Comptes Rendus
Number Theory/Mathematical Analysis
On the modular behaviour of the infinite product (1x)(1xq)(1xq2)(1xq3)
Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 725-730.

Let q=e2πiτ, τ>0, x=e2πiξC and (x;q)=n0(1xqn). Let (q,x)(q,ιqx) be the classical modular substitution given by q=e2πi/τ and ιqx=e2πiξ/τ. The main goal of this Note is to study the “modular behaviour” of the infinite product (x;q), this means, to compare the function defined by (x;q) with that given by (ιqx;q). Inspired by the work [16] of Stieltjes (1886) on some semi-convergent series, we are led to a “closed” analytic formula for the ratio (x;q)/(ιqx;q) by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at logq=0. Thus, we can obtain an expression linking (x;q) to its modular transform (ιqx;q) and which contains, in essence, the modular formulae known for Dedekindʼs eta function, Jacobi theta function and also for certain Lambert series. Among other applications, one can remark that our results allow one to interpret Ramanujanʼs formula (Berndt, 1994) [5, Entry 6, p. 265 & Entry 6′, p. 268] (see also Ramanujan, 1957 [10, pp. 365 & 284]) as being a convergent expression for the infinite product (x;q).

Soit q=e2πiτ, τ>0, x=e2πiξC et (x;q)=n0(1xqn). Soit (q,x)(q,ιqx) la substitution modulaire classique donnée par q=e2πi/τ et ιqx=e2πiξ/τ. Le principal but de la présente Note est dʼétudier le « comportemant modulaire » du produit infini (x;q), cʼest-à-dire, de comparer la fonction définie par (x;q) à celle par (ιqx;q). Inspiré du travail [16] de Stieltjes (1886) sur des séries semi-convergentes, nous somme parvenus à une formule analytique « explicite » pour le rapport (x;q)/(ιqx;q) au moyen du dilogarithme complété par une intégrale du type Laplace, cette dernière admettant une série divergente comme développement taylorien en logq=0. Ceci nous permet dʼobtenir une expression reliant (x;q) à sa transformée modulaire (ιqx;q) qui contient essentiellement les formules modulaires connues pour la fonction eta de Dedekind, la fonction theta de Jacobi et aussi pour certaines séries de Lambert. Parmi dʼautres applications, on remarquera que nos résultats permettent dʼinterpréter une formule de Ramanujan (Berndt, 1994) [5, Entry 6, p. 265 & Entry 6′, p. 268] (voir aussi Ramanujan, 1957 [10, pp. 365 & 284]) comme étant une expression convergente pour le produit infini (x;q).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.06.019

Changgui Zhang 1

1 Laboratoire P. Painlevé CNRS UMR 8524, UFR de mathématiques, université Lille 1 (USTL), cité scientifique, 59655 Villeneuve dʼAscq cedex, France
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Changgui Zhang. On the modular behaviour of the infinite product $ (1-x)(1-xq)(1-x{q}^{2})(1-x{q}^{3})\cdots $. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 725-730. doi : 10.1016/j.crma.2011.06.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.019/

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