Research Article Extensions of Certain Classical Summation Theorems for the Series 2 F 1, 3 F 2, and 4 F 3 with Applications in Ramanujan s Summations

Transcription

1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 00, Article ID , 6 pages doi:0.55/00/ Research Article Extensions of Certain Classical Summation Theorems for the Series F, 3 F, and 4 F 3 with Applications in Ramanujan s Summations Yong Sup Kim, Medhat A. Rakha,, 3 and Arjun K. Rathie 4 Department of Mathematics Education, Wonkwang University, Iksan , Republic of Korea Mathematics Department, College of Science, Suez Canal University, Ismailia 45, Egypt 3 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Muscat, Alkhod 3, Oman 4 Vedant College of Engineering and Technology, Village-Tulsi, Post-Jakhmund, Bundi, Rajasthan State 330, India Correspondence should be addressed to Medhat A. Rakha, medhat@squ.edu.om Received 0 May 00 Revised 7 September 00 Accepted 3 September 00 Academic Editor: Teodor Bulboacă Copyright q 00 Yong Sup Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Motivated by the extension of classical Gauss s summation theorem for the series F given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss s second, and Bailey for the series F, Watson, Dixon andwhipplefortheseries 3 F, and a few other hypergeometric identities for the series 3 F and 4 F 3. As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.. Introduction In 8, Gauss systematically discussed the series n0 a n b n c n z n n! a b aa bb z z, c cc where λ n denotes the Pochhammer symbol defined for λ C by n 0 λ n : λλ λ n n N : {,, 3,...}...

2 International Journal of Mathematics and Mathematical Sciences It is noted that the series. and its natural generalization P F q in.6 are of great importance to mathematicians and physicists. This series. has been known as the Gauss series or the ordinary hypergeometric series and may be regarded as a generalization of the elementary geometric series. In fact. reduces to the geometric series in two cases, when a c and b also when b c and a. The series. is represented by the notation F a, b c z or a, b F z,.3 c which is usually referred to as Gauss hypergeometric function. In., the three elements a, b,andc are described as the parameters of the series, and z is called the variable of the series. All four of these quantities may be real or complex with an exception that c is neither zero nor a negative integer. Also, in., it is easy to see that if any one of the numerator parameters a or b or both is a negative integer, then the series reduces to a polynomials, that is, the series terminates. The series. is absolutely convergent within the unit circle when z < provided that c / 0,,,... Also when z, the series is absolutely convergent if Rc a b > 0, conditionally convergent if < Rc a b 0, z / and divergent if Rc a b. Further, if in., we replace z by z/b and let b, then b n z n /b n z n,and we arrive to the following Kummer s series a n z n c n n! n0 a aa z c cc z..4 This series is absolutely convergent for all values of a, c, andz, real or complex, excluding c 0,,,...and is represented by the notation F a c z or a, F c z,.5 which is called a confluent hypergeometric function. Gauss hypergeometric function F and its confluent case F form the core special functions and include, as their special cases, most of the commonly used functions. Thus F includes, as its special cases, Legendre function, the incomplete beta function, the complete elliptic functions of first and second kinds, and most of the classical orthogonal polynomials. On the other hand, the confluent hypergeometric function includes, as its special cases, Bessel functions, parabolic cylindrical functions, and Coulomb wave function. Also, the Whittaker functions are slightly modified forms of confluent hypergeometric functions. On account of their usefulness, the functions F and F have already been explored to considerable extent by a number of eminent mathematicians, for example, C. F. Gauss, E. E. Kummer, S. Pincherle, H. Mellin, E. W. Barnes, L. J. Slater, Y. L. Luke, A. Erdélyi, and H. Exton.

