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Noncommutative Biorthogonal Polynomials

Noncommutative Biorthogonal Polynomials. by Emily Sergel Thanks to Prof Robert Wilson. Introduction to Orthogonal Polynomials. Suppose we have W(x) so that for any polynomial f(x), ∫ f(x)W(x) dx ≥ 0 Then <f(x),g(x)> = ∫ f(x)g(x)W(x) dx defines an inner product

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Noncommutative Biorthogonal Polynomials

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  1. Noncommutative Biorthogonal Polynomials by Emily Sergel Thanks to Prof Robert Wilson

  2. Introduction to Orthogonal Polynomials • Suppose we have W(x) so that for any polynomial f(x), ∫f(x)W(x) dx ≥ 0 • Then <f(x),g(x)> = ∫f(x)g(x)W(x) dx defines an inner product • A sequence of polynomials {pn(x)} is orthogonal with respect to <·,·> if <pn(x), pm(x)> = 0 whenever n≠m • The nth moment is ∫xnW(x) dx

  3. Introduction cont. • Given a sequence {Si}, we can define an inner product with ith moment Si and we can also create a sequence of orthogonal polynomials in the following way:

  4. The Noncommutative Problem • Suppose {Si} do not commute • We still want to create orthogonal polynomials in this manner but we have a problem with the determinant • Many properties of determinants rely heavily on the commutativity of the matrix entries

  5. Solution! • Much study has gone into developing an appropriate analogue to determinants and quasideterminants appear to be the correct solution. • Quasideterminants preserve several determinant properties: • A matrix with 2 equal rows or columns has all quasideterminants equal to 0 • Cramer’s rule • and many more

  6. Quasideterminants • Each nxn matrix has n2 quasideterminants • The formula has some involved notation but the following are good illustrations:

  7. Constructing Noncommutative Orthogonal Polynomials • Just as we hoped, we can make functions that are orthogonal by looking at a quasideterminant of a particular matrix • The following sequence of polynomials {πn(x)} is orthogonal with moments {Si}

  8. Biorthogonal Polynomials • Let Iab = ∫∫ xaybK(x,y) dα(x)dβ(y) for K satisfying a few additional conditions • {Iab} is the set of bimoments with respect to K(x,y) • Just as before, we can define an inner product and create sets of polynomials with nice properties

  9. Biorthogonal Polynomials cont. • Define {pn(x)},{qn(y)} as follows: • ∫∫ pn(x)qm(y)K(x,y) dα(x)dβ(y) = 0 if n≠m

  10. Recurrence relations • In all three situations, the polynomials generated obey recurrence relations. • For the first case, commutative orthogonal polynomials, the relation is as follows: pn+1(x) = (anx+bn)pn(x) + cnpn-1(x) • The recurrences in the other cases are more complex, but in the same spirit.

  11. Our goals • To understand biorthogonal functions from a more algebraic standpoint • To define something new – noncommutative biorthogonal functions – in a meaningful way • To prove that these new functions have similar constructions and recurrence relations.

  12. References • Bertola, M., M. Gekhtman, and J. Szmigielski. "Cauchy Biorthogonal Polynomials." ArXiv.org. 16 April 2009. 11 June 2009 <arXiv:hep-th/9407124v1>. • Gelfand, Israel, D. Krob, Alain Lascoux, B. Leclerc, V. S. Retakh, and J. Y. Thibon. "Noncommutative Symmetric Functions." ArXiv.org. 20 July 1994. 11 June 2009 <arXiv:hep-th/9407124v1>. • "Orthogonal polynomial." Wikipedia, the free encyclopedia. 27 May 2009. 11 June 2009 <http://en.wikipedia.org/wiki/Orthogonal_polynomial>. • “Quasideterminant.” Wikipedia, the free encyclopedia. 13 May 2009. 11 June 2009 <http://en.wikipedia.org/wiki/Quasideterminant>.

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