A modular-type formula for \((x;q)_\infty \)
- 65 Downloads
Abstract
Let \(q=\text {e}^{2\pi i\tau }, \mathfrak {I}\tau >0\), \(x=\text {e}^{2\pi i{z}}\), \({z}\in \mathbb {C}\), and \((x;q)_\infty =\prod _{n\ge 0}(1-xq^n)\). Let \((q,x)\mapsto ({q_1},{x_1})\) be the classical modular substitution given by the relations \({q_1}=\text {e}^{-2\pi i/\tau }\) and \({x_1}=\text {e}^{2\pi i{z}/{\tau }}\). The main goal of this paper is to give a modular-type representation for the infinite product \((x;q)_\infty \), this means, to compare the function defined by \((x;q)_\infty \) with that given by \(({x_1};{q_1})_\infty \). Inspired by the work (Stieltjes in Collected Papers, Springer, New York, 1993) of Stieltjes on semi-convergent series, we are led to a “closed” analytic formula for the ratio \((x;q)_\infty /({x_1};{q_1})_\infty \) by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at \(\log q=0\). Thus, the function \((x;q)_\infty \) is linked with its modular transform \(({x_1};{q_1})_\infty \) in such an explicit manner that one can directly find the modular formulae known for Dedekind’s Eta function, Jacobi Theta function, and also for certain Lambert series. Moreover, one can remark that our results allow Ramanujan’s formula (Berndt in Ramanujan’s notebooks, Springer, New York, 1994, Entry 6’, p. 268) (see also Ramanujan in Notebook 2, Tata Institute of Fundamental Research, Bombay, 1957, p. 284) to be completed as a convergent expression for the infinite product \((x;q)_\infty \).
Keywords
Modular elliptic functions Jacobi \(\theta \)-function Dedekind \(\eta \)-function Lambert series q-seriesMathematics Subject Classification
11F20 11F27 30D05 33E05 05A30Notes
References
- 1.Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1976)MATHGoogle Scholar
- 2.Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
- 3.Apostol, T.M.: Generalized Dedekind sums and transformation formulae of certain Lambert series. Duke Math. J. 17, 147–157 (1950)MathSciNetCrossRefMATHGoogle Scholar
- 4.Apostol, T.M.: Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums. J. Number Theory 15(1), 14–24 (1982)MathSciNetCrossRefMATHGoogle Scholar
- 5.Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory, 2nd edn. GTM 41. Springer, New York (1990)Google Scholar
- 6.Barnes, E.W.: The theory of the double gamma function. Philos. Trans. R. Soc. Lond. A 196, 265–388 (1901)CrossRefMATHGoogle Scholar
- 7.Berndt, B.C.: Ramanujan’s Notebooks, Part IV. Springer, New York (1994)Google Scholar
- 8.Birkhoff, G.D.: The generalized Riemann problem for linear differential equations and the allied problems for linear difference and \(q\)-difference equations. Proc. Am. Acad. 49, 521–568 (1913)CrossRefMATHGoogle Scholar
- 9.Di Vizio, L., Ramis, J.-P., Sauloy, J., Zhang, C.: Équations aux \(q\)-différences. Gaz. Math. (SMF) 96, 20–49 (2003)MATHGoogle Scholar
- 10.Euler, L.: Introducio in Analysin Infinitorum. Bousquet, Lausanne (1748)Google Scholar
- 11.Guinand, A.P.: On Poisson’s summation formula. Ann. Math. 42(2), 591–603 (1941)MathSciNetCrossRefMATHGoogle Scholar
- 12.Guinand, A.P.: Functional equations and self-reciprocal functions connected with Lambert series. Q. J. Math. Oxf. Ser. 15, 11–23 (1944)MathSciNetCrossRefMATHGoogle Scholar
- 13.Hardy, G.H.: Ramanujan, Twelve Lectures on Subjects Suggested by His Life and Work, Chelsea Publishing Company (originally published by Cambridge University Press), New York (1940)Google Scholar
