On the modular behaviour of the infinite product ( 1 − x ) ( 1 − x q ) ( 1 − x q 2 ) ( 1 − x q 3 ) ⋯

Let q = e 2 π i τ , ℑ τ > 0 , x = e 2 π i ξ ∈ C and ( x ; q ) ∞ = ∏ n ⩾ 0 ( 1 − x q n ) . Let ( q , x ) ↦ ( q ⁎ , ι q x ) be the classical modular substitution given by q ⁎ = e − 2 π i / τ and ι q x = e 2 π i ξ / 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 ( ι q x ; 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 ) ∞ / ( ι q x ; q ⁎ ) ∞ by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at log q = 0 . Thus, we can obtain an expression linking