Stieltjes–Wigert 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

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

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

[math]\displaystyle{ \displaystyle S_n(x;q) = \frac{1}{(q;q)_n}{}_1\phi_1(q^{-n},0;q,-q^{n+1}x), }[/math]

where

[math]\displaystyle{ q = \exp \left(-\frac{1}{2k^2} \right) . }[/math]

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

[math]\displaystyle{ \frac{1}{(-x,-qx^{-1};q)_\infty} }[/math]

and

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

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8 
• 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 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/ 
• Szegő, Gábor (1975), Orthogonal Polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, ISBN 978-0-8218-1023-1 
• Stieltjes, T. -J. (1894), "Recherches sur les fractions continues" (in French), Ann. Fac. Sci. Toulouse VIII (4): 1–122, doi:10.5802/afst.108, http://www.numdam.org/numdam-bin/item?id=AFST_1995_6_4_1_J1_0 
• 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" (in French), Arkiv för matematik, astronomi och fysik 17: 1–15