XAVIER VIENNOT
  • PART I
    • abstract
    • contents
    • Ch0 Introduction to the course
    • Ch1 Ordinary generating functions
    • Ch2 The Catalan garden
    • Ch3 Exponential structures and genarating functions
    • Ch4 The n! garden
    • Ch5 Tilings, determinants and non-intersecting paths
    • list of bijections
    • index
  • PART II
    • abstract
    • contents
    • Ch1 Commutations and heaps of pieces: basic definitions
    • Ch2 Generating functions of heaps of pieces
    • Ch3 Heaps and paths, flows and rearrangements monoids
    • Ch4 Linear algebra revisited with heaps of pieces
    • Ch5 Heaps and algebraic graph theory
    • Ch6 Heaps and Coxeter groups
    • Ch7 Heaps in statistical mechanics
    • Some lectures related to the course
  • PART III
    • abstract
    • contents
    • Ch0 overview of the course
    • Ch1 RSK The Robinson-Schensted-Knuth correspondence
    • Ch2 Quadratic algebra, Q-tableaux and planar automata
    • Ch3 Tableaux for the PASEP quadratic algebra
    • Ch4 Trees and tableaux
    • Ch5 Tableaux and orthogonal polynomials
    • Ch6 Extensions: tableaux for the 2-PASEP quadratic algebra
    • Lectures related to the course
    • references, comments and historical notes
  • PART IV
    • introduction
    • contents
    • Ch0 Overview of the course
    • Ch1 Paths and moments
    • Ch2 Moments and histories
    • Ch3 Continued fractions
    • Ch4 Computation of the coefficients b(k) lambda(k)
    • Ch5 Orthogonality and exponential structures
    • Ch6 q-analogues
    • Lectures related to the course
    • Complements
    • references
  • Epilogue

The Art of Bijective Combinatorics    Part IV 

Combinatorial theory of orthogonal polynomials and continued fractions

The Institute of Mathematical Sciences, Chennai, India  (January-March 2019) 

Chapter 5   Orthogonality and exponential structures

 

Chapter 5a

 February 25 , 2019

 

slides of Ch5a   (pdf 23 Mo )                

video Ch5a  link to YouTube  (from IMSc Matsciencechanel Playlist)

 

This lecture is dedicated to my dear friend Pierre Leroux

 

species and exponential structures   4

Pierre Leroux:  souvenirs ...  5-11

Hypergeometric series and orthogonal polynomials   12

the Askey scheme of hypergeometric orthogonal polynomials   13

notations for hypergeometric power series   15

Gauss hypergeometric series   16

Orthogonal Sheffer polynomials   18

definition of Sheffer polynomials   20

Meixner theorem: characterization of orthogonal Sheffer polynomials   20-21

the five orthogonal Sheffer polynomials   22

Remonding Part I, Ch 3  (species and exponential generating functions)   24

Combinatorial interpretation of Hermite polynomials   50

Mehler identity for Hermite polynomials (Foata)   55

Combinatorial interpretation of Laguerre polynomials   63

Laguerre configuration   66

Combinatorial interpretation of Charlier polynomials   70

Charlier configuration   72

Combinatorial interpretation of Jacobi polynomials (Foata-Leroux)   76

A formula expressing the exponential generating function of Jacobi polynomials as a triple product   76

change of variables with homogenous Jacobi polynomials   80

Jacobi configurations   82

the weight of a Jacobi configurations   87

interpretation of the triple product   92

proof of the triple product formula   92-113

Combinatorial proof of a limit formula (from Jacobi to Laguerre)   

The end   119

 

 

Chapter 5b

 February 28 , 2019

 

slides of Ch5b   (pdf,  20 Mo)                

video Ch5b  link to YouTube  (from IMSc Matsciencechanel Playlist)

 

Back to Ch 5a   3

About teh combinatoiral proof of Mehler formula   12

Reminding Jacobi configurations   23

Combinatorial interpretations of Meixner polynomials (Foata,Labelle)   40 

Meixner configurations   42

limit formula for Meixner formula   51

interpretation of Meixner polynomials with colored premutations   55

a third interpretation of Meixner polynomials   60

Kreweras polynomials   66

Octopus   (Bergeron)   68

interpretation of Gegenbauer   74

interpretation of Meixner-Pollaczek polynomials   78

Pairs of permutations    (J.Labelle-Y.N.Yeh) 81

interpretation of Meixner-Pollaczek polynomials   84

interpretation of Krawtchouk polynomials   87

interpretation of Hahn polynomials   89

the tableau of limit formulae   93

Sheffer polynlomials and delta operators  (summary of Ch 5c)  98

The end 105

  

Chapter 5c

 March 13 , 2019

 

slides of Ch5c   (pdf,  19 Mo)                

video Ch5c  link to YouTube  (from IMSc Matsciencechanel Playlist)

 

Orthogonal Sheffer polynomials   3

Delta operators and umbral operators   8

an example of umbral calculus: Bernoulli polynomials   10-11

Gian-Carlo Rota  13-14

Sheffer polynlomials: definition with delta operators   16

binomial type polynomials: definition  17

shift-invariant operators  18

delta operatiors: definition   19

basic sequence for Q delta operators   20

isomorphism shift-invariant operators -- formal powers series   21

Sheffer polynomials: definition with 2 delta operators S and Q   22

exponential generating function for Sheffer polynomials   24

inverse sequence of a Sheffer sequence   26

Riordan arrays   28

Appell sequence  30

Inverse sequence of orthogonal polynlomials (from Ch 1d)   32

Reversing the paths interpreting mu_n,i   39

Laguerre histories and restricted Laguerre histories  (Ch2b, 19-23)   44

Delta operators Q and S for Laguerre polynomials   49

Delta operators Q and S for general Sheffer polynomials   54

Delta operators Q and S for the 5 Sheffer orthogonal polynomials   62

in conclusion: delta operators S and Q interperted with left and right subtrees   68

The end 73