3 International Journal of Mathematics and Mathematical Sciences 3 by A natural generalization of F is the generalized hypergeometric series p F q defined a p pf q a z a n (a p )n z n n0 b n (b q n! )n..6 b b q The series.6 is convergent for all z < if p q and for z < ifp q while it is divergent for all z, z / 0ifp>q. When z withp q, the series.6 converges absolutely if R q p b j a j > 0, j j.7 conditionally convergent if < R q p b j a j 0, z/.8 j j and divergent if R q p b j a j. j j.9 It should be remarked here that whenever hypergeometric and generalized hypergeometric functions can be summed to be expressed in terms of Gamma functions, the results are very important from a theoretical and an applicable point of view. Only a few summation theorems are available in the literature and it is well known that the classical summation theorems such as of Gauss, Gauss s second, Kummer, and Bailey for the series F, and Watson, Dixon, and Whipple for the series 3 F play an important role in the theory of generalized hypergeometric series. It has been pointed out by Berndt, that very interesting summations due to Ramanujan can be obtained quite simply by employing the above mentioned classical summation theorems. Also, in a well-known paper by Bailey 3, a large number of very interesting results involving products of generalized hypergeometric series have been developed. In 4 a generalization of Kummer s second theorem was given from which the well-known Preece identity and a well-known quadratic transformation due to Kummer were derived.

4 4 International Journal of Mathematics and Mathematical Sciences. Known Classical Summation Theorems As already mentioned that the classical summation theorems such as those of Gauss, Kummer, Gauss s second, and Bailey for the series F and Watson, Dixon, and Whipple for the series 3 F play an important role in the theory of hypergeometric series. These theorems are included in this section so that the paper may be self-contained. In this section, we will mention classical summation theorems for the series F and 3F. These are the following. Gauss theorem 5: a, b F c ΓcΓc a b Γc aγc b. provided Rc a b > 0. Kummer theorem 5: a, b F a b Γ a bγ /a Γ /a bγ a.. Gauss s second theorem 5: a, b F a b Γ/Γ/a /b / Γ/a /Γ/b /..3 Bailey theorem 5: a, a F c Γ/cΓ/c / Γ/c /aγ/c /a /..4 Watson theorem 5: 3F a b, c Γ/Γc /Γ/a /b /Γc /a /b / Γ/a /Γ/b /Γc /a /Γc /b /.5 provided Rc a b >.

5 International Journal of Mathematics and Mathematical Sciences 5 Dixon theorem 5: 3F a b, a c.6 Γ /aγ a bγ a cγ /a b c Γ aγ /a bγ /a cγ a b c provided Ra b c >. Whipple theorem 5: 3F e, f πγeγ ( f ) c Γ/a /eγ ( /a /f ) Γ/b /eγ ( /b /f ).7 provided Rc > 0andRe f a b c > 0witha b ande f c. Other hypergeometric identities 5: a, a, b 3F a, a b a, 4F 3 a, a b, a c Γ a bγ/a / Γ aγ/a b /, Γ a bγ a cγ/a /Γ/a b c / Γ aγ a b cγ/a b /Γ/a c /.8.9 provided Ra b c >. It is not out of place to mention here that Ramanujan independently discovered a great number of the primary classical summation theorems in the theory of hypergeometric series. In particular, he rediscovered well-known summation theorems of Gauss, Kummer, Dougall, Dixon, Saalschütz, and Thomae as well as special cases of the well-known Whipple s transformation. Unfortunately, Ramanujan left us little knowledge as to know how he made his beautiful discoveries about hypergeometric series.

6 6 International Journal of Mathematics and Mathematical Sciences 3. Ramanujan s Summations The classical summation theorems mentioned in Section have wide applications in the theory of generalized hypergeometric series and other connected areas. It has been pointed out by Berndt that a large number of very interesting summations due to Ramanujan can be obtained quite simply by employing the above mentioned theorems. We now mention here certain very interesting summations by Ramanujan. i For Rx > /, ( ) x x x x x x x x, ( ) 3 4 ( ) π Γ 3/ ii For Rx > 0, x x x x x x x Γ x, 3.3 Γx ( ) ( ) ( ) π Γ 3/ iii For Rx > 0, x 3 x 5 x x x x 4x Γ 4 x 4xΓ x. 3.5 iv For Rx > /4, x x [ x x x x x 4x Γ 4 x Γ4x. 3.6 Γ 4 x

