Scaled Asymptotics For Some $q$-Series As $q$ Approaching Unit

Abstract

In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\to 1$. These $q$-series generalize Ramanujan function $A_q(z)$; Jackson's $q$-Bessel function $J_{\nu }^{(2)}$(z;q), Ismail-Masson orthogonal polynomials($q^{-1}$-Hermite polynomials) $h_n(x|q)$, Stieltjes-Wigert orthogonal polynomials $S_n(x;q)$, $q$-Laguerre orthogonal polynomials $L_n^{(\alpha )}(x;q)$ and confluent basic hypergeometric series.

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