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1 Available online at Advances in Applied Science Reseach, 204, 5(:48-52 Tansfomations of q-seies using summations fomulae Padhyot Kuma Misha and Ravinda Kuma Yadav ISS: CODE (USA: AASRFC Depatment of Mathematics, Sobhasaia Goup of Institutions, Gokulpua, Bajajgam, Sika ABSTRACT In this pape, making use of cetain known identities and some known summations of tuncated hypegeometic seies, an attempt has been made to established some new tansfomations fo basic hypegeometic seies. Key Wods: Tansfomations fomulae, q-hypegeometic seies, tuncated seies AMS SUBJECT CLASSIFICATIO: 33A30, 33D5. ITRODUCTIO Thoughout this pape we shall adopt the following notations and definitions. Fo any numbes a and q eal o complex and <, ; ;>0 ; 0 (. Accodingly, we have ; Also,,, ; ; ; ; ; Following Gaspe and Rahman [5], we define a basic hypegeometic seies, Φ!",, ;;# ' $,$,$ $ % (,',' * ' + ;, -. - / 3 4! (.2! Whee 0< < and 5<6+.,,/ (,/,/ * / 0 ;, - Results equied in sequel In this pape we have established cetain tansfomation fomulae fo basic hypegeometic functions by make use of some diffeent summations of tuncated seies and following identity, m A( Bm ( + C ( m (. m 0 0 m 0 0 C m q m (2. m 48
2 whee, ( aq e( e b A( ( q e( e ab, q m B ( m m q abq2 e, ; q e m C ( m m e abq, ; q q e m C ( e abq, ; q Agawal, R. P.II [3]; Eq.0, (79 we shall use following summations in ou analysis. Φ ( γ ( γ β ( β ( γ ( γ β γ β, q γ, β, q; q n, β, q; q 3 2 γ, δ 0 q, γ, δ ; Agawal, R.P. II, [3]; Eq.8, (79 Φ,,, q, q, z; z q zq 4 3 q q z q z Agawal, R. P. [2]; App II, (23 q q, n+ q q q n n ( ( ( n n ( n+ [ q; [ zq; q [ q; z ; q Z (2.2 (2.3, q, q, β, γ, δ; q Φ 6 5 q q q,,,, β γ δ β γ δ [ q; βq; [ γ q; δ q; q q q ; q ; q ; q [ q; β γ δ whee βγδ ( ( β ( γ ( δ n ( q (2.4 Agawal, R. P. [2]; App II, (25 γ, δ ; q γδ ( γ ( δ n n ( γδ ( Φ [ ] 2 q q q γ q; q δ q; (2.5 γδ q; q [ q; 49
3 Agawal, R. P. [2]; App II, (8 METHODS AD DISCUSSIO In this section we shall establish ou main tansfomation fomulae. (i Putting, ( ; q ( β; q q ( γ ; ( δ; ( q; in (2. and using (2.2 we obtain the tans fomation: ( aq e( e b( q γ ( γ β Φ ( ( ( γ ( γ β a, b, q; q 2 q e e abq q q 3 2 abq e, e ( aq e( e b( q γ ( γ β( ( β γ βq ( q e( e ab( q γ ( γ β q γ a, b, q, βq, q; q Φ abq βq e,, γ, e γ ( q γ ( γ β (, β; e abq γ βq ( γ ( γ β, ;, ; + q q q q q γ a, b,, β, q; q Φ 5 4 e abq (3.,, γ, δ q e (ii Putting, z q q q q q q z ( ; ( q ; ( q ; ( z; ( ; ( ; ; ( ; in (2. and using (2.3 we obtain the tansfomation: a, b, q, zq; q ( q e( e b Φ 2 ( q e( e ab z 4 3 abq q e,, e z, q, q, z, q; z + Φ e abq 5 3, ; q,, q e zq 50
4 a, b,, q, q, z, q; z Φ 7 5 e abq q,,,, q e z (iii Putting, ( ; ( ; ( ; ( β; ( γ ; ( δ; q q q q q q q q q q q q ; q ; q ; q ; q ; q β γ δ ( ( in (2. and using (2.4 we obtain the tansfomation: a, b, q, βq, γ q, δ q; q ( aq e( e b Φ 2 ( q e( e ab 6 5 abq q q q e,,,, e β γ δ,,,,,, ; q q q β γ δ q q + Φ 7 5 q q q e abq,,,,, ; q β γ δ q e a, b,, q, q, β, γ, δ, q; q Φ 9 7 e abq q q q,,,,,, q e β γ δ Povided that, βγδ (3.2 (3.3 (iv Putting, ( γ ; q ( δ; q q ( q; ( γδ q; in (2. and using (2.5, we obtain the tansfomation:, b, γ q, δ q ( aq e( e b Φ 2 ( q e( e ab 4 3 abq e,, γδ q ( a b, ; γ, δ, q ; q + Φ e abq 3 γ δ q e, ; q a, b, γ, δ, q; q Φ 5 3 e abq (3.4,, γ δ q q e COCLUSIO Results founds in the section 3 ae vey useful and inteesting summations fomulae in the light of basic hypegeometic functions by make use of some diffeent summations of tuncated seies by using q-seies. REFERECES [] Agawal, R.P., Manocho, H. L., Sinivas Rao K., Allied publishes limited, ew Delhi 200. (pp [2] Agawal, R.P.: Genealized hypegeometic seies and its applications to the theoy of Combinatoial analysis and patition theoy, (Unpublished monogaph,
5 [3] Agawal, R.P., ew Age Intenational (P Limited, Vol. III, ew Delhi, 996. [4] Slate, L. J., Cambidge Univesity Pess, London, 966, pp [5] Gaspe, G. and Rahman, M., Cambidge Univesity Pess, 990, pp 6-7. [6] Andews, G. E. and Wanaa, S. O., The Ramanujan J. Vol. 4, o., 2007, pp [7] Bhagava S., Vasuki K. R. and Seeamuthy T. G., Indian J. Pue applied Mathematics., 35/8, 2004, pp003. [8] Stanton, D., q-seies and patitions, Spinge-Velag, ew Yok, 989. [9] Andews, L. C., Macmillan, ew Yok, 985. [0] Ramanujan, S., The lost otebook and othe unpublished papes, aosa, ew Delhi,
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