Stieltjes–Wigert polynomials explained

Stieltjes–Wigert polynomials should not be confused with Stieltjes 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]

w(x) = (k)/(\sqrt{\pi) } x^ \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 S _{n(x;q)} = (1)/((q;q) _{n}) {} _{1\phi} ^{- n} ^{1 (q} , 0 ;q, - q ^{n+1} x),

where

q = \exp \left( - (1)/(2 k ^{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

(1)/(( - x, - qx ^{-1} ;q) _{infty}) and(k)/(\sqrt{\pi) } x^ \exp \left(-k^2 \log^2 x \right) .

References

• • • • • • Xiang-Sheng . Wang. Roderick . Wong. Uniform asymptotics of some q-orthogonal polynomials. J. Math. Anal. Appl. . 2010 . 364 . 1. 79–87 . 10.1016/j.jmaa.2009.10.038 .

Notes and References

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 S_n(x;q)=p_n(q^−1/2x) for p_n(x) in Szegő (1975), Section 2.7.