7 International Journal of Mathematics and Mathematical Sciences 7 v For Rx >, x x 3 5x x x x ( ) ( ( ) ( 3 4 ) ) 0, 3.7 π 4Γ 4 3/4, π 5/ 8 Γ 3/4, 3.8 ( ) ( ) 3 4 π Γ 4 3/4. vi For Rx < /3, ( x 3 ( ) xx 3!) 6sinπx/ sinπxγ3 /x! π x Γ3/x cos πx. 3.9 vii For Rx > /, x x 3 5x x. x x x 3.0 viii For Rx > /, [ x [ x x x 3 5 x x x x. 3. We now come to the derivations of these summation in brief. It is easy to see that the series 3. corresponds to, x F x 3. which is a special case of Gauss s summation theorem. for a, b x and c x.

8 8 International Journal of Mathematics and Mathematical Sciences The series 3. corresponds to, F 3.3 which is a special case of Kummer s summation theorem. for a b /. Similarly the series 3.3 corresponds to, x F 3.4 x which is a special case of Kummer s summation theorem. for a, b x. The series 3.4 corresponds to, F 3.5 which is a special case of Gauss s second summation theorem.3 for a b / or Bailey s summation theorem.4 for a /andc. Also, it can easily be seen that the series 3.5 to 3.9 correspond to each of the following series:,, x 3, x, x, 3F 3, x,, x 3F, 3F x, x, x,,, 4 3F 5,, 4, 4 3F 5, 4 4, 5, 4,, x, x, x 3F, 3F,,, 3.6 which are special cases of classical Dixon s theorem.6 for i a, b /, c x, ii a, b c x, iii a, b 3/, c x, iv a b /, c /4, v a /, b c /4, vi a b c /, and vii a b c x, respectively.

9 International Journal of Mathematics and Mathematical Sciences 9 The series 3.0 which corresponds to 3,, x 3F, x 3.7 is a special case of.8 for a, b x, and the series 3. which corresponds to 3,, x, x 4F 3, x, x 3.8 is a special case of.9 for a, b c x. Thus by evaluating the hypergeometric series by respective summation theorems, we easily obtain the right hand side of the Ramanujan s summations. Recently good progress has been done in the direction of generalizing the abovementioned classical summation theorems..7 see 6. In fact, in a series of three papers by Lavoie et al. 7 9, a large number of very interesting contiguous results of the above mentioned classical summation theorems..7 are given. In these papers, the authors have obtained explicit expressions of a, b F, a b i a, b F, a b i a, a i F c each for i 0, ±, ±, ±3, ±4, ±5, and 3F a b i, c j 3.

10 0 International Journal of Mathematics and Mathematical Sciences for i, j 0, ±, ± 3F a b i, a c i j 3.3 for i 3,,, 0,, j 0,,, 3, and 3F e, f 3.4 for a b i j, e f c i for i, j 0, ±, ±, ±3. Notice that, if we denote 3.3 by f i,j, the natural symmetry f i,j f ij, j a, c, b 3.5 makes it possible to extend the result to j,, 3. It is very interesting to mention here that, in order to complete the results 3.3 of 77 matrix, very recently Choi 0 obtained the remaining ten results. For i 0, the results 3.9, 3.0, and 3. reduce to.,.3, and.4, respectively, and for i j 0, the results 3., 3.3, and3.4 reduce to.5,.6, and.7, respectively. On the other hand the following very interesting result for the series 3 F written here in a slightly different form is given in the literature e.g., see 3F a, b, d c, d [ Γc Γc a b Γc a Γc b c a b ab d 3.6 provided Rc a b > 0andRd > 0. For d c, we get Gauss s summation theorem.. Thus3.6 may be regarded as the extension of Gauss s summation theorem.. Miller very recently rederived the result 3.6 and obtained a reduction formula for the Kampé defériet function. For comment of Miller s paper, see a recent paper by Kim and Rathie 3. The aim of this research paper is to establish the extensions of the above mentioned classical summation theorem. to.9. In the end, as an application, certain very interesting summations, which generalize summations due to Ramanujan have been obtained. The results are derived with the help of contiguous results of the above mentioned classical summation theorems obtained in a series of three research papers by Lavoie et al The results derived in this paper are simple, interesting, easily established, and may be useful.

