Stieltjes–Wigert polynomials

Template:Distinguish Template:For In mathematics, Stieltjes–Wigert polynomials (named after Thomas Jan Stieltjes and Carl Severin Wigert) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function

$w(x)=\frac{k}{\sqrt{\pi }}{x}^{-1/2}\exp (-k^2{\log }^{2}x)$

on the positive real line x > 0.

The moment problem for the Stieltjes–Wigert polynomials is indeterminate; in other words, there are many other measures giving the same family of orthogonal polynomials (see Krein's condition).

Koekoek et al. (2010) give in Section 14.27 a detailed list of the properties of these polynomials.

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

$S_n(x;q)=\frac{1}{(q;q{)}_n}{}_1{\varphi }_1(q^{-n},0;q,-q^{n+1}x),$

where

$q=\exp (-\frac{1}{2k^2}).$

Orthogonality

Since the moment problem for these polynomials is indeterminate there are many different weight functions on [0,∞] for which they are orthogonal. Two examples of such weight functions are

$\frac{1}{(-x,-q{x}^{-1};q{)}_{\mathrm{\infty }}}$

and

$\frac{k}{\sqrt{\pi }}{x}^{-1/2}\exp (-k^2{\log }^{2}x).$

References

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