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Stieltjes–Wigert polynomials

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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

Contents

w ( x ) = k π 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

S n ( x ; q ) = 1 ( q ; q ) n 1 ϕ 1 ( q − n , 0 ; q , − q n + 1 x ) ,

where

q = exp ⁡ ( − 1 2 k 2 ) .

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

1 ( − x , − q x − 1 ; q ) ∞

and

k π x − 1 / 2 exp ⁡ ( − k 2 log 2 ⁡ x ) .

References

Stieltjes–Wigert polynomials Wikipedia
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