# Combinatorics of Orthogonal Polynomials

## Table of Contents

## 1. General Information

### 1.1. Class information

- Class code: Topics in Algebra MTH4031-41
- Instructor: Jang Soo Kim (김장수)
- email: name@skku.edu, where name=jangsookim
- Classroom: 31351A
- Class time: Tue 16:30-17:45, Thu 15:00-16:15
- Office hour: Tue 14:00-15:00
- Lecture homepage: https://jangsookim.github.io/lectures/OPS/ops.html

### 1.2. Overview

Orthogonal polynomials are classical objects arising from the study of continued fractions. They have become an important subject in many areas of mathematics. In this class we will learn fascinating combinatorial properties of orthogonal polynomials.

### 1.3. Useful sources

- Viennot's lecture videos https://viennot.org/abjc4-contents.html
- Viennot. Une théorie combinatoire des polynomes orthogonaux généraux. Lecture Notes, UQAM, 1983
- Chihara. An introduction to orthogonal polynomials. Gordon and Breach Science Publishers, New York, 1978
- Ismail. Classical and Quantum Orthogonal Polynomials in One Variable, 2005
- Stanley. Enumerative Combinatorics, Volumes 1 and 2

### 1.4. Grading

- The grade will be determined by class participation and homework.
- Attendance: 50
- Homework: 50 (student presentation may be included)

### 1.5. Language

- The class will be taught in English.

## 2. Lecture notes

- Lecture notes (This file will be updated regularly.)

## 3. Homework

- Homework problems (This file will be updated regularly.)
- Homework solutions with grading criteria (This file will be updated regularly.)
- Please email your homework as a single pdf file to the instructor by the due date.

## 4. Student presentation

- Dates: Nov 28, Nov 30, Dec 5, Dec 7
- Each student will give a presentation for 20 min.
- You can use a board or a beamer.
- Possible papers to present:
- de Medicis, Stanton. 1996. Combinatorial orthogonal expansions
- Kim. 1997. On Combinatorics of Al-Salam Carlitz Polynomials
- Kim, Stanton. Combinatorics of orthogonal polynomials of type R1
- Kim, Stanton, Zeng. 2006. The combinatorics of the Al-Salam-Chihara q-Charlier polynomials
- Kim, Stanton. 2014. Moments of Askey–Wilson polynomials
- Donghyun Kim. Combinatorial formulas for the coefficients of the Al-Salam-Chihara polynomials
- Kasraoui, Stanton, Zeng. 2011. The combinatorics of Al-Salam–Chihara q-Laguerre polynomials
- Ismail, Kasraoui, Zeng. 2013. Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials
- Kamioka. 2007. A combinatorial representation with Schröder paths of biorthogonality of Laurent biorthogonal polynomials
- Kamioka. 2008. A combinatorial derivation with Schröder paths of a determinant representation of Laurent biorthogonal polynomials
- Ismail, Stanton, Viennot. 1987. The combinatorics of the q-Hermite polynomials and the Askey-Wilson integral
- Hwang, Kim, Oh, Yu. 2020. On linearization coefficients of q-Laguerre polynomials
- Corteel, Nunge. 2020. Combinatorics of the 2-species exclusion processes, marked Laguerre histories, and partially signed permutations
- Corteel, Stanley, Stanton, Williams. 2012. Formulae for Askey-Wilson moments and enumeration of staircase tableaux
- Corteel, Williams. 2007. A Markov chain on permutations which projects to the PASEP
- Corteel, Williams. 2011. Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials
- Jang, Kim, Kim, Song, Song. 2023. Negative moments of orthogonal polynomials
- Foata. 1978. A combinatorial proof of the Mehler formula
- Ito. Generalized Schroder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials
- Josuat-Verges. 2011. Combinatorics of the three-parameter PASEP partition function
- Josuat-Verges. 2011. Rook placements in Young diagrams and permutation enumeration
- Josuat-Verges, Rubey. 2011. Crossings, Motzkin paths and moments
- Josuat-Verges. 2010. A q-enumeration of alternating permutations
- Eu, Fu. 2005. A Simple Proof of the Aztec Diamond Theorem
- Lin, Kim. 2022. A combinatorial bijection on k-noncrossing partitions

## 5. Schedule and links to lecture videos

- lecture notes, video : Introduction,
- lecture notes, video : Sign-reversing involutions,
- lecture notes, video : The moment functional and orthogonality, Existence of OPS,
- lecture notes, video : The fundamental recurrence,
- lecture notes, video : Christoffel-Darboux identities and zeros of OP,
- lecture notes, video : Formal power series and generating functions,
- lecture notes, video : Dyck paths, Motzkin paths, set partitions, matchings,
- lecture notes, video : Permutations,
- lecture notes, video : Combinatorial models for OPS 1,
- lecture notes, video : Combinatorial models for OPS 2,
- lecture notes, video : Moments of Chebyshev and Hermite polynomials,
- lecture notes, video : Moments of Charlier and Laguerre polynomials,
- lecture notes, video : Duality between mixed moments and coefficients,
- lecture notes, video : Special Cases: \( e_k \), \( h_k \), and binomial coefficients,
- lecture notes, video : Special Cases: q-binomial coefficients and Stirling numbers,
- lecture notes, video : Lindstrom-Gessel-Viennot lemma,
- lecture notes, video : Hankel determinants of moments,
- lecture notes, video : Duality between mixed moments and coefficients,
- lecture notes, video : Continued fractions 1,
- lecture notes, video : Continued fractions 2,
- lecture notes, video : Determinants and disjoint cycles,
- lecture notes, video : Symmetric orthogonal polynomials,
- lecture notes, video : Linearization coefficients 1,
- lecture notes, video : Linearization coefficients 2,