Let , , and . Let be the classical modular substitution given by and . The main goal of this Note is to study the “modular behaviour” of the infinite product , this means, to compare the function defined by with that given by . Inspired by the work [16] of Stieltjes (1886) on some semi-convergent series, we are led to a “closed” analytic formula for the ratio by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at . Thus, we can obtain an expression linking to its modular transform 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 .
Soit , , et . Soit la substitution modulaire classique donnée par et . Le principal but de la présente Note est dʼétudier le « comportemant modulaire » du produit infini , cʼest-à-dire, de comparer la fonction définie par à celle par . Inspiré du travail [16] de Stieltjes (1886) sur des séries semi-convergentes, nous somme parvenus à une formule analytique « explicite » pour le rapport au moyen du dilogarithme complété par une intégrale du type Laplace, cette dernière admettant une série divergente comme développement taylorien en . Ceci nous permet dʼobtenir une expression reliant à sa transformée modulaire 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 .
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Changgui Zhang 1
@article{CRMATH_2011__349_13-14_725_0, author = {Changgui Zhang}, title = {On the modular behaviour of the infinite product $ (1-x)(1-xq)(1-x{q}^{2})(1-x{q}^{3})\cdots $}, journal = {Comptes Rendus. Math\'ematique}, pages = {725--730}, publisher = {Elsevier}, volume = {349}, number = {13-14}, year = {2011}, doi = {10.1016/j.crma.2011.06.019}, language = {en}, }
TY - JOUR AU - Changgui Zhang TI - On the modular behaviour of the infinite product $ (1-x)(1-xq)(1-x{q}^{2})(1-x{q}^{3})\cdots $ JO - Comptes Rendus. Mathématique PY - 2011 SP - 725 EP - 730 VL - 349 IS - 13-14 PB - Elsevier DO - 10.1016/j.crma.2011.06.019 LA - en ID - CRMATH_2011__349_13-14_725_0 ER -
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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