In mathematics, Stieltjes–Wigert polynomials (named after

Thomas Jan Stieltjes Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics a ...

and Carl Severin Wigert) are a family of basic hypergeometric

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

in the basic

Askey scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , ...

, for the weight function :

w(x) = \frac x^ \exp(-k^2\log^2 x)

on the positive real line ''x'' > 0. The

moment problem In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ''μ'' to the sequences of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) ...

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 In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums : \left\, to be dense in a weighted L2 space on the real line. It was discovered by Mark Krein in the 1940s. A corollary, also called K ...

). 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 function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...

s and the

Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \ ...

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

\displaystyle   S_n(x;q) = \frac_1\phi_1(q^,0;q,-q^x),

where :

q = \exp \left(-\frac \right) .

Orthogonality

Since the

for these polynomials is indeterminate there are many different weight functions on ,∞for which they are orthogonal. Two examples of such weight functions are :

\frac

and :

\frac x^ \exp \left(-k^2 \log^2 x \right) .

Notes

References

* * * * * * * {{DEFAULTSORT:Stieltjes-Wigert polynomials Orthogonal polynomials Special hypergeometric functions