Stieltjes–Wigert polynomials

Not to be confused with Stieltjes polynomial.

For the generalized Stieltjes–Wigert polynomials, see q-Laguerre polynomials.

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \displaystyle S_n(x;q) = \frac{1}{(q;q)_n}{}_1\phi_1(q^{-n},0;q,-q^{n+1}x)

(where q = e^{​−1⁄_(2k2)}).

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

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

and

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