On a reciprocity theorem of Ramanujan.Abstract In this paper, we obtain a new proof of Ramanujan's receprocity theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. using an Heine's transformation which is a generalization gen·er·al·i·za·tion n. 1. The act or an instance of generalizing. 2. A principle, a statement, or an idea having general application. of Jacobi's triple product identity. We show that the reciprocity reciprocity In international trade, the granting of mutual concessions on tariffs, quotas, or other commercial restrictions. Reciprocity implies that these concessions are neither intended nor expected to be generalized to other countries with which the contracting parties theorem leads to a q-integral extension of the classical gamma function In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. For a complex number z with positive real part it is defined by 1. Introduction In his "lost" notebook [10], Ramanujan recorded the following theorem: Theorem 1.1. If a, b [not equal to] [-q.sup.-n] and |q| < 1, then [rho](a, b) - [rho](b, a) = (1/b - 1/a) [(aq/b).sub.[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ]][(bq/a).sub.[infinity]][(q).sub.[infinity]] / [(-aq).sub.[infinity]][(-bq).sub.[infinity]] (1.1) where [rho](a, b) = (1 + 1/b) [[infinity].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (n=0)] [(-1).sup.n][q.sup.n(n+1)/2][a.sup.n][b.sup.-n]/ [(-aq).sub.n], (1.2) [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], and [(a).sub.n] := [(a; q).sub.n] = [(a; q).sub.[infinity]]/[([aq.sup.n]; q).sub.[infinity]], -[infinity] < n < [infinity]. The first proof of Theorem 1.1 was given by Andrews [1]. Recently Berndt et al. [3] have given three different proofs of (1.1). For the first proof they have used Ramanujan's [sub.1][[psi PSI - Portable Scheme Interpreter ].sub.1] summation formula and Heine's transformation which is similar to the proof given earlier by Somashekara and Fathima [12]. The second proof depends upon a transformation formula of Fine [5] and triple product identity and the last proof is purely combinatorial. The purpose of this paper is to provide a simple proof of Theorem 1.1 which is a generalization of Jacobi's triple product identity [8] and has an application to sums of three squares [3]. In Section 3 we show that (1.1) is a q-integral extension of [GAMMA](x) = [[integral].sup.[infinity].sub.0] [e.sup.-t][t.sup.x-1]dt. In the last section, we deduce de·duce tr.v. de·duced, de·duc·ing, de·duc·es 1. To reach (a conclusion) by reasoning. 2. To infer from a general principle; reason deductively: some new eta-function identities which complement the works of Berndt and Zhang [4], Fine [5], Koohler [9], Zucker [14], and Vasuki [13]. 2. Proof of Theorem 1.1 In this section we give a new proof of Theorem 1.1 using Heine's transformation [11]: [[infinity].summation over (n=0)] [(a).sub.n][(b).sub.n]/[(q).sub.n][(c).sub.n] [z.sup.n] = [(c/b).sub.[infinity]] [(bz).sub.[infinity]]/[(c).sub.[infinity]][(z).sub.