SOME UNIFIED AND GENERALIZED KUMMER S FIRST SUMMATION THEOREMS WITH APPLICATIONS IN LAPLACE TRANSFORM TECHNIQUE

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1 Asia Pacific Journal of Mathematics, Vol. 3, No. 1 16, 1-3 ISSN SOME UNIFIED AND GENERAIZED KUMMER S FIRST SUMMATION THEOREMS WITH APPICATIONS IN APACE TRANSFORM TECHNIQUE M. I. QURESHI 1 AND M. S. BABOO, 1 Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia A Central University, New Delhi -115, India School of Basic Sciences and Research, Sharda University, Greater Noida, Uttar Pradesh, 136, India Corresponding author Abstract. Some significant hypergeometric summation theorems with suitable convergence conditions are obtained in the present study, which are analogous to Kummer s summation theorem F 1 1 recorded by Prudnikov et al. and derived by Choi, Kim et al., Rakha-Rathie and Rathie-Kim. By means of these summation theorems we also find the aplace transforms of Kummer s confluent hypergeometric function 1 F 1 in closed form. 1 Mathematics Subject Classification. 33B15, 33C5, 44A. Key words and phrases. beta and gamma functions; Gauss and Kummer hypergeometric functions; egendre duplication formul principle of analytic continuation; aplace transforms. 1. Introduction, Definitions and Preliminaries In the usual notation, let R and C denote the sets of real and complex numbers, respectively. Also let N N {, N {1,, 3,... N \{, Z {, 1,,... Z {, Z {1,, 3,... and Z Z N being the sets of integers. Here, in our present investigation, we propose to explore several summation and other related formulas for the Gauss and Kummer hypergeometric functions which are, respectively, in the cases p 1 q 1 and p q 1. c 16 Asia Pacific Journal of Mathematics 1

2 Here generalized hypergeometric function p F q with p numerator parameters α 1, α,..., α p and q denominator parameters β 1, β,..., β q, is defined by see, for example, 15, p. 41 et.seq.; see also 14, pp. 71 7: pf q α 1, α,..., α p ; β 1, β,..., β q ; z n p α j n j1 q β j n p, q N ; p q + 1 ; p q and z < ; p q + 1 and z < 1; p q + 1, z 1 and Rω > ; p q + 1, z 1, z 1 and Rω > 1, where ω : q β j j1 p j1 α j j1 z n n! α j C j 1,,..., p; β j C\Z j 1,,..., q. In terms of Gamma function Γz, the widely-used Pochhammer symbol λ ν λ, ν C is defined, in general, by 1 Γ λ + ν 1.1 λ ν : Γ λ λ λ λ + n 1 ν ; λ C \ { ν n N; λ C. It is understood conventionally that : 1 and assumed tacitly that the Γ quotient exists see, for details, e st t α1 Γα s α Rs >, < Rα < or Rs, < Rα < 1. Most of the elementary functions as well as special functions of mathematical physics and other areas of applied sciences are special or limited cases of the Gauss hypergeometric function F 1 α, β; γ; z. For a detailed history of this function, especially about the origin of term hypergeometric for it, by Kummer 8, see among other places 16, p

3 Euler s Beta-type integral representation for the Gauss hypergeometric function F 1 is given by 14, p. 65, Equation F 1 α, β; γ; z Γγ ΓβΓγ β t β1 1 t γβ1 1 zt α arg1 z π ɛ < ɛ < π; Rγ > Rβ > ; α C, or, equivalently 14, p. 65, Equation 1.51, 1.4 F 1 α, β; γ; z Γγ ΓαΓγ α t α1 1 t γα1 1 zt β arg1 z π ɛ < ɛ < π; Rγ > Rα > ; β C. The familiar Beta function Bα, β 14, p.8, Equation is defined by 1.5 Bα, β t α1 1 t β1 min {Rα, Rβ > ΓαΓβ Γα + β α, β C\Z and egendre s duplication formula 15, p. 3, Equation 1.15 is given by 1.6 πγz ΓzΓ z1 z + 1 z C\Z. In addition to the Gauss summation theorem 15, p. 3, Equation 1.7 for F 1 1, there are numerous closed-form summation theorems for F 1 z for different values of the argument z, see 1,, 3, 1, 13. Here, for the purpose of our present investigation, we choose to recall the following summation theorem, which is due to Kummer 8, p. 134, Entry 1. Kummer s first summation theorem 6, p. 85, Equation 1.3 : 1.7 F 1 a, a b + 1; 1 a π Γa b + 1 Γ 1+aΓ a b + 1 Γ1 + a bγ1 + a Γ1 + a bγ1 + a a b C\Z ; Rb < 1. 1

4 . Known Results Analogous to Kummer s First Summation Theorem 1.7 In the year 199, following summation theorems were recorded by Prudnikov et al. 9, p. 489, Entries , ,.1 F 1 1 ma Γ1 aγm a + 1 m+1 { 1 r a r a m; Γm a a m r Ra < m ; m a, 1 a, a m, a C\Z ; m N {1, 1,. F 1 1 ma Γ1 aγm a 1 m { 1 r 1 + a m r a + m; Γm a 1 a r1 r Ra < m ; m a, 1 a, m a, 1 + a m C\Z ; m N\{1,. In the year 7, the following summation theorems were given by Choi-Rathie and Malani 5, pp , Equations.,.3.3 F 1 a, 1 + a b m; 1 Rb < m Γ1 + a b m Γa { m r ; a, 1 + a b m C\Z ; m N, Γ r+a Γ r+a + 1 b m.4 F 1 a, 1 + a b + m; Rb < m+ 1 Γ1 + a b + m Γa1 b m { m 1 r Γ r+a r Γ r+a + 1 b ; a, 1 b, 1 + a b + m C\Z ; m N. In the year 11, the following summation theorems were given by Rakha-Rathie 11, p. 88, Theorems 4, 3.5 F 1 a, 1 + a b m; 1 Rb < m a Γ 1 Γ1 + a b m Γ 1+am bγ am + 1 b { m Γ am+r+1 r ; 1 + a b m, 1 + a m b C\Z ; m N, b Γ 1+a+rm.6 F 1 a, 1 + a b + m; 1 a Γ 1 Γb mγ1 + a b + m ΓbΓ a+m+1 bγ a+m b + 1 { m Γ 1 r a+m+r+1 b r Γ 1+a+rm 13

5 Rb < +m ; b, b m, 1 + a b + m, 1 + a + m b C\Z ; m N. 3. New Results Analogous to Kummer s First Summation Theorem 1.7 Any values of parameters and variables leading to the result which do not make sense are tacitly excluded, then we have 3.1 F 1 a, a b m; 1 Γa b m Γa Rb < 1m { m Γ a+r a+r+1 r Γ a+rbm + Γ ; a, a b m C\Z ; m N, Γ a+r+1bm 3. F 1 a, a b + m; F F 1 Rb < m+1 n, a m; Γa b + m Γab m 1 Ra < 1mn n, a + m; Ra < m+1n 1 { m 1 r Γ a+r + 1r Γ a+r+1 r Γ a+rb Γ a+r+1b ; a, b, a b + m C\Z ; m N, Γm a Γn m+n+1 { 1 r m n 1 r Γ r+n r!γ rnam ; n, m a C\Z ; m + n N {1, Γ1 aγm a ΓnΓm a n mn1 { 1 + n mr Γ r+n r!γ n+r+a ; n, m a n, 1 a, m a C\Z ; m n N, n+1m n, Γm a { 1 r m n 1 r Γ r+n 3.5 F 1 1 a + m; Γn r!γ ra+mn Ra < m+1n ; n, m a C\Z ; n m N {1. Remark 1: If we put n 1 in summation theorems 3.3 and 3.4, we get the results analogous to known summation theorems.1 and. respectively. 14

