Research Article On a New Summation Formula for

Transcription

1 International Mathematics and Mathematical Sciences Volume 2011, Article ID , 7 pages doi: /2011/ Research Article On a New Summation Formula for 2ψ 2 Basic Bilateral Hypergeometric Series and Its Applications D. D. Somasheara, K. Narasimha Murthy, and S. L. Shalini Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore , India Correspondence should be addressed to K. Narasimha Murthy, simhamurth@yahoo.com Received 7 December 2010; Accepted 24 January 2011 Academic Editor: J. Dyda Copyright q 2011 D. D. Somasheara 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 wor is properly cited. We have obtained a new summation formula for 2 ψ 2 bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions, q-gamma, and q-beta function identities. 1. Introduction The summation formulae for hypergeometric series form a very interesting and useful component of the theory of basic hypergeometric series. The q-binomial theorem of Cauchy 1 is perhaps the first identity in the class of the summation formulae, which can be stated as a q 0 z az z, z < 1, q < 1, 1.1 where a : a; q : 1 aq, a : a; q : 0 a, is an integer. aq 1.2

2 2 International Mathematics and Mathematical Sciences For more details about the q-binomial theorem and about the identities which fall in this sequel, one may refer to the boo by Gasper and Rahman 2. Another famous identity in the sequel is the Ramanujan s 1 ψ 1 summation formula 3 a z az q q/az b/a, b z b b/az q/a b a < z < 1, q < There are a number of proofs of the 1 ψ 1 summation formula 1.3 in the literature. For more details, one refers to the boo by Berndt 4 and a recent paper of Johnson 5. In this paper, we derive a new summation formula for 2 ψ 2 basic bilateral hypergeometric series using the 1 ψ 1 summation formula 1.3 by the method of parameter augmentation. We then use the formula to derive the q-analogue of Gauss summation formula and to obtain a number of etafunction, q-gamma, and q-beta function identities, which complement the wors of Bhargava and Somasheara 6, Bhargava et al. 7, Somasheara and Mamta 8, Srivastava 9, and Bhargava and Adiga 10. First, we recall that q-difference operator and the q-shift operator are defined by D q f a f a f aq a, ζ f a f aq, 1.4 respectively. In 11, Chen and Liu have constructed an operator θ as θ ζ 1 D q, 1.5 and thereby they defined the operator E bθ as E bθ 0 bθ q 1 /2 q; q. 1.6 Then, we have the following operator identities 12, Theorem 1 : E bθ { at; q } at, bt; q, E bθ { as, at; q } as,at,bs,bt; q, abst/q; q abst q <

3 International Mathematics and Mathematical Sciences 3 Further, the Dedeind etafunction is defined by η τ : e πiτ/12 1 e 2πiτ : q 1/24 q; q, where q e 2πiτ,andIm τ > 0. Jacson 13 defined the q-analogue of the gamma function by Γ q x q; q qx ; q 1 q 1 x, 0 <q< In his paper on the q-gamma and q-beta function, Asey 14 has obtained q-analogues of several classical results about the gamma function. Further, he has given the definition for q-beta function as q n 1 B q x, y 1 q q nx n 0 q x y In fact, he has shown that B q x, y Γ q x Γ q y Γ q x y In Section 2, we prove our main result. In Section 3, we deduce the well-nown q- analogue of the Gauss summation formula and some etafunction, q-gamma, and q-beta function identities. 2. Main Result Theorem 2.1. If 0 < z < 1, q < 1, then a bc/azq b c z az q q/az b/a c/a bc/azq z b c b/az c/az q/a. 2.1 Proof. Ramanujan s 1 ψ 1 summation formula 1.3 can be written as a q/b z b q/a 0 1 b az q q/az b/a. 2.2 az z b b/az q/a

4 4 International Mathematics and Mathematical Sciences This is the same as { a z bq } b az 0 q/az az q z q/a 1 { b a 1 q 1 /2 q/a }. 1 { bq az } b az 2.3 On applying E cθ to both sides with respect to b,weobtain { bq a z b/az cq c/az } 0 bcq /azq 1 q 1 /2 1 { bq b/az cq c/az } 1 q/a az bcq /azq az { q q/az b c }. z q/a a a 2.4 Multiplying 2.4 throughout by { bc/azq / b c },weobtain 0 a bc/azq z a bc/azq z b c b 1 c az q q/az b/a c/a bc/azq z b c b/az c/az q/a, 2.5 which yields Some Applications of the Main Identity The following identity is the well-nown q-analogue of the Gauss summation formula. Corollary 3.1 see 15. If q < 1, γ/αβ < 1,then α β q γ 0 γ αβ γ/α γ γ/β γ/αβ. 3.1 Proof. Putting a α, b q, c γ,andz γ/αβ in 2.1,weobtain 3.1.

