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

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Ch. 18, Orthogonal polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Szegő, Gábor (1975), Orthogonal Polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, ISBN 978-0-8218-1023-1, MR 0372517
• Stieltjes, T. -J. (1894), "Recherches sur les fractions continues", Ann. Fac. Sci. Toulouse (in French), VIII (4): 1–122, doi:10.5802/afst.108, JFM 25.0326.01, MR 1344720
• Wang, Xiang-Sheng; Wong, Roderick (2010). "Uniform asymptotics of some q-orthogonal polynomials". J. Math. Anal. Appl. 364 (1): 79–87. doi:10.1016/j.jmaa.2009.10.038.
• Wigert, S. (1923), "Sur les polynomes orthogonaux et l'approximation des fonctions continues", Arkiv för matematik, astronomi och fysik (in French), 17: 1–15, JFM 49.0296.01