11 International Journal of Mathematics and Mathematical Sciences 4. Results Required The following summation formulas which are special cases of the results. to.7 obtained earlier by Lavoie et al. 7 9 will be required in our present investigations. i Contiguous Kummer s theorem 9: a, b F a b Γ/Γ a b a b [ Γ/a /Γ/a b Γ/aΓ/a b 3/ a, b F 3 a b, 4. Γ/Γ3 a b a b b [ a b Γ/a /Γ/a b Γ/aΓ/a b 3/. ii Contiguous Gauss s Second theorem 9: a, b F a b 3 Γ/Γ/a /b 3/Γ/a /b / Γ/a /b 3/ 4. [ /a b Γ/a /Γ/b / Γ/aΓ/a.

12 International Journal of Mathematics and Mathematical Sciences iii Contiguous Bailey s theorem 9: a, 3 a F c Γ/ΓcΓ a c 3 Γ3 a [ c Γ/c /a /Γ/c /a Γ/c /aγ/c /a 3/. 4.3 iv Contiguous Watson s theorem 7: 3F a b, c ab Γc /Γ/a /b /Γc /a /b / Γ/ΓaΓb [ Γ/aΓ/b Γ/a /Γ/b / Γc /a /Γc /b / Γc /a Γc /b 4.4 provided that Rc a b >. 3F a b 3, c ab Γc /Γ/a /b 3/Γc /a /b / a b a b Γ/ΓaΓb [ ac a bc b c Γ/aΓ/b 8 Γc /a /Γc /b / Γ/a /Γ/b / Γc /aγc /b 4.5

13 International Journal of Mathematics and Mathematical Sciences 3 provided that Rc a b >. 3F a b 3, c ab Γc /Γ/a /b 3/Γc /a /b / a b a b Γ/ΓaΓb [ a b Γ/aΓ/b Γc /a /Γc /b / 4c a b 3Γ/a /Γ/b / Γc /aγc /b 4.6 provided that Rc a b >. v Contiguous Dixon s theorem 8: 3F a b, a c c Γ a bγ a c b c Γa c Γa b c [ Γ/a c 3/Γ/a b c Γ/a /Γ/a b Γ/a c Γ/a b c 5/ Γ/aΓ/a b 3/ 4.7 provided that Ra b c > 4. 3F a b, a c b Γ a cγ a b b Γa b Γa b c [ Γ/a b Γ/a b c 3/ Γ/aΓ/a c / Γ/a b 3/Γ/a b c Γ/a /Γ/a c 4.8

14 4 International Journal of Mathematics and Mathematical Sciences provided that Ra b c > 3. 3F a c, 3 a b b Γ a cγ3 a b b b c Γa b 3Γa b c 3 [ a c b 3Γ/a b Γ/a b c 5/ Γ/aΓ/a c 3/ a b Γ/a b 3/Γ/a b c 3 Γ/a /Γ/a c 4.9 provided that Ra b c > 3. vi Contiguous Whipple s theorem 9: a, a, c 3F e, c e ΓeΓc eγe c a Γe aγe cγc e a [ Γ/e /a /Γc /e /a Γ/e /a /Γc /e /a Γ/e /aγc /e /a 3/ Γ/e /aγc /e /a / 4.0 provided that Rc > 0. a, 3 a, c 3F e, c e ΓeΓc e Γe c a c a a Γe aγe cγc e a [ c eγ/e /a /Γc /e /a Γ/e /a 3/Γc /e /a e Γ/e /aγc /e /a 3/ Γ/e /a Γc /e /a / 4. provided that Rc > 0.

