2011/12/21 by Changgui Zhang, Zhang, Changgui
Mathematics · #11F-xx #33D-05 #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Probability and Statistical Research #Quantum Algebra (math.QA) #math.NT #math.QA #msc:11F-xx #msc:33D-05
paper · pdf · doi:10.48550/arxiv.1112.4979
Comptes rendus - Mathématique, 2011
arxiv created 2011/12/21 · openalex publication_date 2011/12/21 · arxiv updated 2011/12/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let q=e2πiτ, \Imτ>0, x=e2πiξ∈\CC and (x;q)_∞=∏n≥ 0(1-xqn). Let (q,x)↦(q^*,ιq x) be the classical modular substitution given by q^*=e-2πi/τ and ιq x=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 (ιq x;q^*)_∞. Inspired by the work of Stieltjes on some semi-convergent series, we are led to a "closed" analytic formula for (x;q)_∞ by means of the dilogarithm combined with a Laplace type integral that admits a divergent series as Taylor expansion at log q=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 Dedekind's eta function, Jacobi theta function and also for certain Lambert series. Among other applications, one can remark that our results allow to obtain a Ramanujan's asymptotic formula about (x;q)_∞ for q→ 1.