[infinity]] [[infinity].summation over (n=0)] [(abz/c).sub.n][(b).sub.n]/[(bz).sub.n][(q).sub.n] [(c/b).sup.n] (|z| < 1; |q| < 1). (2.1) Our proof is similar to the proof of Ramanujan's [sub.1][[psi].sub.1] summation formula given by Ismail [6]. Proof. we have [rho](a,[-q.sup.m]) = (1 - 1/[q.sup.m]) [[infinity].summation over (n=0)] [q.sup.n(n+1)/2][(a/[q.sup.m]).sup.n]/ [(-aq).sub.n], = (1 - 1[q.sup.m]) (1 + a) [m.summation over (n=0)] [(1/ [q.sup.m-1]).sub.n] [(-a).sup.n] (2.2) Here we have used Heine transformation (2.1) with b = q, c = 0, a = 1/[q.sup.m-1] and z = -a. and [rho]([-q.sup.m], a) = (1 + 1/a) [[infinity].summation over (n=0)] [q.sup.n(n+1)/2] [([q.sup.m]/a).sup.n]/[([q.sup.m+1]).sub.n] = (1 + 1/a)[(q).sub.m][(a/[q.sup.m]).sup.m][q.sup.m(m-1)/2] [[infinity].summation over (n=m)] [q.sup.n(n-1)/2][(q/a).sup.n][(q).sub.n] (2.3) From (2.2) and (2.3), we have [rho](a,[-q.sup.m]) - [rho]([-q.sup.m], a) = -(1 + 1/a)[(q).sub's][(a/[q.sup.m]).sup.m][q.sup.m(m-1)/2] [[infinity].summation over (n=0)] [q.sup.n(n-1)/2][(q/a).sup.n]/[(q).sub.n] = -(1 + 1/a)[(q).sub.m][(a/[q.sup.m]).sup.m][q.sup.m (m-1)/2][(q/a).sub.[infinity]]. (2.4) which is same as right side of (1.1) with b = [-q.sup.m] (m = 0, 1, 2, 3, ...). Both sides of (1.1) agree when b is replaced by [-q.sup.m], which are analytic in b, for b close to zero. Zero is the limit point of the sequence {[-q.sup.m]}(m = 0, 1, 2, 3, ...). Since the two functions are analytic in an open set around b = 0 and are equal at infinitely many points in this set, they must be equal there. Analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series gives (1.1) for a, b [not equal to] [-q.sup.-n]. Corollary corollary: see theorem. (Jacobi's triple product identity). [(-qb; q).sub.[infinity]][(-1/b; q).sub.[infinity]][(q; q).sub.[infinity]] = [[infinity].summation over (n=-[infinity])] [q.sup.n(n+1)/2].[b.sup.n] (2.5) where b [not equal to] 0 and |q| < 1. Proof. Using (2.1) we deduce that -[rho](b, a) = (1 + 1/b) [[infinity].summation over (n=1)] [(-1/a).sub.n] [(-b).sup.-n] and [rho](a, b) = (1 + 1/b) [[infinity].summation over (n=0)] [(-1/a).sub.-n][(-b).sup.-n] combining these two and using (1.1) we have (1 + 1/b) [[infinity].summation over (n=-[infinity])][ (-1/a).sub.n][(-b).sup.n] = (1/b - 1/a) [(aq/b).sub.[infinity]][(bq/a).sub.[infinity]][(q).sub.[infinity]] /[(-aq).sub.[infinity]][(-bq).sub.[infinity]]. (2.6) Changing b to -ab and putting a = 0, we obtain (2.5). 3. Connection with the q-Gamma Function Jackson [7] defined the q-analogue of the gamma function by [[GAMMA].sub.q](x) = [(q).sub.[infinity]]/[([q.sup.x]).sub.[infinity]] [(1 - q).sup.1-x], 0 < q < 1. (3.1) Jackson [7] also defined a q-integral by, [[integral].sup.a.sub.0] f(t)[d.sub.q](t) = a(1 - q) [[infinity].summation over (n=0)] f([aq.sup.n])[q.sup.n] (3.2) and [[integral].sup.[infinity].sub.0] f(t)[d.sub.q](t) = (1 - q) [[infinity].summation over (n = -[infinity])] f([q.sup.n]).[q.sup.n]. (3.3) In his recent work on the q-Gamma and q-Beta functions, Askey [2] has obtained analogues of several classical results about the gamma function including the Bohr-Mollerup theorem, the duplication formula and an asymptotic formula for large x. Using the q-binomial theorem [(at).sub.[infinity]]/[(a).sub.[infinity]] = [[infinity].summation over (n=0)] [(t).sub.n]/[(q).sub.n] x [a.sup.n] (3.4) and the definition of the q-gamma function, Askey [2] observed that [[GAMMA].sub.q](x) = [(q).sub.[infinity]][(1 - q).sup.1-x] [[infinity].summation over (n=0)] [q.sup.nx]/[(q).sub.n]. (3.5) In this section we establish a q-integral extension of (1.1) From (2.6) we have [[infinity].summation over (n=-[infinity])] [(-b).sup.n]/ [([-q.sup.n]/a).sub.