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
Weeks 1-3:

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.

Week 4:

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.

Week 5:

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