Stieltjes–Wigert polynomials

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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 [1]

 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[edit]

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by[2]

\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[edit]

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)

Notes[edit]

  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.

References[edit]