#!:4!@#                                                               1F1( − , ;  ; −1 − B# – y(1-z))  dz                                          ……. (2.6)  Finally, applying (1.3) to the integral part in the last identity, we have  2F2(, ,  ; , b ; +y) = /:  ∑.!1#;.#;   <!(<)* − − =#<                                                              2F2(, − , b ; b,  + ? ; −-y)                                                        ……. (2.7)  This completes the proof of (2.1).  Acknwledgemennt Corresponding Auther (A.P.)is thankful to all reviewers and Authers whose references are taken to prepare this article.  References [1] Choi Junesang and Rathie Arjun K., “Another method for a kummer-type transformation for a 2F2 hypergeometric function”, Commun. Korean Mathematical Society, Vol. 22, No. 3, July (2007) pp. 369-371. [2] Erd´elyi  A., “Higher transcendental functions”, McGraw Hill, New York, Vol-1 (1953).  [3] Paris R. B., “A Kummer-type transformation for a 2F2 hypergeometric function”, J. Comput. Appl. Math. 173 (2005) pp.  379-382. [4] Slater L. J., “Confluent Hypergeometric Functions”, Cambridge University Press, Cambridge, 1960. [5] “Generalized Hypergeometric Functions”, Cambridge University Press, Cambridge,(1966).     This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  More information about the publisher can be found in the IISTE’s homepage:  http://www.iiste.org  CALL FOR PAPERS The IISTE is currently hosting more than 30 peer-reviewed academic journals and collaborating with academic institutions around the world.  There’s no deadline for submission.  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A Method of Transformation for Generalized Hypergeometric Function 2F2

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A Method of Transformation for Generalized Hypergeometric Function 2F2

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