Mean-preserving spread: Difference between revisions

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{{distinguish|Stieltjes polynomial}}
{{for|the generalized Stieltjes–Wigert polynomials|q-Laguerre 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 <ref>Up to a constant factor this is ''w''(''q''<sup>-1/2</sup>''x'') for the weight function ''w'' in Szegő (1975), Section 2.7.
See also Koornwinder et al. (2010), Section 18.27(vi).</ref>
:<math> w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)</math>
on the positive real line ''x''&nbsp;>&nbsp;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.
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==Definition==
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The polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by<ref>Up to a constant factor ''S''<sub>''n''</sub>(''x'';''q'')=''p''<sub>''n''</sub>(''q''<sup>-1/2</sup>''x'') for ''p''<sub>''n''</sub>(''x'') in Szegő (1975), Section 2.7.</ref>
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:<math>\displaystyle  S_n(x;q) = \frac{1}{(q;q)_n)}{}_1\phi_1(q^{-n},0;q,-q^{n+1}x) </math>
(where ''q''&nbsp;=&nbsp;''e''<sup>{{frac|&minus;1|(2''k''<sup>2</sup>)}}</sup>).


==Orthogonality==
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Since the moment problem for these polynomials is indeterminate there are many different weight functions on [0,∞] for which they are orthogonal.  
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Two examples of such weight functions are
:<math>\frac{1}{(-x,-qx^{-1};q)_\infty}</math>
and
:<math>\frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2 \log^2 x)</math>


==Notes==
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{{reflist}}
 
==References==
 
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
*{{dlmf|title=Ch. 18, Orthogonal polynomials|url=http://dlmf.nist.gov/18|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | publisher=Colloquium Publications 23, American Mathematical Society, Fourth Edition| isbn=978-0-8218-1023-1 | mr=0372517 | year=1975}}
*{{Citation | last1=Stieltjes | first1=T. -J. | title=Recherches sur les fractions continues | url=http://www.numdam.org/numdam-bin/item?id=AFST_1995_6_4_1_J1_0 | language=French | jfm=25.0326.01 | mr = 1344720 | year=1894 | journal=Ann. Fac. Sci. Toulouse | volume=VIII | pages=1–122}}
*{{Citation | last1=Wigert | first1=S. | title=Sur les polynomes orthogonaux et l'approximation des fonctions continues | language=French | jfm=49.0296.01 | year=1923 | journal=Arkiv för matematik, astronomi och fysik | volume=17 | pages=1–15}}
 
{{DEFAULTSORT:Stieltjes-Wigert polynomials}}
[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]

Latest revision as of 12:14, 8 October 2014


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