# Stieltjes–Wigert polynomials

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## 4 matching pages

##### 1: 18.27 $q$-Hahn Class
###### §18.27(vi) Stieltjes–WigertPolynomials
18.27.18 $S_{n}\left(x;q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell^{2}}(-x)^{\ell}}{\left(q;% q\right)_{\ell}\left(q;q\right)_{n-\ell}}=\frac{1}{\left(q;q\right)_{n}}{{}_{1% }\phi_{1}}\left({q^{-n}\atop 0};q,-q^{n+1}x\right).$
18.27.19 $\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\,\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},$
18.27.20 $\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)\,% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.$
##### 2: 18.1 Notation
• StieltjesWigert: $S_{n}\left(x;q\right)$.

• ##### 3: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
18.29.2 $Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},$ $n\to\infty$; $z,a,b,c,d,q$ fixed.
For a uniform asymptotic expansion of the StieltjesWigert polynomials, see Wang and Wong (2006). …
##### 4: Bibliography W
• Z. Wang and R. Wong (2006) Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.