[infinity]] = (1/b - 1/a)/(1 + 1/b) [(aq/b).sub.[infinity]][(bq/a).sub.[infinity]][(q).sub.[infinity]] /[(-1/a).sub.[infinity]][(-aq).sub.[infinity]] [(-bq).sub.[infinity]] (3.6) Changing b to [-q.sup.b] and using (3.3) and (3.1) in the above equation we obtain, [[integral].sup.[infinity].sub.0] [x.sup.b-1]/[(-x/a).sub.[infinity]] [d.sub.q](x) = [(1 - q).sup.b] [([-aq.sup.1-b]).sub.[infinity]][([q.sup.b]/a).sub.[infinity]]/ [(-1/a).sub.[infinity]][(-aq).sub.[infinity]] [[GAMMA].sub.q](b) (3.7) Putting a = 1/(1 - q) and letting q [right arrow] 1, we deduce that [GAMMA](b) = [[integral].sup.[infinity].sub.0] [e.sup.-x][x.sup.b-1]dx. 4. Some eta-Function Identities The Dedekind eta-function is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.1) where q = [e.sup.2[pi]i[tau]] and Im([tau]) > 0. The eta-function is useful in the study of modular forms In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally . From (1.1) and (1.2) we have [(aq/b).sub.[infinity]][(b/a).sub.[infinity]][(q).sub.[infinity]]/ [(-aq).sub.[infinity]][(-bq).sub.[infinity]] = (b + 1) [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(n+1)/2][a.sup.n][b.sup.-n]/[(-aq).sub.n] +(b + b/a) [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(n+1)/2][b.sup.n][a.sup.-n]/[(-bq).sub.n] (4.2) Putting a = [q.sup.1/2], b = [-q.sup.2] and then changing q to [q.sup.2] in (4.2) we obtain, [[eta].sup.2](2[tau])/[eta]([tau])[eta](4[tau]) = [q.sup.-1/24]/(1 + q)(1 - [q.sup.2]) [[infinity].summation over (n=0)] [q.sup.n(n-2)]/[([-q.sup.3]; [q.sup.2]).sub.n] - [q.sup.71/24]/(1 - [q.sup.2])(1 - [q.sup.4]) [[infinity].summation over (n=0)] [q.sup.n(n+4)]/[([q.sup.6]; [q.sup.2]).sub.n]. (4.3) Putting a = [q.sup.3/2], b = [-q.sup.2] and then changing q to [q.sup.2] in (4.2) we obtain, [eta](4[tau])[eta](2[tau])/[eta]([tau]) = [q.sup.29/24]/(1 + [q.sup.3])[(1 + q).sup.2] [[infinity].summation over (n=0)] [q.sup.n(n+3)]/[([-q.sup.5]; [q.sup.2]).sub.n] - [q.sup.-19/24] /(1 - [q.sup.2])(1 + q) [[infinity].summation over (n=0)] [q.sup.n(n-1)]/[([q.sup.3]; [q.sup.2]).sub.n]. (4.4) Putting a = [q.sup.-3/2], b = q and then changing q to [q.sup.2] in (4.2) we obtain, [eta]([tau])/[eta](2[tau]) = [q.sup.-71/24](1 + q) [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(n-4)]/ [(1/q; [q.sup.2]).sub.n] + [q.sup.119/24][(1 + q).sup.2]/(1 + [q.sup.2]) [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(n+6)]/[([-q.sup.4]; [q.supp.2]).sub.n]. (4.5) Putting a = [q.sup.-1/3], b = [q.sup.1/3] and then changing q to [q.sup.3] in (4.2) we obtain, [[eta].sup.2]([tau])[eta](6[tau])/[eta](3[tau])[eta](2[tau]) = [q.sup.1/8] (1 + [q.sup.3])/1 + q) [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(3n-1)/2]/[([-q.sup.2]; [q.sup.3]).sub.n] + [q.sup.9/8] [[infinity].summation over (n=0)] [(-1).sup.n][q.sup.n(3n+7)/2]/[([-q.sup.4]; [q.sup.3]).sub.n]. (4.6) Received July 31, 2004, Accepted October 20, 2004. References [1] G. E. Andrews, Ramanujan's "lost" notebook. I: Partial [theta Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ]-functions, Adv. in Math. 41 (1981), 137-172. [2] R. Askey, The q-gamma and q-beta functions, Applicable Anal anal (a´n'l) relating to the anus. a·nal adj. 1. Of, relating to, or near the anus. 2. . 