6 m+ 1, Γm a { 1 r m r Γ r F 1 1 a m; r!γ r1am Ra < m ; m a C\Z ; m N {1,, m 1, m a 1Γ1 a { mr Γ r F 1 1 a + m; r!γ r+3a Ra < m ; 1 a, m a 1 C\Z ; m N\{1. Remark : The summation theorems.3,.4,.5,.6, 3.1 and 3. are the unifications and generalizations of summation theorems given in terms of gamma function recorded by Prudnikov et al. 9, p. 489, Entries , , and and other summation theorems given by Choi 4, p.4, Table 1, p. 5, Table ; m, 1,,..., 9., Kim et al. 7, p.17, Table 3 ; m, 1,,..., 5. and Rathie-Kim 1, pp , Table 1 ; m, 1,,..., 5., are written in terms of greatest integer function, absolute valued function and gamma function. The coefficients associated with summation theorems discussed in the references 4, 7 and 1, are not the functions of m. For particular values of m, coefficients are calculated and are arranged in tabular form. The summation theorems of the references 4, 7 and 1 are not general in nature. 4. Proof of Summation Theorems a, In order to evaluate F 1 1, apply Euler-Beta type integral representation 1.3 for F 1, we a b m; get F 1 a, a b m; 1 Γa b m ΓaΓb m t a1 1 t bm1 1 + t b Γa b m t a1 1 t bm1 1 + t m+1 ΓaΓb m { Γa b m 1 m t a1 1 t bm1 m r 1 r t r 1 + t ΓaΓb m r! Γa b m ΓaΓb m m r 1 r r! t r+a1 1 t bm1 1 + t 15

7 Γa b m ΓaΓb m Γa b m ΓaΓb m m 1 t r+a1 1 t bm1 + r m 1 y r+a 1 1 y bm1 + r t r+a 1 t bm1 dy y r+a y bm1 dy which, in view of Beta function definition 1.5, yields the desired formula 3.1 by appealing also to the principle of analytic continuation. Similarly we can derive other results of section Applications of Summation Theorems For the classical aplace transform defined by {ft e st ft Fs, whenever the integral exists in the ebesgue sense, it is easily seen for Kummer s confluent hypergeometric function 1 F 1 that see, for example, 15, p. 19, Equation 4.16 {t λ11f 1 µ; ν; zt 5.1 Γλ s λ F 1 λ, µ; e st t λ1 1F 1 ν; z, s µ; ν; zt s > z ; s z, Rν µ λ > ; s z, s z, Rν µ λ > 1 ; Rλ > ; ν C\Z and Rs > max{rz,. In this section, we apply the summation formulas.3,.4,.5,.6, 3.1, 3., 3.3, 3.4 and 3.5 in order to derive several closed-form expressions for the aplace transforms of Kummer s confluent hypergeometric function 1 F 1 with suitable convergence conditions for validity of the results. In each of the following results, any exceptional values of the parameters and variables, which would make the results invalid, are tacitly excluded. If we assume λ b, µ a, ν 1 + a b m and z s in equation 5.1 and use summation theorem.3 we obtain 16

8 {t b11f a b m; st e st t b1 1F a b m; st 5. Γb Γ1 + a b m s b Γa Rb < m { m r Γ r+a Γ r+a + 1 b m ; a, 1 + a b m C\Z ; m N ; Rb >, Rs >. If we choose λ a, µ b, ν 1 + a b m and z s in equation 5.1 and apply summation theorem.3 we deduce {t a11f a b m; st e st t a1 1F a b m; st 5.3 Rb < m Γ1 + a b m s a { m r Γ r+a Γ r+a + 1 b m ; a, 1 + a b m C\Z ; m N ; Ra >, Rs >. If we let λ b, µ a, ν 1 + a b + m and z s in equation 5.1 and use summation theorem.4 we obtain {t b11f a b + m; st e st t b1 1F a b + m; st 5.4 Γb Γ1 + a b + m s b Γa1 b m Rb < m+ { m 1 r Γ r+a r Γ r+a + 1 b ; a, 1 b, 1 + a b + m C\Z ; m N ; Rb >, Rs >. If we choose λ a, µ b, ν 1 + a b + m and z s in equation 5.1 and apply summation theorem.4 we get {t a11f a b + m; st e st t a1 1F a b + m; st 5.5 Γ1 + a b + m s a 1 b m { m 1 r Γ r+a r Γ r+a + 1 b 17

9 Rb < m+ ; a, 1 b, 1 + a b + m C\Z ; m N ; Ra >, Rs >. If we set λ a, µ b, ν 1+abm and z s in equation 5.1 and use summation theorem.5 we obtain {t a11f a b m; st e st t a1 1F a b m; st 5.6 Γa a Γ 1 Γ1 + a b m { m Γ am+r+1 b s a Γ 1+am bγ am + 1 b r Γ 1+a+rm Rb < m ; 1 + a b m, 1 + a m b C\Z ; m N ; Ra >, Rs >. If we assume λ b, µ a, ν 1 + a b m and z s in equation 5.1 and apply summation theorem.5 we obtain {t b11f a b m; st e st t b1 1F a b m; st 5.7 Γb a Γ 1 Γ1 + a b m { m Γ am+r+1 b s b Γ 1+am bγ am + 1 b r Γ 1+a+rm Rb < m ; 1 + a b m, 1 + a m b C\Z ; m N ; Rb >, Rs >. If we let λ a, µ b, ν 1+ab+m and z s in equation 5.1 and use summation theorem.6 we find {t a11f a b + m; st e st t a1 1F a b + m; st 5.8 Γa a Γ 1 Γb mγ1 + a b + m { m Γ a+m+r+1 s a ΓbΓ a+m+1 bγ a+m 1 r b b + 1 r Γ 1+a+rm Rb < +m ; b, b m, 1 + a b + m, 1 + a + m b C\Z ; m N ; Ra >, Rs >. 18

10 If we select λ b, µ a, ν 1 + a b + m and z s in equation 5.1 and apply summation theorem.6 we get {t b11f a b + m; st e st t b1 1F a b + m; st 5.9 Rb < +m a Γ 1 Γb mγ1 + a b + m s b Γ a+m+1 bγ a+m b + 1 { m Γ a+m+r+1 1 r b r Γ 1+a+rm ; b, b m, 1 + a b + m, 1 + a + m b C\Z ; m N ; Rb >, Rs >. If we choose λ b, µ a, ν a b m and z s in equation 5.1 and use summation theorem 3.1 we find {t b11f 1 a b m; st e st t b1 1F 1 a b m; st 5.1 Γb Γa b m s b Γa Rb < 1m { m Γ a+r a+r+1 r Γ a+rbm + Γ ; a, a b m C\Z ; m N ; Rb >, Rs >. Γ a+r+1bm If we let λ a, µ b, ν abm and z s in equation 5.1 and apply summation theorem 3.1 we find {t a11f 1 a b m; st e st t a1 1F 1 a b m; st { Γa b m m Γ a+r 5.11 a+r+1 s a r Γ a+rbm + Γ Γ a+r+1bm Rb < 1m ; a, a b m C\Z ; m N ; Ra >, Rs >. If we let λ b, µ a, ν a b + m and z s in equation 5.1 and use summation theorem 3. we obtain 19

11 {t b11f 1 a b + m; st e st t b1 1F 1 a b + m; { 5.1 Γb Γa b + m m 1 r Γ a+r + 1r Γ a+r+1 s b Γab m r Γ a+rb Γ a+r+1b Rb < m+1 ; a, b, a b + m C\Z ; m N ; Rb >, Rs >. If we choose λ a, µ b, ν a b + m and z s in equation 5.1 and apply summation theorem 3. we obtain {t a11f Rb < m+1 a b + m; Γa b + m s a b m st e st t a1 1F 1 a b + m; st st { m 1 r Γ a+r + 1r Γ a+r+1 r Γ a+rb Γ a+r+1b ; a, b, a b + m C\Z ; m N ; Ra >, Rs >. If we let λ a, µ n, ν a m and z s in equation 5.1 and use summation theorem 3.3 we get {t a11f 1 n; a m; st e st t a1 1F 1 n; a m; 5.14 Γa m+n+1 Γm a { 1 r m n 1 r Γ r+n s a Γn r!γ rnam Ra < 1mn ; n, m a C\Z ; m + n N {1 ; Ra >, Rs >. If we assume λ n, µ a, ν a m and z s in equation 5.1 and apply summation theorem 3.3 we obtain {t n11f 1 a m; 5.15 Ra < 1mn st Γm a s n m+n+1 e st t n1 1F 1 a m; { 1 r m n 1 r Γ r+n r!γ rnam st st ; m a C\Z ; m + n N {1 ; Rs >.

