A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

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1 Commun. Koren Mth. So , No. 1, pp A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS Hrsh Vrdhn Hrsh, Yong Sup Kim, Medht Ahmed Rkh, nd Arjun Kumr Rthie Astrt. In 181, Guss otined fifteen ontiguous funtions reltions. Lter on, 1847, Henie gve their -nlogue. Reently, good progress hs een done in finding more ontiguous funtions reltions y employing results due to Guss. In 1999, Cho et l. hve otined 4 new nd interesting ontiguous funtions reltions with the help of Guss s 15 ontiguous reltions. In ft, suh type of 7 reltions exists nd therefore the rest 48 ontiguous funtions reltions hve very reently een otined y Rkh et l.. Thus, the pper is in ontinution of the pper [16] pulished in Computer & Mthemtis with Applitions , In this pper, first we otined 15 -ontiguous funtions reltions due to Henie y following different method nd then with the help of these 15 -ontiguous funtions reltions, we otin 7 new nd interesting - ontiguous funtions reltions. These -ontiguous funtions reltions hve wide pplitions. 1. Introdution The study of si hypergeometri series -series or -hypergeometri series essentilly strted in 1748 when Euler [5], onsidered the infinite produt ; 1 1 k+1 1 s generting funtion of pn, the numer of k0 prtitions of positive integer n into positive integers. One hundred yers lter, the si hypergeometri series uired n independent sttus when Heine [5], onverted simple oservtion tht 1.1 lim into systemti theory of si hypergeometri series prllel to the theory of Guss hypergeometri series. Reeived April 18, 015; Revised Novemer 18, Mthemtis Sujet Clssifition. 33C05, 33D15. Key words nd phrses. si hypergeometri series, -ontiguous funtions reltions, Guss s ontiguous funtions reltions Koren Mthemtil Soiety

2 66 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE The si hypergeometri series hve mny signifint pplitions in severl res of pure nd pplied mthemtis inluding the theory of prtitions, omintoril identities, numer theory, finite vetor spes, Lie theory, mthemtil physis nd sttistis. -integrls re the most ommon pplitions etween -series, Lie lgers nd their root systems. In 181, Guss [6], presented to the Royl Soiety of Sienes t Göttingen his fmous pper Guss 1813 in whih he onsidered the infinite series z z z 3 +, s funtion of,,,z, where it is ssumed tht 0, 1,,..., so tht no zero ftors pper in the denomintors of the terms of the series. He showed tht the series onverges solutely for z < 1, nd for z 1 when R > 0, gve its ontiguous reurrene reltions, nd derived his fmous formul 1.3 F,;;1 ΓΓ, R > 0, Γ Γ for the sum of his series when z 1 nd R > 0. Although Guss used the nottion F,,,z for his series, it is now ustomry to use F,;;z or either of the nottions 1.4 F 1,;;z or F 1 [, for his series nd for its sum when it onverges, euse these nottions seprte the numertor prmeters, from the denomintor prmeter nd the vrile z. Two hypergeometri funtions with the sme rgument z re sid to e ontiguous if their prmeters, nd differ y integers. Guss derived nlogous reltions etween F 1 [,;;z] nd ny two ontiguous hypergeometris in whih prmeter hs een hnged y ±1. Rinville [14] generlized this to ses with more prmeters. Applitions of ontiguous reltions rnge from the evlution of hypergeometri series to the derivtion of summtion nd trnsformtion formuls for suh series, they n e used to evlute hypergeometri funtion tht is ontiguous to hypergeometri series whih n e stisftorily evluted. Contiguous reltions re lso used to mke orrespondene etween Lie lgers nd speil funtions. The orrespondene yields formuls of speil funtions [13]. Guss [6] defined s ontiguous to F 1,;;z or simply F,;;z eh of the six funtions otined y inresing or deresing one of the prmeters y unity. He lso proved tht etween F nd ny two of its ontiguous funtions, there exists liner reltion with oeffiients t most liner nd otined his fifteen interesting nd useful results [8]. ;z ],

