Iteatioal Joual of Matheatical Achive-3(5,, 8-8 Available olie though www.ija.ifo ISSN 9 546 CERTAIN NEW CONTINUED FRACTIONS FOR THE RATIO OF TWO 3 ψ 3 SERIES Maheshwa Pathak* & Pakaj Sivastava** *Depatet of Matheatics, Uivesity of Petoleu & Eegy Studies, Dehadu-487, Idia **Depatet of Matheatics, Motilal Nehu Natioal Istitute of Techology, Allahabad-4, Idia (Received o: 8-4-; Accepted o: 9-5- ABSTRACT I the peset pape we have developed cetai ew cotiued factios epesetatios fo the atios of two basic bilateal hypegeoetic seies 3 ψ 3. Keywods: Basic hypegeoetic seies, Basic bilateal hypegeoetic seies, Cotiued factio. AMS Subject Classificatio: 33D5, A55.. INTRODUCTION Now a days cotiued factios have bee becoe cete of attactio fo pue ad applied atheaticias. The Idia atheaticia Ayabhata(d. 55 AD [4] used a cotiued factio to solve a liea ideteiate equatio. The developet of cotiued factios theoy was stopped fo oe tha a thousad yeas due to the specific applicatios. Majo ipacts of cotiued factios i the diffeet field of atheatics have bee oticed fo the begiig of th cetuy. The Idia geius Raauja [6, 7] had show ew facet of cotiued factios i the field of qq-seies. Raauja ad his woks o cotiued factios had ispied a ube of atheaticias wokig i the field of qqseies to cay out eseaches i this diectio. Thee ae a ube of atheaticias aely R. P. Agawal [, ], G. E. Adews ad D. Bowa [3], B. C. Bedt [5], N. A. Bhagiathi [6,7], S. Bhagava, C. Adiga ad D. D. Soashekaa [8], R. Y. Deis [9,, ], R. Y. Deis ad S. N. Sigh [], Maheshwa Pathak ad Pakaj Sivastava [4], P. Rai [5], S. N. Sigh [8, 9, ], Pakaj Sivastava [], A. Vea, R. Y. Deis ad K. Siivasa Rao [] etc. have established seveal esults fo basic hypegeoetic seies φ, 3 φ ad basic bilateal hypegeoetic seies ψ, 3 ψ 3 i tes of cotiued factios. I the peset pape cetai ew esults fo the quotiets of two 3ψ seies i tes of cotiued factios have bee established followig the techiques developed by Pakaj 3 Sivastava [] ad also cetai special cases have bee deduced.. DEFINITIONS AND NOTATIONS We shall use the followig qq-sybols: Fo q < ad q < s ( a; q (, s s ( a; q (, s s s ( ;, ( ; ( a; q ( ( a ( ;. Coespodig autho: Maheshwa Pathak* *Depatet of Matheatics, Uivesity of Petoleu & Eegy Studies, Dehadu-487, Idia Iteatioal Joual of Matheatical Achive- 3 (5, May 8
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 A geealized basic hypegeoetic seies with base qq is defied as: A φ zz <, qq <. A a, a,..., a A ( a; q...( aa; q z z, b, b,..., b A ; q... A; q ( qq ; A geealized basic bilateal hypegeoetic seies with base qq is defied as: a, a,..., a A ( a; q...( aa; q z Aψ A z, b, b,..., b A ; q... A; q bb bb bb AA /aa aa aa AA < zz <, qq <. A fiite cotiued factio is a expessio of the type: aa, aa, aa 3, aa 4, ae eal o coplex ubes. It is a ifiite cotiued factio whe : a a a a a a a a a a a a a a 3 5 7 9 4 6 8... I ode to establish ai esults, we shall ake use of the followig kow esults: S. N. Sigh [] has established the followig geeal tasfoatio foula betwee basic ad basic bilateal hypegeoetic seies:. ψ 3 3 a b b az q, cq, dq, q,..., q q,,, ; q b b b az b a z q b q q b, c, d,,...,,, az, ; q b b b b az a ( bc( bd ; q... ; q a, bcq, bdq, b q 3φ c d b b ( ( ( / ;...( / ;,..., bc, bd, b,..., b z ( Bhagava, Adiga ad Soashekaa [8] have established the followig esults fo the quotiets of two 3 φ seies, which ae as follows: a, b, cq φ de / abc 3 dq, e ( d E a, b, cq ( d ( dq( e / cq 3φ de / abc d, e F E F ( dq ( dq ( e / cq ( dq ( ad E ( deq / abc( ( ( dq c,,,,... F e cq dq a dq b cq ( / [ ( / ][ ( / ](,,,,..., IJMA. All Rights Reseved 83