15 International Journal of Mathematics and Mathematical Sciences 5 5. Main Summation Formulas In this section, the following extensions of the classical summation theorems will be established. In all these theorems we have Rd > 0. i Extension of Kummer s theorem: a, b, d 3F a b, d [ Γ/Γ a b a b/d a b Γ/aΓ/a b 3/ a/d. Γ/a /Γ/a b 5. ii Extension of Gauss s second theorem: a, b, d 3F a b 3, d Γ/Γ/a /b 3/Γ/a /b / Γ/a /b 3/ { } /a b ab/d a b /d. Γ/a /Γ/b / Γ/aΓ/b 5. iii Extension of Bailey s theorem: a, a, d 3F c, d { Γ/Γc c /d Γ/c /aγ/c /a / c/d Γ/c /a Γ/c /a / }. 5.3

16 6 International Journal of Mathematics and Mathematical Sciences iv Extension of Watson s theorem: First Extension:, d 4F 3 a b, c, d ab Γc /Γ/a /b /Γc /a /b / Γ/ΓaΓb { Γ/aΓ/b Γc /a /Γc /b / c d/dγ/a /Γ/b / Γ/c /a Γc /b } 5.4 provided Rc a b >. Second Extension:, d 4F 3 a b 3, c, d ab Γc /Γ/a /b 3/Γc /a /b / a b a b Γ/ΓaΓb { } Γ/aΓ/b Γ/a /Γ/b / α β Γc /a /Γc /b / Γc /aγc /b 5.5 provided Rc a b >, α and β are given by α ac a bc b c ab 4c a b, d [ β 8 a b. d 5.6 v Extension of Dixon s theorem:, d 4F 3 a b, a c, d α Γ a cγ a bγ3/ /a b cγ/ b a Γ/aΓ/a c /Γ a b cγ/a b 3/ β a Γ/Γ a cγ a bγ /a b c b Γ/a /Γ /a bγ /a cγ a b c 5.7

17 International Journal of Mathematics and Mathematical Sciences 7 provided Ra b c >, α and β are given by β a b a b c α a b, d [ ( ) a a b c d a b c. 5.8 vi Extension of Whipple s theorem: a, a, c, d 4F 3 e, c e, d a Γe Γe cγc e Γe a Γe c Γc e a {( c e ) Γ/e /a Γc /e /a / d Γ/e /aγc /e /a / ( e ) } Γ/e /a /Γc /e /a d Γ/e /a /Γc /e /a 5.9 provided Rc > 0. vii Extension of.8: a, b, d 3F a b, d ( a ) Γ a bγ /a ( a ) Γ a bγ/a / d Γ aγ /a b d Γ aγ/a b /. 5.0 viii Extension of.9:, d 4F 3 a b, a c, d ( a ) Γ /aγ a bγ a cγ /a b c d Γ aγ a b cγ /a bγ /a c ( a ) Γ/ /aγ a bγ a cγ/ /a b c d Γ aγ a b cγ/ /a bγ/ /a c 5. provided Ra b c >.

18 8 International Journal of Mathematics and Mathematical Sciences 5.. Derivations In order to derive 5., it is just a simple exercise to prove the following relation: a, b, d 3F a b, d a, b a, b ab F d a b F a b 3 a b. 5. Now, it is easy to see that the first and second F on the right-hand side of 5. can be evaluated with the help of contiguous Kummer s theorems 4., and after a little simplification, we arrive at the desired result 5.. In the exactly same manner, the results 5. to 5. can be established with the help of the following relations: a, b, d 3F a b 3, d a, b a, b F ab da b 3 F a b 3 a b 5 a, a, d 3F c, d, a, a a, a F a a d c F, c c, d 4F 3 a b, c, d a, b, c 3F abc dc a b 3F ab, c a b 3, c,

19 International Journal of Mathematics and Mathematical Sciences 9, d 4F 3 a b 3, c, d a, b, c 3F ab da b 3 3F, a b 3, c a b 5, c, d 4F 3 a b, a c, d a, b, c abc 3F da ba c 3F, a b, a c 3a b, a c a, a, c, d 4F 3 e, c e, d a, a, c a, a, c ac a 3F dec e 3F, e, c e e, c e a, b, d 3F a b, d a, b a, b ab F d a b F, a b a b, d 4F 3 a b, a c, d a, b, c abc 3F da ba c 3F a b, a c a b, a c 5.3 and using the results 4..4, 4.3.5, , , , 4.., 4.,and.6, 4.7, respectively.