- 14.Ismail, M.E.H., Zhang, C.: Zeros of entire functions and a problem of Ramanujan. Adv. Math. 209, 363–380 (2007)MathSciNetCrossRefMATHGoogle Scholar
- 15.Jackson, F.H.: The basic gamma-function and the elliptic functions. Proc. R. Soc. Lond. A 76, 127–144 (1905)CrossRefMATHGoogle Scholar
- 16.Martinet, J., Ramis, J.-P.: Problèmes de modules pour des équations différentielles non linéaires du premier ordre. Inst. Hautes Études Sci. Publ. Math. 55, 63–164 (1982)CrossRefMATHGoogle Scholar
- 17.McIntosh, R.J.: Some asymptotic formulae for \(q\)-shifted factorials. Ramanujan J. 3, 205–214 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 18.Ramanujan, S.: In: Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M. (eds.) Collected Papers of Srinivasa Ramanujan. Chelsea Publishing Company, New York (Second printing of the 1927 original) (1927)Google Scholar
- 19.Ramanujan, S.: Notebook 2. Tata Institute of Fundamental Research, Bombay (1957, reprint by Springer) (1984)Google Scholar
- 20.Ramis, J.-P.: Séries divergentes et théories asymptotiques. Bull. Soc. Math. Fr. 121 (Panoramas et Synthèses, suppl.), 74 (1993)Google Scholar
- 21.Ramis, J.-P., Sauloy, J., Zhang, C.: Local analytic classification of \(q\)-difference equations. http://front.math.ucdavis.edu/0903.0853. arXiv: 0903.0853 (2009)
- 22.Sauloy, J.: Galois theory of Fuchsian \(q\)-difference equations. Ann. Sci. École Norm. Sup. 36, 925–968 (2003)MathSciNetCrossRefMATHGoogle Scholar
- 23.Selberg, A.: Reflections on Ramanujan centenary. In: Berndt, B.C., Rankin, R.A. (eds.) Ramanujan: Essays and Surveys, pp. 203–214. History of Mathematics 22. Hindustan Book Agency, New Delhi (1989)Google Scholar
- 24.Serre, J.-P.: Cours d’arithmétique. Deuxième édition revue et corrigé. Presses Universitaires de France, Paris (1977)Google Scholar
- 25.Shintani, T.: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(1), 167–199 (1977)MathSciNetMATHGoogle Scholar
- 26.Siegel, C.L.: A simple proof of \(\eta (-1/\tau )=\eta (\tau ){\sqrt{\tau /2}}\). Mathematica 1, 4 (1954)Google Scholar
- 27.Stieltjes, T.J.: Collected Papers, vol. II. Springer, New York (1993)MATHGoogle Scholar
- 28.Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1937)MATHGoogle Scholar
- 29.Ueno, K., Nishizawa, M.: Mltiple gamma functions and multiple \(q\)-gamma functions. Publ. RIMS 33, 813–838 (1997)CrossRefMATHGoogle Scholar
- 30.Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Reprinted Cambridge University Press, New York (1962)Google Scholar
- 31.Zagier, D.: The dilogarithm function in geometry and number theory. In: Number Theory and Related Topics, vol. 12, pp. 231–249. Tata Institute of Fundamental Research Studies in Mathematics/Oxford University Press, Bombay/Oxford (1989)Google Scholar
- 32.Zhang, C.: Développements asymptotiques \(q\)-Gevrey et séries \(Gq\)-sommables. Ann. Inst. Fourier 49, 227–261 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 33.Zhang, C.: Sur les fonctions \(q\)-Bessel de Jackson. J. Approx. Theory 122, 208–223 (2003)MathSciNetCrossRefMATHGoogle Scholar
- 34.Zhang, C.: On the modular behaviour of the infinite product \((1-x)(1-xq)(1-xq^2)(1-xq^3)...\), C. R. Acad. Sci. Paris Ser. I 349, 725–730 (2011)MathSciNetCrossRefMATHGoogle Scholar
- 35.Zhou, S., Luo, Z., Zhang, C.: On summability of formal solutions to a Cauchy problem and generalization of Mordell’s theorem. C. R. Math. Acad. Sci. Paris 348, 753–758 (2010)MathSciNetCrossRefMATHGoogle Scholar