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 a, b, c φ deq / abc 3 dq, e ( d[ ( de / abc] D a, b, c ( d[ ( de / abc] ( dq 3φ de / abc d, e C D C ( dq [ ( de / abc] ( dq ( dq [ ( de / abc] (3 ad C eq dq a dq b dq c [ ( / ][ ( / ][ ( / ],,,,... D ( de / abc( ( ( cq,,,,...,, c 3φ de / abcq d, eq a, b, c 3φ de / abc d, e ( e / [ ( de / abcq] c[( de / abcq ( e / c][ ( d / c] ( de / abcq( ( ( c ( d[ ( de / abcq] ( B A B A [ ( de / abcq] ( [ ( de / abcq] ( (4 A ( de / abcq( ( cq,,,3,... ad B [ ( dq / a][ ( eq / a],,,3,... The idetity due to Gaspe ad Raha [3, III.9, pp. 4] is: a, b, c ( e / a, de c; q a, d, d / c φ de / abc φ e / a d, e d, de c (5 3 3 (, e de / abc; q 3. MAIN RESULTS We have established the followig esults which ae as follows: a,, ψ q, b (, b q ; / ; ( b a, (, / ; ; q 3ψ 3 q, b, b 3 3 E F E F...... ( ( b/ ( ( / q (6 ad E ( q / ( ( b q q b q,,,,... F / ( / a( / (,,,,..., IJMA. All Rights Reseved 84
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 a,, ψ q,, bb( ( a( b a, (, b( b ( ab 3ψ 3 q, b, b 3 3 ( b( ab D C D C ( b( a b ( ( ( b / a ( ( ( b / a (7 ad C q ( / a( (,,,,... ( D / a( ( q ( b q,,,,... b /,, 3ψ 3 q,, b a,, 3ψ 3 q, b, b ( a( b ( ab ( b b ( / ( / ( q ( / ( q q ( ( / ( B A B A, (8 ( / ( ( / ( ad A (/ ( b q ( b q,,, 3,... b q a B ( / a ( /,,,3,... 4. PROOF OF MAIN RESULTS The poof of ai esults 6, 7 ad 8 ae give below: Poof of (6: Takig, cc dd i tasfoatio foula (, we get ψ 3 3 z q, b, b az q q,,, ; q,, az, ; q b az a a,, b q az b a ; q ; q a,, 3φ z q q / ; / ; b, b (9 eplacig bb by bb qq ad puttig zz /aq i (9, futhe takig zz /aq i (9 ad takig atio of these two ad akig use of the esult ( ad afte siplificatio, we obtai the esult (6., IJMA. All Rights Reseved 85
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 Poof of (7: Takig zz /aq i (9, we get ψ ( q,,/, q / a; q ( a,, 3 3 q, b, b q, q,/ q, / a; q ; q ; q a,, 3φ b b ( / ; / ;, akig use of esult (5 i ( ad the eplace bb, bb by bb qq, bb qq i that esult. Agai we use the esult (5 i ( ad take the atio of these two esults, fially akig use of the esult (3 ad afte siplificatio; we get the esult (7. Poof of (8: Replacig aa by aaaa ad bb by bb qq i (, we get ψ / b ( q,,/,/ a; q ( /,,/, / ;,, 3 3 q,, b q q b q ; q ; q, b q, 3φ q / ; / ;, b ( takig the atio of ( ad ( ad akig use of esult (4 ad afte siplificatio, we get the esult (8. 5. SPECIAL CASES: I this sectio, we shall coside cetai applicatios of the ai esults obtaied i the 3. ( Puttig i esult (6, we get a, ψ q, b q ; b / ; ( b Eo ( ( ( ( ( ( ( b/ ( ( / ; b; q b ( ( b/ a, ψ q, b F E F ( ad ( ( ( ( E q / b q b q b q,,,,... ( ( ( ( F b/ / a,,,,... ( Puttig we get i esult ( ad akig use of the Raauja ψ suatio foula [3, (5.., pp. 6],, IJMA. All Rights Reseved 86
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 a, / b ψ q, b a ( qb; ( ( b ( ( b / qq, / a,, q b G ( q, qb / a,/ a, b ; q Ho G H.. (4 ( ( ( b/ ( ad ( ( ( ( G q / a b q b q b,,,,... ( ( ( ( H b/ / a,,,,... (3 Puttig b a ad b i esult (7, we get, ψ, b ( ( ( a( a b ( ( ( b / a q aq b ( a b (( a( b /, / ( ( b b q b ( ψ, b / qb q ( a b a ab D C D C...... ( ( ( b / a (4 ad ( ( ( ( C q q b q (4 Puttig bb i esult (6, we get,,,,... ( ( ( ( D b / a q,,,,... a,, φ ( b Eo ( b ( ( b/ a,, 3φ b, b 3 b, b, IJMA. All Rights Reseved 87
