The idea of orthogonal polynomials is well established and many applications exist.
As stated on the above page, we can construct polynomials that are orthogonal with respect to some inner product by taking the determinant of a particular matrix.
In 1994, Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon generalized this idea to non-commutative orthogonal polynomials by instead examining a particular quasidetermiant of the same matrix.
In the recent paper, "Cauchy Biorthogonal Polynomials", Bertola, Gekhtman and Szmigielski generalize orthogonal polynomials in a different way to biorthogonal polynomials.
The goal of my REU project is to explore this new concept and to again generalize to noncommutative biorthogonal polynomials. I am working with Prof Wilson in pursuing this goal.
In all existing generalizations, the polynomials have satisfied either 3 or 4 term recurrence relations.
Our best hope is that we can define noncommutative biorthogonal polynomials in a way that will agree with both the definition of noncommutative polynomials and that of the biorthogonal polynomials while still preserving some sort of recurrence relations.
As part of the REU, each student gives 2 presentations. The first is an introduction to their project. The second summarizes their results. Below is my first presentation. At the end of the summer I will post my second presentation.Introductory Presentation
I spent this time learning the background material. Perhaps the first 2 weeks were spent understanding quasideterminants. The last understanding the existing proofs for properties of orthogonal polynomials and noncommutative polynomials.
I am currently attempting to understand the proofs in the paper "Cauchy Biorthogonal Polynomials". It seems that this paper only proves a recurrence relation for biorthogonal polynomials with what the authors call the "Cauchy Kernel". This might mean that recurrence relations are not possibe at my desired level of generality. It could also mean that I will need to spend some time proving a general recurrence relation in the commutative case.
I am also starting to work on finding the proper way of defining noncommutative biorthogonal polynomials. I am generally following the paper "Noncommutative Symmetric Functions" but I am beginning to think the anti-automorphism * is not necessary here or that the set-up in "Cauchy Biorthogonal Polynomials" must be slightly adapted.
I now feel comfortable with the paper "Cauchy Biorthogonal Polynomials" but it has not enlightened my current pursuit. I am still working on a general recurrence relation for the commutative case. If I find such a thing and prove it I will begin work on a proof for the noncommutative case. At this point it seems more likely that I will show that a finite termed recurrence is not possible with total generality or identify cases in which some recurrence does hold. I will most likely spend the rest of this and perhaps next week on the recurrence formulas. I feel that given a week I could write up what I believe the definition for the noncommutative biorthogonal polynomials should be and give justification along with proof of the orthogonality condition.
Week 6: I no longer believe a general recurrence is possible. I am, however, generalizing a particular case (the so called "cauchy kernel"). I have proved the cauchy kernel recurrence for the noncommutative case as well as recurrences for various other kernels in both the commutative and noncommutative biorthogonal case. I hope to spend some time working towards a version of Favard's Theorem.
Week 7: I no longer believe that a generalization of Favard's theorem is possible for biorthogonal polynomials. A proof which is extremely similar for the orthogonal case does work for the noncommutative orthogonal case. I spent this week working on Favard's (with failure) and also putting together slides for the presentation on Friday.
Bertola, Gekhtman, Szmigielski "Cauhy Biorthogonal Polynomials" Link
Chihara "An Introduction to Orthogonal Polynomials" ISBN 0-677-04150-0.
Gelfand, Gelfand, Retakh, Wilson "Quasideterminants" Link
Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon "Noncommutative Symmetric Functions" Link
Wilson "Three Lectures on Quasideterminants" Link