3 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 67 Thirty three yers fter Guss s pper, Heine [7], introdued the series z z +, where it is ssumed tht 0, 1,,.... This series onverges solutely for z < 1 when < 1 nd it tends t lest term-wise to Guss series s 1, euse of 1.1. The series 1.5 is usully lled Heine s series or, in view of the se, the si hypergeometri series or -hypergeometri series. Anlogous to Guss nottion, Heine [5], used the nottion Φ,,,,z for his series. It is now ustomry to define the si hypergeometri series y 1.6 where Φ,;;,z or Φ 1,;;,z [ ], ; n ; n or Φ 1 ;,z z n, ; n ; n 1.7 ; n n0 { 1, n 0, n 1, n 1,,..., is the -shifted ftoril nd it is ssumed tht m for m 0,1,... Some other nottions tht hve een used for the produt ; n re,n, [] n nd even n when 1.7 is not used nd the se is not displyed. Generlizing Heine s series, we shll define n r Φ s si hypergeometri series y rφ s 1,,..., r ; 1,,..., s ;,z [ ] 1, r Φ,..., r s ;,z 1,..., s 1 ; n ; n r ; [ n 1 n ] 1+s r n z n 1 ; n ; n s ; n n0 with n nn 1, where 0 when r > s+1. For more detil, see [5]. We lso define 1.8 ; 1 k k0 for < 1. Sine produts of -shifted ftorils our so often, to simplify them we shll freuently use the more ompt nottions ,,..., m ; n 1 ; n ; n m ; n, 1,,..., m ; 1 ; ; m ;, 1 n n+1 n 1 n,

4 68 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE n n n 1 1 n 1. Reently, Wei et l. [17] re-estlished fifteen interesting three-term reltions for the 1 series y the method of ompring oeffiients. In suh reltions, their limiting ses reover Guss fifteen ontiguous reltions for F 1 series. The pper is orgnized s follows. In Setion, fifteen -ontiguous funtion reltions due to Henie [7] will e derived y nother method. In Setion 3, we estlish 4 -ontiguous funtion reltions nd in Setion 4 s n pplition of our results, we otined further 48 -ontiguous reltions. In Setion 5, speil ses of our results.13.7, nd re given. Finlly, s n pplition we otin 18 -summtion formuls in losed forms whih will e given in Setion 6. The results derived in this pper re simple, interesting, esily estlished nd my e useful.. Heine -ontiguous funtions reltions In this setion, the following 15 -ontiguous funtions reltions due to Henie [7] will e derived y nother method. These re { 1 + z 1 z 1 z 1 1 z 1 1 z + z + z {1 + +z {1 + {1 + z. { + z +1 z +1 z +1 z + z

5 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 69 {1 + +z +1 z { 1 + z 1 z 1 z 1 { + z + z { ++ z 1 z.1. Derivtions of.13.7 The derivtions of the 15 -ontiguous funtions reltions due to Heine.13.7 re strightforwrd, y expressing on the right-hnd-side of these reltions s series nd then simplifying using the identities So we will derive only three of these reltions, nd the rest n e derived on similr lines. Derivtion of.15: In order to derive.15, it is suffiient to show tht 1 1 z z z. 1 Now, we strt with the right-hnd side of the ove eution 1 z z z. 1 Expressing s series, we hve { n n 1 z n n n z 1 z n n n n n n0 + 1 n0 n n z n n n + z n. n n n n n0 Using the identities , we hve n0 n 1 n n z n n0 n n { n n z z n 1 n + n n 1 n +. n0

6 70 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE After simplifition, we get n 1 n n z n n n z z n 1 n 1 n n0 n n n0 n n 1 n [ ] n n z n {1 n 1 n n n n0 1 n n z n. n n n0 Finlly summing up the series, we get 1 whih is the left-hnd side. Derivtion of.1: In order to derive.1, it is suffiient to show tht +z. Now, we strt with the right-hnd side of the ove eution +z. Expressing s series, we hve n n z n n n z n n n n n n0 +z n n z n. n n n0 Proeeding s efore, we hve n n n n 1 n 1 zn n0 n n n0 n n 1 n 1 zn + n n z n+1 n0 n n n n n n z n{ 1 n 1 n0 n0 1 n n 1 n 1 1 n 1 1 n 1 After little simplifition, we get { n n z n n ++ n 1 n n 1 n 1 1 n 1 n0.