12 If we let λ a, µ n, ν a + m and z s in equation 5.1 and use summation theorem 3.4 we get {t a11f 1 n; a + m; st e st t a1 1F 1 n; a + m; 5.16 Γa mn1 Γm aγ1 a { n + 1 mr Γ r+n s a ΓnΓm a n r!γ n+r+a Ra < m+1n ; n, m a n, 1 a, m a C\Z ; m n N ; Ra >, Rs >. If we choose λ n, µ a, ν a + m and z s in equation 5.1 and apply summation theorem 3.4 we find {t n11f Ra < m+1n a + m; st Γm aγ1 a s n Γm a n mn1 e st t n1 1F 1 a + m; { n + 1 mr Γ r+n r!γ n+r+a st st ; m a n, 1 a, m a C\Z ; m n N ; Rs >. If we let λ a, µ n, ν a + m and z s in equation 5.1 and use summation theorem 3.5 we get {t a11f 1 n; a + m; st e st t a1 1F 1 n; a + m; 5.18 Γa n+1m Γa + m { 1 r m n 1 r Γ r+n s a Γn r!γ ra+mn Ra < m+1n ; n, m a C\Z ; n m N {1 ; Ra >, Rs >. If we select λ n, µ a, ν a + m and z s in equation 5.1 and apply summation theorem 3.5 we deduce {t n11f a + m; st Γa + m s n n+1m e st t n1 1F 1 a + m; { 1 r m n 1 r Γ r+n r!γ ra+mn st st 1

13 Ra < m+1n ; m a C\Z ; n m N {1 ; Rs >. We conclude our present investigation by observing that several other corollaries and consequences of the remaining summation formulas.1,., 3.6, 3.7 of sections, 3 and its applications in aplace transforms of Kummer s confluent hypergeometric functions, can also be deduced in an analogous manner. References 1 Andrews, G. E., Askey, R. and Roy, R. ; Special Function, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, Bailey, W. N. ; Generalized Hypergeometric Series, Cambridge Math. Tract No. 3, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert-Hafner, New York, Carlson, B. C. ; Special Functions of Applied Mathematics, Academic Press, New York, San Francisco and ondon, Choi, J. ; Contiguous Extensions of Dixon s Theorem on the Sum of a 3 F, Journal of Inequalities and Applications, Art. ID , 1, Choi, J., Rathie, A. K. and Malani, S. ; Kummer s Theorem and its Contiguous Identities, Taiwanese Journal of Mathematics , Choi, J., Rathie, A. K. and Srivastava, H. M. ; A Generalization of a Formula Due to Kummer, Integral transforms and Special Functions, 11 11, Kim, Y. S., Rathie, A. K. and Cvijović, D. ; New aplace Transforms of Kummer s Confluent Hypergeometric Functions, Mathematical and Computer Modelling, 55 1, Kummer, E. E. ; Über die hypergeometrische Reihe 1 + α.β 1.γ αα + 1.ββ + 1 x + x + 1..γγ + 1 αα + 1α +.ββ + 1β + x , 1..3.γγ + 1γ + J. Reine Angew. Math , and 17 17; see also Collected papers, Vol. II: Function Theory, Geometry and Miscellaneous Edited and with a Foreword by André Weil, Springer-Verlag, Berlin, Heidelberg and New York, Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. ; Integrals and Series, Volume III : More Special Functions Nauka, Moscow, 1986 In Russian; Translated from the Russian by G.G.Gould Gordon and Breach Science Publishers, New York, Rainville, E. D. ; Special Functions, The Macmillan Company, New York, 196 ; Reprinted by Chelsea Publ. Co., Bronx, New York, Rakha, M. A. and Rathie, A. K. ; Generalizations of Classical Summation Theorems for the Series F 1 and 3 F with Applications, Integral Transforms and Special Functions, 11 11, Rathie, A. K. and Kim, Y. S. ; Further Results on Srivastava s Triple Hypergeometric Series H A and H C, Indian J. Pure Appl. Math , Slater,. J. ; Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.

14 14 Srivastava, H. M. and Choi, J. ; Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, ondon and New York, Srivastava, H. M. and Manocha, H.. ; A Treatise on Generating Functions, Halsted Press Ellis Horwood imited, Chichester, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, Whittaker, E. T. and Watson, G. N. ; A Course of Modern Analysis, Fourth Edition, Cambridge University Press, Cambridge, ondon, and New York,