A New Product Formula for $$(z;q)_\infty $$, with Applications to Asymptotics

Abstract

<jats:p> We express the <jats:italic>q</jats:italic> -Pochhammer symbol <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(z;q)_\infty $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> as an infinite product of gamma functions, analogously to how Narukawa expressed the elliptic gamma function as an infinite product of hyperbolic gamma functions. This identity is used to obtain asymptotic expansions when <jats:italic>q</jats:italic> tends to 1. </jats:p>