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 ad F E F (5 ( ( ( b/ ( ( ( ( ( E q / b q b q b q,,,,... ( ( ( ( F b/ / a /,,,,... (5 Puttig a αq ad b i esult (7, we get α q,, φ αq, αq( b ( b( αqb ( ( ( αqb b αq b αq,, 3φ αq b, b 3 D C D C o (6 ( ( ( b / αq ( ( ( b / αq α > ad ( α ( ( ( C q /,,,,... ( α ( α ( ( D b / q q q,,,,... (6 Puttig b i esult (8, we get /, / ( / ( ( ( /, ψ ( q b b b ( a /(/ a q a, a b ψ q, b B A, ( ( / ( ( / ( ( / ( ( B A ad A (/ ( b q,,, 3,... ( /,,,3,... B b q a (7, IJMA. All Rights Reseved 88
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 (7 Puttig bb ad bb i esult (8, we get, φ ( b /(/ a q a, ( b ( / ( φ b B A B A......, ( / ( ( / ( ad REFERENCES A (/ ( b q,,, 3,... B b q a ( /,,,3,... / ( ( / ( ( [] Agawal, R. P., Pade appoxiats cotiued factios ad Heie's qq-seies, J. Math. Phys. Sci. 3 6 (99, 8-9. [] Agawal, R. P., Resoace of Raauja's atheatics Vol.III, New Age Iteatioal Pvt. Ltd. Publishes, New Delhi, 998. [3] Adews, G.E. ad Bowa, D., A full extesio of the Roges-Raauja cotiued factio, Poc. Ae. Math. Soc. ( 3 (995, 3343-335. [4] Beziski, Claude, Histoy of cotiued factios ad Pade appoxiats, Spige-Velag, New Yok, 98. [5] Bedt, B. C., Raauja's Notebook pat III, Spige-Velag, 99. [6] Bhagiathi, N. A., O cetai ivestigatios i qq-seies ad cotiued factios, Math. Studet -4 56 (988, 58-7. [7] Bhagiathi, N. A., O basic bilateal hypegeoetic seies ad cotiued factios, Math. Studet -4 56 (988, 35-4. [8] Bhagava, S., Adiga, C., Soashekaa, D. D., O cetai cotiued factios elated to 3 φ basic hypegeoetic fuctios, J. Math. Phys. Sci. (987, 63-69. [9] Deis, R. Y., O basic hypegeoetic fuctio ad cotiued factios, Math. Studet -4 5 (984, 9-36. [] Deis, R. Y., O cetai qq-seies ad cotiued factios, Math. Studet 44 (983, 7-76. [] Deis, R. Y., O geealizatio of cotiued factio of Gauss, Iteat. J. Math. & Math. Sci. (4 3 (99, 74-746. [] Deis, R. Y. ad Sigh, S. N., Cetai esult ivolvig qq-seies ad cotiued factios, Fa East J. Math. Sci. (FJMS (5 3 (, 73-737. [3] Gaspe, G. ad Raha, M., Basic Hypegeoetic Seies, Cabidge Uivesity Pess, New Yok, 99. [4] Pathak, Maheshwa ad Sivastava, Pakaj, A ote o cotiued factios ad 3 ψ 3 seies, Ital. J. Pue Appl. Math. 7 (, 9-. [5] Rai, P., Cetai bilateal extesios of the Roges-Raauja cotiued factio, Gaita ( 57 (6, 73-87. [6] Raauja, S., Note book of Raauja II, T.I.F.R. Bobay, 957., IJMA. All Rights Reseved 89 (8
Maheshwa Pathak* & Pakaj Sivastava**/ Cetai ew cotiued factios fo the atio of two 3 ψ 3 seies / IJMA- 3(5, May-, Page: 8-8 [7] Raauja, S., The lost ote book ad othe upublished papes, Naosa, New Delhi 988. [8] Sigh, S. N., O qq-hypegeoetic fuctio ad cotiued factios, Math. Studet -4 56 (988, 8-84. [9] Sigh, S. N., Basic hypegeoetic seies ad cotiued factios, Math. Studet -4 56 (988, 9-96. [] Sigh, S. N., Cetai tasfoatio foulae fo basic ad bibasic hypegeoetic seies, Poc. Nat. Acad. Sci. Idia (III 65(A (995, 39-39. [] Sivastava, Pakaj, Cetai cotiued factios fo quotiets of two 3 ψ 3 seies, Poc. Nat. Acad. Sci. Idia (IV 78(A (8, 37-33. [] Vea, A., Deis, R. Y. ad Siivasa Rao, K., New cotiued factios ivolvig basic hypegeoetic 3 φ fuctio, Jou. Math. Phy. Sci. 6 (987, 585-59. Souce of suppot: Nil, Coflict of iteest: Noe Declaed, IJMA. All Rights Reseved 8