7 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 71 whih is the left-hnd side. Derivtion of.7: In order to derive.7, it is suffiient to show tht z + z. 1 Now, we strt with the right-hnd side of the ove eution [ z Expressing s series, we hve n n n0 n n0 n + ] z n n n z n n z n + 1 ] ++ n n z n. n n n0 n n z n n n n0 Proeeding s efore, we get n n 1 n 1 z n n0 n n [ n n z n+1 1 n 1 + n n 1 n 1 n0 ++ ]. After little simplifition, we get n n 1 n 1 z n n n z n+11 n 1 n n0 n n n0 n n 1 n n n z n n n n0 Finlly, summing up the series, we get whih is the left-hnd side. Remrks. For note on these ontiguous funtions reltions, see pper y Kim et l. [11].

8 7 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE 3. New results In this setion, the following 4 results will e estlished with the help of the results given in Setion. These re , , , , z { + z +, z 1 z { z 1 +1 z, 1 1 z+ z, z z + 1 z+ z, z z + z { + + z, { z, { +1, +1 z, z { + + z,,

9 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 73 z+, z+, { 1 + z + z, 1 z { + z +, z { + z + z 1, z { z 1 +1 z, +, 1 +1, { 1 + z + z, z { + z +, 3.1. Derivtions of results 3.8 to 3.51 The derivtions of our new -ontiguous funtions reltions re uite strightforwrd y lgeri mnipultions. For exmple, if wish to derive the result 3.8, then in eution.13, reple y nd then multiply y, we get 1 1, fter rerrngement of the terms, we esily get 3.8. In similr mnner, other results n e esily otined. The sheme is outlined in Tle-1 inluding tht of 3.8.

10 74 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE Tle 1. Derivtions of Derivtion of In eution Ation Reple y / nd multiply y 3..1 Reple y / nd multiply y 3.3. Reple y / nd multiply y 3.4. Reple y nd divide y Reple y / nd multiply y Reple y / nd multiply y Reple y nd divide y Reple y / nd multiply y Reple y nd multiply y Reple y / nd multiply y Reple y / nd multiply y Reple y nd divide y Reple y nd divide y Reple y / nd multiply y Reple y nd divide y Reple y nd divide y Reple y nd divide y Reple y / nd multiply y Reple y / nd divide y Reple y / nd divide y Reple y / nd divide y Reple y nd divide y Reple y nd divide y Reple y nd divide y 4. Applitions In this setion we shll otin 48 more ontiguous funtions reltions with the help of the results otined in Setions nd 3. These re {1 +z 1 1 z+1 z, {1 +z 1 1 z+1 z, { 1 1 +z 1 z+1,

11 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS { 1 1 +z 1 z+1, { 1 z 1 { 1 + z +1 z, { 1 z 1 { 1 + z +1 z, 1 { 1 z 1 z + 1, { +z 1 z + 1 z +, { +1 { +z z +1 1 z, +1 [ z { +1 ] 1 1 { +1 z +1 1 zz, {1 z 1 1 z+1, [ { 1 z 1 + z { 1 + z ],, 1 1 { 1 + z z +1 z z,

12 76 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE [ zz + { 1 + z { + ] 1 z { + z +1 1 z, [ z + { 1 + z { + z ] 1 z { + z +1 1 z, {1 +z 1 1 z +1, { +z 1 1 z +, { +z z, z 1 1 { + z z, z{ +z { + z 1 1 z, z, z{ +z { + z 1 1 z, { z 1 1 z,

13 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 77 { z, 1 1, [{1 + { + +z { + + ] 1 z{ +, [ z { ] { z, [{1 + +z 1 ] 1, [ { {1 + +z 1 + z 1 z ] { 1 + z 1 z z [ { + z1, +1 { +1 ] { zz, 4.83 [ 3 1 +z { ] { 1 + z 1 z z, 4.84 {1 + 1 z, 4.85 [ 3 1

