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Stieltjes–Wigert polynomials
In mathematics, Stieltjes–Wigert polynomials (named after T. J. Stieltjes and Carl Severin Wigert) are family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function
\( w(x) = \frac{1}{\sqrt{\pi}}k\exp(-k^2\log(x)^2) \)
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).
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
\( \displaystyle S_n(x;q) = \frac{1}{(q;q)_n)}{}_1\phi_1(q^{-n},0;q,-q^{n+1}x) \)
(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
\( \frac{k}{\sqrt{\pi}}\exp(-k^2(\log x)^2) \)
References
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248
Szegő, Gábor (1939), Orthogonal Polynomials, Colloquium Publications - American Mathematical Society, ISBN 978-0-8218-1023-1, MR 0372517
Stieltjes, T. -J. (1894), "Recherches sur les fractions continues" (in French), Ann. Fac. Sci. Toulouse VIII: 1–122, JFM 25.0326.01, MR 1344720
Wigert, S. (1923), "Sur les polynomes orthogonaux et l'approximation des fonctions continues" (in French), Arkiv för matematik, astronomi och fysik 17: 1–15, JFM 49.0296.01
When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.
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