8 (1978), 125-141. [3] B. C. Berndt, S. H. Chan, B. P. Yeap, and A. J. Yee, A reciprocity theorem for certain q-series found in Ramanujan's lost notebook Srinivasa Ramanujan's lost notebook is the manuscript in which Ramanujan, a widely admired Indian mathematician from Cambridge University, recorded the mathematical discoveries of the last year of his life. It was rediscovered by George Andrews in 1976, in a box of effects of G. N. , Ramanujan J. (to appear). [4] B. C. Berndt and L. C. Zhang, Ramanujan's identities for eta-functions, Math. Ann. 292 (1992), 561-573. [5] N. J. Fine, Basic Hypergeometric Series In mathematics, the basic hypergeometric series, also sometimes called the hypergeometric q-series, are q-analog generalizations of ordinary hypergeometric series. Two basic series are commonly defined, the unilateral basic hypergeometric series, and the and Applications, American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards to mathematicians. , Providence, Rhode Island  “Providence” redirects here. For other uses, see Providence (disambiguation). Providence is the capital and the most populous city of the U.S. , 1988. [6] M. E.-H. Ismail, A simple proof of Ramanujan's [sub.1][[psi].sub.1] sum, Proc. Amer. Math. Soc. 63 (1977), 185-186. [7] F. H. Jackson, On q-definite integrals, Quart quart: see English units of measurement. . J. Pure Appl. Math. 41 (1910), 193-203. [8] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Gesammelte Werke, Erster Band, G. Reimer, Berlin, 1881. [9] G. Kohler, Some eta-identities arisisng from theta series, Math. Scand. 66 (1990), 147-154. [10] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi New Delhi (dĕl`ē), city (1991 pop. 294,149), capital of India and of Delhi state, N central India, on the right bank of the Yamuna River. , 1988. [11] L. J. Rogers, On a three-fold symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. in the elements of Heine's series, Proc. London Math. Soc. (Ser. 3) 21 (1983),171-179. [12] D. D. Somashekara and S. N. Fathima, An Interesting generlization of Jacobi's triple product identity, Far East J. Math. Sci. 9 (2003), 255-259. [13] K. R. Vasuki, Basic bilateral hypergeometric series In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. The series, if convergent, will define a hypergeometric function summation formula and its applications, Math. Student 68 (1999), 211-218. [14] I. J. Zucker, Further relations amongst infinite series infinite series In mathematics, the sum of infinitely many numbers, whose relationship can typically be expressed as a formula or a function. An infinite series that results in a finite sum is said to converge (see convergence). One that does not, diverges. and products. II: The evaluation of 3-dimenstional lattice (theory) lattice - A partially ordered set in which all finite subsets have a least upper bound and greatest lower bound. This definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not sum, J. Phys. A: Math. Gen. 23 (1990), 117-132. C. Adiga * and N. Anitha ([dagger]) Department of Studies in Mathematics, University of Mysore University of Mysore is a public university in India. It has its main campus in the city of Mysore and extension campuses in the neighbouring districts of Hassan, and Mandya. Manasagangotri, Mysore 570006, India * E-mail:adiga@yahoo.com ([dagger]) E-mail:lohith@yahoo.com |
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