14 78 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE z { + + ] 1 z 1 1 z, [ { { +1 + z ] 1 z { + 3, [ { z { ] { 1 + z 1 z z, { z +1, { z { +z { +z z [{ + z { + z z ] z { + z 1 1 1, [{ + z { +,,, z] z { + 3, [ { {1 + +z + z

15 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS z] 1 z { + z 1 1, { +z 1 z, [ {1 + +z { 1 + z 1 z] { 1 + z 1 z z, [ 1 z { 1 + z { 1 + z ] 1 1 { 1 + z +1 z z, { +z z z, [{ + z { + z ] z { + 1 1, 4.1. Derivtion of results The derivtions of these new ontiguous funtion reltions re uite stright forwrd. For exmple, y lgeri mnipultion, if we wish to derive 4.5, we onsider Henie s -ontiguous reltion.13 nd the result 3.34 nd eliminte, we esily rrive 4.5. Similrly other results n lso e otined. The sheme is outlined in Tle nd Tle Speil ses i In.13.7, if we tke 1, we get the orresponding ontiguous funtions reltions in hypergeometri series, due to Guss given in [15].

16 80 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE Tle. Derivtions of S. No. If we tke Henie s reltion nd reltion if we eliminte we get the result ii In , if we tke 1, then fter little simplifition, we get the orresponding ontiguous funtion reltions in hypergeometri series otined y Cho et l. [3]. iii In , if we tke 1, then fter little simplifition, we get the orresponding ontiguous funtion reltions in hypergeometri series otined very reently y Rkh et l. [16].

17 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 81 Tle 3. Derivtions of S. No. If we tke Henie s reltion nd reltion if we eliminte we get the result Applitions Our min im in this setion is to otin losed form for the following summtion formuls for the -series,, i, ;,, i,3,4,5,6,,, i, ;,, i 0,1,,3,4,, i,, ;,, i 1,,3,4, nd, i,, ;,, i 1,,3,4.

18 8 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE 6.1. Min results In this setion, the results to e estlished re ,,, ;, ; 1 ; ;,, 3, { ; ;, ; ; ; 3 1+ ;,, 4, ; { ; ; 3 ; ; ; ; 4 ; ;, ; 1 4 ; ; ; ; { ; ; 4 ; 6 1 ; 5 3 ;,, 5, ;, ; ; 5 ; ; 3 ; ; { ; ; 4 ; ; ; 6 5 ; ; ;

19 6.104 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 83,, 6, ;, ; ; 6 ; ; 3 ; { ; ; ; 4 ; ; ; ; 7 5 ;,, ;,, ; ; ; 1,,, ; ; 1 ; 6 5 ; ; { ; ; + 1 ; ; ;, ;,,, ; ; 1 ; ;, { 1 ; ; ; + 1 { ; 1 1 ; ;,, ;, 3, ; ; ; ; ; ; ;

20 84 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE ; ; 1 3 ; { ; ; ; ; 3 ;,, 4, 4 ; ; 1 4 ; ; 4 ;, 3 ; ; 3 ; ; 3 ; { ; 4 ; ; ; ; ; ; ; ; 3 ; ; ,, ;,, 6.111,, ; 1 ; 1 ;, ; ; ;, ; 1 ; ; {, 3,, { ; ; ; ; [ ; 1 + ] 1 ;, ; { 1 +1

21 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 85 where nd ; ; 1 ; { ; X ; X 1 ; 1 { 1 { 1 Y { { 1, 4,, ;, ; Y ; ; 1 { 3 ; ; 3 ; X ; +1 1 ; Y where X 1 X Y 1 nd Y X Y 1 lso, X 1 nd Y 1 n e otined from X 1 nd Y 1 y hnging to.,, ;, 6.115, ; { ; ; ; ; + ;,, ;,, ; ; ; ; [ ; 4 3 ; ; ] {1 3,, ; + 1

22 86 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE where nd where nd, 3,, ; ; ; X 3 ;, { ; 4 ; X 3 + ; ; Y 3, 5 { Y { , 4,, ; ; ; ;, { ; 6 X 4 ; X 4 + ; 1 4 X Y 3 Y 4 Y X 3 ; Y 4, 5 lso, X 3 nd Y 3 n e otined from X 3 nd Y 3 y hnging to. 6.. Derivtions In order to strt the derivtions of , the ontiguous funtions reltions 3.4, 3.8, 4.1 nd 4.3 together the Biley-Dhum s summtion formul [, 4], viz ; ; 6.118,, ;,, ; ; ; will e reuired in our investigtions. The derivtions of re uite strightforwrd. So we shll derive only 6.100, 6.107, nd nd the rest n e derived on similr lines. In order to derive the result 6.100, we use the result 3.31 in the form ,.

23 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 87 In 6.119, if we selet 1,, ;,, 6.10 then fter simplifition, we get,,, 1 ;,,, ;,, 1 1,,, ;, Now, it is esy to see tht, first 1 on the right-hnd-side of 6.10 n e evluted with the help of the Biley-Dhum s summtion formul nd seond 1 n lso e evluted with the help of Biley-Dhum s summtion formul y simply hnging y, we get,,, ;, ; ; Noting tht ; ; ; ; ;, nd 1 ; ; 3 ; ; ; we get ; 1,,, ; ; ;, ; 1 ; ; ; 1 1 ; ; 3 ; ; ;,

24 88 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE 6.15 whih on simplifition gives,,, ;, ; 1 ; ; {, ; ; 3 ;. This ompletes the derivtion of In order to derive 6.107, we use 3.35 nd selet,, 1 ;,,, then fter little simplifition, we,, 6.16 ;, ,,, ;,,, ;,,. Now, the first 1 on the right-hnd side of 6.16 n e evluted with the help of the -ontiguous Kummer s formul nd the seond 1 n lso e evluted with the -ontiguous Kummer s formul y simply hnging y, we get,, 6.17 ;,, ; ; { ; ; 1 ; ; ; ; ; ; ; ; ; ; 1 { 1 ; + 1 ; ; ;.

25 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS Noting tht 1+ we get,,, ; 1 ; 1 ; ; ;, ; + ; ; 1 ; ; Finlly, using 6.1, we get,, ;, 6.131, ; ; 1 ; ; + ; ; 1 ; ` 1 1 ; ; { ; 1 ; 1 ; ; whih fter simplifition gives,,, ; ;, ; + 1 { ; 1 ; + 1 ; ; ; ; { ; 1 ; + 1 ; { ; 1 ; + 1 ; ; { ; ; 1 1 ; ; ; 1 ; ; ; whih ompletes the derivtion of ; ; ; ;,. ; ;

26 90 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE In order to derive the result 6.111, we use the result 4.5 nd selet 1,,, ;,, then fter little simplifition, we,, 6.13 ;, 1 1,, ;,, 1 1 1,,, ;, Now, it is esy to see tht, first 1 on the right-hnd side of 6.13 n e evluted with the help of the summtion formul nd seond 1 n lso e evluted with the help of the formul y simply hnging y, we get 6.133,,, ;, ; 1 ; ; 1 { ; ; nd ; 1 ; ; 1 ; Noting tht 1 ; ; 1. { ; ; ; ; ; 1 ; ;. ; 1 ; ;, ;,

27 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS together with 6.1, we get,, ;,, ; 1 1 ; 1 ; { ; 1 ; + ; ; +1 ; ; 1 ; ; whih fter simplifition 6.135,,, ; 1 1 ; ;, ; 1 ; ; { [ ; 1 + ] 1. ; { 1 +1 This ompletes the derivtion of In order to derive the result 6.115, we use the result 4.74 nd selet,, 1 ;,, then fter little simplifition, we,, ;, ,,, ;,, ,, ;,, Now, it is esy to see tht, first 1 on the right-hnd side of n e evluted with the help of the summtion formul nd seond 1 n lso e evluted with the help of the summtion formul y simply hnging y, we get,,, ; ; ; ;, { ; ; + ; 3 ;.

28 9 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE ; ; Noting tht, 1 ; ; we get,,, ; ; ; + ; ;, { ; 3 ; + ; ; 1 4 ; 4 ;. ; {1 4 ; + ; 3 ; ; ; { ; 3 whih fter simplifition gives,, ;,, ; +1 ; ; [ ; ; 4 ; ; ; 3 ; ]. This ompletes the derivtion of Similrly, other results n lso e otined {1 4 ; + 1 Remrks. 1 For -Guss seond, Kummer nd Biley summtion formuls, we refers the pper y Andrews [1]. The results nd were lso otined y Kim et l. [1] y following different method. 3 For -ontiguous Guss s seond summtion formuls we prefer the pper y Kim nd Rthie [10]. 7. Conluding remrk In ddition to 15 -ontiguous funtions reltions ville in the literture, we hve, in this pper, otined 7 new nd interesting -ontiguous funtions reltions. These reltions hve wide pplitions. Severl new nd interesting results y employing the -ontiguous funtions reltions given in the pper re under investigtions nd will form prt of suseuent pper in this diretion.

29 A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS 93 Referenes [1] G. E. Andrews, On the -nlog of Kummer s theorem nd pplitions, Duke Mth. J , [] W. N. Biley, A note on ertin -identities, Qurt. J. Mth , [3] Y. J. Cho, T. Y. Seo, nd J. Choi, Note on ontiguous funtions reltions, Est Asin Mth. J , no. 1, [4] J. A. Dhum, The si nlog of Kummer s theorems, Bull. Amer. Mth. So , [5] G. Gsper nd M. Rhmn, Bsi hypergeometri series, Cmridge University Press, [6] C. F. Guss, Disuisitiones generles ir seriem infinitm..., Comm. So. Reg. Si. Gött. Re., Vol. II; reprinted in Werke , [7] E. Henie, Untersuhungen uer die Rehie, J. Reine Angew. Mth , [8] A. K. Irhim nd M. A. Rkh, Contiguous reltions nd their omputtions for F 1 hypergeometri series, Comput. Mth. Appl , no. 8, [9] F. H. Jkson, Trnsformtion of -series, Messenger of Mth , [10] Y. S. Kim nd A. K. Rthie, Another method for proving -nlogue of Guss s summtion theorem, Fr Est J. Mth. Si. 5 00, no. 3, [11] Y. S. Kim, A. K. Rthie, nd J. Choi, Three term ontiguous funtionl reltions for si hypergeometri series Φ 1, Commun. Koren Mth. So. 0005, no., [1] Y. S. Kim, A. K. Rthie, nd C. H. Lee, On -nlog of Kummer s theorem nd its ontiguous results, Commun. Koren Mth. So , no. 1, [13] W. Miller, Jr., Lie theory nd generliztions of hypergeometri funtions, SIAM J. Appl. Mth , [14] E. D. Rinville, The ontiguous funtion reltions for pf with pplitions to Btemn s Jn u,v nd Rie s H nζ,p,v, Bull. Amer. Mth. So , [15], Speil Funtions, The Mmilln Compny, New York, [16] M. A. Rkh, A. K. Rthie, nd P. Chopr, On ontiguous funtion reltions, Comput. Mth. Appl , [17] C. Wei nd D. Gong, -Extensions of Guss fifteen ontiguous reltion for F 1 series, Commun. Computer Informtion Siene , no., Hrsh Vrdhn Hrsh Deprtment of Mthemtis Amity Shool of Engineering nd Tehnology Amity University Jipur, Rhsthn Stte, Indi E-mil ddress: hrshvrdhnhrsh@gmil.om Yong Sup Kim Deprtment of Mthemtis Edution Wonkwng University Iksn , Kore E-mil ddress: yspkim@wonkwng..kr

30 94 H. V. HARSH, Y. S. KIM, M. A. RAKHA, AND A. K. RATHIE Medht Ahmed Rkh Deprtment of Mthemtis Fulty of Siene Suez Cnl University Ismili, 415, Egypt E-mil ddress: medht Arjun Kumr Rthie Deprtment of Mthemtis Shool of Mthemtil nd Physil Sienes Centrl University of Kerl Periye P.O. Distl. Ksrgod, , Kerl Stte, Indi E-mil ddress: