# Stieltjes–Wigert polynomials

(Redirected from Stieltjes-Wigert polynomial)

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 [1]

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

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[2]

${\displaystyle \displaystyle S_{n}(x;q)={\frac {1}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n},0;q,-q^{n+1}x),}$

where

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

## 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

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

and

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

## Notes

1. ^ Up to a constant factor this is w(q−1/2x) for the weight function w in Szegő (1975), Section 2.7. See also Koornwinder et al. (2010), Section 18.27(vi).
2. ^ Up to a constant factor Sn(x;q)=pn(q−1/2x) for pn(x) in Szegő (1975), Section 2.7.