Stieltjes-Wigert Polynomial

Orthogonal Polynomials associated with Weighting Function

$$w(x)=\pi^{-1/2} k\mathop{\rm exp}\nolimits (-k^2\ln^2 x)=\pi^{-1/2}k x^{-k^2\ln x}$$ (1)

for $x\in(0,\infty)$ and $k>0$. Using

$$\left[{\matrix{n\cr \nu\cr}}\right] = {(1-q^n)(1-q^{n-1})\cdots (1-q^{n-\nu+1})\over (1-q)(1-q^2)\cdots (1-q^\nu)}$$ (2)

where $0<\nu<n$,

$$\left[{\matrix{n\cr 0\cr}}\right]=\left[{\matrix{n\cr n\cr}}\right]=1,$$ (3)

and

$$q=\mathop{\rm exp}\nolimits [-(2k^2)^{-1}].$$ (4)

Then

$$p_n(x)=(-1)^n q^{n/2+1/4}[(1-q)(1-q^2)\cdots(1-q^n)]^{-1/2}\... ... \left[{\matrix{n\cr \nu\cr}}\right] q^{\nu^2}(-q^{1/2} x)^\nu$$ (5)

for $n>0$ and

$$p_0(x)= q^{1/4}.$$ (6)

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 33, 1975.