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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 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). : 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 :\displaystyle S_n(x;q) = \frac{1}{(q;q)_n}{}_1\phi_1(q^{-n},0;q,-q^{n+1}x),

where

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

Notes

References

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References

Up to a constant factor ''S''_''n''(''x'';''q'')=''p''_''n''(''q''^−1/2''x'') for ''p''_''n''(''x'') in Szegő (1975), Section 2.7.

Wikipedia Source

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orthogonal-polynomials special-hypergeometric-functions

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