5 International Mathematics and Mathematical Sciences 5 Corollary 3.2. If q < 1, then q 2 ; q 4 1 q 4 1 q 3 ; q 4 q q 1/8 η 7 2τ η 3 τ η 2 4τ, q 2 ; q 2 1 q q 2q 3/4 η6 4τ q; q 2 η 3 2τ, q; q 2 q 2 ; q 2 1 q q 2q 3/4 η8 4τ q; q 2 2 η 7 2τ, n 1 q q; q 2 q 2 ; q 2 1 q 2 2 q 2 ; q 2 2 q 2 ; q 6 1 q 6 1 q 5 ; q 6 q η 2τ η 2 τ, q 3 q 1/4 η4 2τ η 2 τ, 1 q 2 2 q 2 ; q 2 1 q 2 ; q q q2 ; q 2 2 q η2 4τ η 4 2τ Proof. Putting a z q 1/4, b q 3/4, c q 5/4, and then changing q to q 4 in 2.1, weobtain q; q 4 q 2 ; q 4 q; q 4 q3 ; q 4 q 1 q 2 ; q 4 4 q 4 ; q 4 2 q3 ; q q; q 4 3 Simplifying the right hand side and then using 1.8, weobtain 3.2. Similarly, putting a q 1/2, z q 1/2, b q 3/2, c q 3/2, and then changing q to q 2, we obtain 3.3. Putting a q 1/2, z q 1/2, b c q 3/2, and then changing q to q 2,weobtain 3.4. Putting a q 1/2, z q 1/2, b q, c q 2, and then changing q to q 2,weobtain 3.5. Putting a q 1/6,z q 1/2, b q 7/6, c q 5/6, and then changing q to q 6,weobtain 3.6. Finally, putting a 1, z q, b c q 2, and then changing q to q 2,weobtain 3.7. Corollary 3.3. If 0 <q<1, 0 <x, y<1, and 0 <x y<1, then 2 Bq 3 Γ q 1 x y Γq x y 1 q x, y 1 q y 2 q 1 x q x y q 1 y 2 q y. 3.9 Proof. Putting a q 1 x, z q y,andb c q 1 y in 2.1, weget q 1 x q x y q 1 y 2 q 1 x y q y q q x y q y q 1 y q 1 y On using 1.9, 1.10,and 1.11,weobtain 3.9. q x y q x

6 6 International Mathematics and Mathematical Sciences Corollary 3.4. If 0 <q<1, 0 <x, y<1, and 1 <x y<2, then Bq 2 Γ q y x Γq x y 1 Γq x y 1 x, y Γ q y Γq 1 x q x q x y 1 q y 2 q y Proof. Putting a q x, z b c q y,in 2.1, and then using 1.9, 1.10, and 1.11, we obtain Corollary 3.5. If 0 <x, y<1, and 0 <x y<1, then B 3 x, y Γ 1 x y Γ x y 1 x x y y y 2 y 1 x 1 x y Proof. Letting q 1in 3.9,weobtain Corollary 3.6. If 0 <x, y<1, and 1 <x y<2, then B 2 x, y Γ y x Γ x y 1 Γ x y 1 Γ y Γ 1 x x x y y 2. 0 y 1 1 x 2 x y 3.13 Proof. Letting q 1in 3.11,weobtain Corollary 3.7. If 0 <q<1, 0 <x, y<1, and 0 <x y<1, then B q x, y Γq y Γq 1 y 0 q 1 x y q y 2 q x. q 3.14 Proof. Putting a q 1 x y, z q x,andb c q in 2.1, weobtain Acnowledgment The authors would lie to than the referees for their valuable suggestions which considerably improved the quality of the paper. References 1 A.-L. Cauchy, Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par une équation linéaire, et sur diverses transformations de produits composés d un nombre indéfini de facteurs, Comptes Rendus de l Académie des Sciences, Paris, vol. 17, pp , 1893, Oeuvres de Cauchy, 1re érie, T. VIII, Gauthier-Villars, Paris, France.

7 International Mathematics and Mathematical Sciences 7 2 G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 96 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, S. Ramanujan, Noteboos, vol. 2, Tata Institute of Fundamental Research, Bombay, India, B. C. Berndt, Ramanujan s Noteboos. Part III, Springer, New Yor, NY, USA, W. P. Johnson, How Cauchy missed Ramanujan s 1 ψ 1 summation, American Mathematical Monthly, vol. 111, no. 9, pp , S. Bhargava and D. D. Somasheara, Some eta-function identities deducible from Ramanujan s 1 ψ 1 summation, Mathematical Analysis and Applications, vol. 176, no. 2, pp , S. Bhargava, D. D. Somasheara, and S. N. Fathima, A new transformation for basic bilateral hypergeometric series, Mathematics Student, vol. 74, no. 1 4, pp , D. D. Somasheara and D. Mamta, On some identities of Ramanujan found in his lost noteboo, Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp , H. M. Srivastava, Some convolution identities based upon Ramanujan s bilateral sum, Bulletin of the Australian Mathematical Society, vol. 49, pp , S. Bhargava and C. Adiga, A basic bilateral series summation formula and its applications, Integral Transforms and Special Functions, vol. 2, no. 3, pp , W. Y. C. Chen and Z.-G. Liu, Parameter augmentation for basic hypergeometric series. I, in Mathematical Essays in Honor of Gian-Carlo Rota Cambridge, MA, 1996, B.E.SaganandR.P.Stanley, Eds., vol. 161 of Progress in Mathematics, pp , Birhäuser, Boston, Mass, USA, Z.-G. Liu, Some operator identities and q-series transformation formulas, Discrete Mathematics,vol. 265, no. 1 3, pp , F. H. Jacson, On q- definite integrals, Pure and Applied Mathematics Quarterly, vol. 41, pp , R. Asey, The q-gamma and q-beta functions, Applicable Analysis, vol. 8, no. 2, pp , E. Heine, Untersuchungen über die Reihe 1 1 q α 1 q β / 1 q 1 q γ X 1 q α 1 q α 1 1 q β 1 q β 1 / 1 q 1 q 2 1 q γ 1 q γ 1 X 2..., Journal fur die Reine und Angewandte Mathemati, vol. 34, pp , 1847.

8 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization