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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.

Mathematics Encyclopedia

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