Koornwinder polynomials

Koornwinder polynomials

In mathematics, Koornwinder polynomials are a family of orthogonal polynomials in several variables, named for their discoverer Tom H. Koornwinder, that generalize the Askey-Wilson polynomials. They can also be viewed as Macdonald polynomials attached to the non-reduced root system of type BC, and in particular satisfy analogues of Macdonald's "conjectures". In addition, the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Koornwinder polynomials.

The Koorwinder polynomial in "n" variables associated to the partition λ is the unique Laurent polynomial invariant under permutation and inversion variables, with leading monomial "x"λ, and orthogonal with respect to the density

: prod_{1le i

on the unit torus

: |x_1|=|x_2|=cdots|x_n|=1.,

where the parameters satisfy the constraint

:|a|,|b|,|c|,|d|,|q|,|t|<1,

and (x;q)_infty denotes the infinite q-Pochhammer symbol.

References

*Citation | last1=Koornwinder | first1=Tom H. | title=Askey-Wilson polynomials for root systems of type BC | series=Contemp. Math. | volume=138 | year=1992 | pages=189-204

*Citation | last1=Koornwinder | first1=Tom H. | title=Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I | id=MathSciNet | id = 0340673 | year=1974 | journal=Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. | volume=36 | pages=48–58

*Citation | last1=Stokman | first1=Jasper V. | title=Laredo Lectures on Orthogonal Polynomials and Special Functions | publisher=Nova Sci. Publ. | location=Hauppauge, NY | series=Adv. Theory Spec. Funct. Orthogonal Polynomials | id=MathSciNet | id = 2085855 | year=2004 | chapter=Lecture notes on Koornwinder polynomials | pages=145–207


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Macdonald polynomials — In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal polynomials in several variables, introduced by Macdonald (1987). Macdonald originally associated his polynomials with weights λ of finite root systems and used just …   Wikipedia

  • 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… …   Wikipedia

  • Tom H. Koornwinder — is a Dutch mathematician at the Korteweg de Vries Institute of Mathematics who introduced Koornwinder polynomials.References* [http://www.science.uva.nl/ thk home page] *MathGenealogy|id=52441 …   Wikipedia

  • Hermite polynomials — In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical… …   Wikipedia

  • Classical orthogonal polynomials — In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical… …   Wikipedia

  • Chebyshev polynomials — Not to be confused with discrete Chebyshev polynomials. In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre s formula and which can be defined… …   Wikipedia

  • Dual Hahn polynomials — In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by for 0≤n≤N where λ(x)=x(x+γ+δ+1).… …   Wikipedia

  • Continuous Hahn polynomials — In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by Roelof Koekoek, Peter A.… …   Wikipedia

  • Continuous dual Hahn polynomials — In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by Roelof Koekoek, Peter A …   Wikipedia

  • Continuous q-Jacobi polynomials — In mathematics, the continuous q Jacobi polynomials P(α,β) n(x|q), introduced by Askey Wilson (1985), are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”