20 0 International Journal of Mathematics and Mathematical Sciences 5.. Special Cases In 5., if we take d a b, we get Kummer s theorem.. In 5., if we take d /a b, we get Gauss s second theorem.3. 3 In 5.3, if we take d c, we get Bailey s theorem.4. 4 In 5.4, if we take d c, we get Watson s theorem.5. 5 In 5.5, if we take d /a b, we again get Watson s theorem.5. 6 In 5.7, if we take d a b, we get Dixon s theorem.6. 7 In 5.9, if we take d e, we get Whipple s theorem.7. 8 In 5.0, if we take d /a, weget.8. 9 In 5., if we take d /a, weget Generalizations of Summations Due to Ramanujan In this section, the following summations, which generalize Ramanujan s summations 3. to 3., will be established. In all the summations, we have d>0. i For Rx > /: ( x x ( )( d d ) ( ) d d ) ( x x x x 3 ( 3 4 )( d ) ( ) d [( ) π 3d d d ) x {dx x}, dxx 6. 4 Γ /4 ( ) d. Γ 3/4 6. ii For Rx > 0: ( x x )( ) ( )( d x x d d x x 3 d [( ) Γ/Γ x x x d Γ/Γ x ( ) ( ) d ( ) 3 ( d d 4 3d ) ) [ π ( ) d dγ 3/4, Γx / ( d ) Γ /4

21 International Journal of Mathematics and Mathematical Sciences iii For Rx > 0: ( )( ) x d ( )( x x d 3 x d 5 x x 3 d ( ) x x d xx π ( ) 4 d ) ΓxΓx x Γ x /. 6.5 iv For Rx > /4: x x x ( d d ) x x x x x 3 ( ) x x d Γx Γx 3/ π πγx 8 ( d d ) x Γ xγx ΓxΓ x / [ 3x 4x. d 6.6 v For Rx > : ( ) ( ) x d x x d 3 5 x d x x 3 d ( Γx Γx 3 x ), 4x Γ x / d ( ) ( d 5 d 3 ( ) 4π d ( d 9d )( ) ) 9 ( ) 3 ( d 4 3d ( ), 4d π 3Γ 4 3/4 ) )( ) 3 4 ( 5d 5 9 3d 5π 3/ ( ) 5 48 Γ 3/4 4d 5 48 π 5/ Γ 3/4 ( 3 8d 3 ), 6.7 ( ) 3 ( ) d d π Γ 4 3/4 3 ( 3 4 ( d ) 3 ( ) d 3d ) Γ 3/4 π 3.

22 International Journal of Mathematics and Mathematical Sciences vi For Rx < /3: ( ) ( ) n3 d nn 3 (d ) d! 3.4 d! /d n Γ/Γ3/ n/ n Γn/Γ/ n/γ3/ n/γ n 6.8 n n/d n 3nΓ/Γ 3n/. n Γ n/γn/ /Γ n vii For Rx > /: ( ) ( ) x d x x d 3 5 x ( x d x x 3 d d ) Γ x π d Γx /. viii For Rx > /: ( ) x d x x ( d x d x x d ( ) πγ xγx / d ΓxΓ x / ) d Γ xγx. Γ xγx Derivations The series 6. corresponds to 3F, x, d x, d 6. which is a special case of extended Gauss s summation theorem 3.6 for a,b x and c x. The series 6. corresponds to 3F,, d 6., d which is a special case of extended Kummer s summation theorem 5. for a b /. Similarly the series 6.3 corresponds to 3F, x, d 6.3 x, d which is a special case of extended Kummer s theorem 5. for a andb x.

23 International Journal of Mathematics and Mathematical Sciences 3 The series 6.4 corresponds to,, d 3F, d 6.4 which is a special case of extended Gauss s second summation theorem 5. for a b / or extended Bailey s summation theorem 5.3 for a /, c. Also, it can be easily seen that the series 6.5 to 6.8 which correspond to,, x, d 4F 3 3,, x, d, x, x, d 4F 3, x, x, d 3,, x, d 4F 3,, x, d,, 4, d 4F 3 5,, 4, d, 4, 4, d 4F 3 5 4, 5, 4, d 4F 3,,, d,,, d n, n, n, d 4F 3,, d 6.5 are special cases of extended Dixon s theorem 5.7.

24 4 International Journal of Mathematics and Mathematical Sciences The series 6.9 corresponds to, x, d 3F x, d 6.6 which is a special case of 5.0 for a andb x. And the series 6.0 corresponds to, x, x, d 4F 3 x, x, d 6.7 which is a special case of 5. for a, b x c. 7. Concluding Remarks Various other applications of these results are under investigations and will be published later. Further generalizations of the extended summation theorem 5. to 5.9 in the forms a, b, d 3F a b i, d, a, b, d 3F a b 3 i, d, 7. a, a i, d 3F c, d

25 International Journal of Mathematics and Mathematical Sciences 5 each for i 0, ±, ±, ±3, ±4, ±5, and, d 4F 3, a b i, c j, d, d 4F 3, a b i, a c i j, d 7. each for i, j 0, ±, ±, ±3, and, d 4F 3 e, f, d, 7.3 where a b i j, e f c j for i, j 0, ±, ±, ±3 are also under investigations and will be published later. Acknowledgments The authors are highly grateful to the referees for carefully reading the manuscript and providing certain very useful suggestions which led to a better presentation of this research article. They are so much appreciated to the College of Science, Sultan Qaboos University, Muscat - Oman for supporting the publication charges of this paper. The first author is supported by the Research Fund of Wonkwang University 0 and the second author is supported by the research grant IG/SCI/DOMS/0/03 of Sultan Qaboos University, OMAN. References C. F. Gauss, Disquisitiones generales circa seriem infinitam αβ/.γx αα ββ /..γγ x αααβββ/..3.γγ γ x 3 etc. pars prior, Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, vol., 8, reprinted in Gesammelte Werke, vol. 3, pp and 07 9, 866. B. C. Berndt, Ramanujan s Note Books, Vol. II, Springer, New York, NY, USA, W. N. Bailey, Products of generalized hypergeometric series, Proceedings of the London Mathematical Society, vol., no. 8, pp. 4 54, Y. S. Kim, M. A. Rakha, and A. K. Rathie, Generalization of Kummer s second theorem with application, Computational Mathematics and Mathematical Physics, vol. 50, no. 3, pp , W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 3, Stechert-Hafner, New York, NY, USA, M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series F and 3 F with applications, to appear in Integral Transform and Special Functions. 7 J.-L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Watson s theorem on the sum of a 3 F, Indian Journal of Mathematics, vol. 3, no., pp. 3 3, 99.

26 6 International Journal of Mathematics and Mathematical Sciences 8 J.-L. Lavoie, F. Grondin, A. K. Rathie, and K. Arora, Generalizations of Dixon s theorem on the sum of a 3 F, Mathematics of Computation, vol. 6, no. 05, pp , J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple s theorem on the sum of a 3F, Journal of Computational and Applied Mathematics, vol. 7, no., pp , J. Choi, Contiguous extensions of Dixon s theorem on the sum of a 3 F, Journal of Inequalities and Applications, vol. 00, Article ID 58968, 7 pages, 00. A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Gordon and Breach Science, New York, NY, USA, 986. A. R. Miller, A summation formula for Clausen s series 3 F with an application to Goursat s function F x, Journal of Physics A, vol. 38, no. 6, pp , Y. S. Kim and A. K. Rathie, Comment on: a summation formula for Clausen s series 3 F with an application to Goursat s function F x, Journal of Physics A, vol. 4, no. 7, Article ID 07800, 008.