• XAVIER VIENNOT
  • Foreword
    • Preface
    • Introduction
    • Acknowledgements
    • Lectures for wide audience
  • PART I
    • Preface
    • 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
    • Lectures related to the course
    • List of bijections
    • Index
  • PART II
    • Preface
    • 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
    • Lectures related to the course
  • PART III
    • Preface
    • 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
    • Preface
    • 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 Ekalavya  (IMSc Media Center)
video Ch5a  link to YouTube  (from IMSc Matsciencechanel Playlist)
video Ch5a link to bilibili
This lecture is dedicated to my dear friend Pierre Leroux
species and exponential structures   4     1:48
Pierre Leroux:  souvenirs ...  5-11     2:20
Hypergeometric series and orthogonal polynomials   14     7:36
the Askey scheme of hypergeometric orthogonal polynomials   15     7:41
definiiton: hypergeometric power series   16     7:51
notations for hypergeometric power series   17     8:41
Gauss hypergeometric series   18     9:52
Vandermonde-Chu, Kummer, Pfaff-Saalschütz identities   19      11:04
Orthogonal Sheffer polynomials   20     12:04
definition of Sheffer polynomials   21     12:12
Meixner theorem: characterization of orthogonal Sheffer polynomials   22-23     13:36
the five orthogonal Sheffer polynomials   24     14:23
Remonding Part I, Ch 3  (species and exponential generating functions)   26     15:00
Combinatorial interpretation of Hermite polynomials   52     39:32
Mehler identity for Hermite polynomials (Foata)   57     42:48
Combinatorial interpretation of Laguerre polynomials   65     49:51
Laguerre configuration   68     51:26
Combinatorial interpretation of Charlier polynomials   72     55:48
Charlier configuration   74     56:23
Combinatorial interpretation of Jacobi polynomials (Foata-Leroux)   79     59:22
A formula expressing the exponential generating function of Jacobi polynomials as a triple product   81     1:00:36
change of variables with homogenous Jacobi polynomials   82     1:01:31
Jacobi configurations   84     1:04:00
the weight of a Jacobi configurations   89     1:07:09
combinatorial interpretation of Jacobi polynomials   93     1:11:19
interpretation of the triple product   94     1:12:12
proof of the triple product formula   95-113     1:14:42

relation between  Phi_alpha  and  Phi_a     99     1:16:00
expression for  Phi_alpha   101     1:17:28

Jacobi contraction     103     1:19:33
Jacobi arborescence   107     1:24:03
end of the proof   113     1:27:32
a formula of Leroux and Strehl   114     1:27:58
Combinatorial proof of a limit formula (from Jacobi to Laguerre)   115     1:28:13
The end   119     1:30:35

Chapter 5b

 February 28 , 2019
slides of Ch5b   (pdf,  20 Mo)                
video Ch5b  link to Ekalavya  (IMSc Media Center)
video Ch5b  link to YouTube  (from IMSc Matsciencechanel Playlist)
video Ch5b link to bilibili
Back to Ch 5a   3     0:22
About the combinatoiral proof of Mehler formula for Hermite polynomials   12     7:22
comparison bijective proofs and analytic proofs of Mehler formula  (Watson, Erdélyi, Mehler, ...)     18-21     11:11
multilinear extensions (Foata, Garsia)   22     15:39
Reminding Jacobi configurations   23     18:13
Combinatorial interpretations of Meixner polynomials (Foata, J.Labelle)   40     29:01
Meixner configurations   42     30:18
the weight of a Meixner configuration   45     33:06
proposition: combinatoiral interpretation  of Meixner polynomials   47     34:28
limit formula for Meixner formula   51     39:52
interpretation of Meixner polynomials with colored premutations   55     43:32
a third interpretation of Meixner polynomials   60     50:12
Kreweras polynomials   66     55:53
Octopus   (Bergeron)   68     1:00:27
interpretation of Gegenbauer   74     1:04:20
interpretation of Meixner-Pollaczek polynomials   78     1:07:48
Pairs of permutations    (J.Labelle-Y.N.Yeh) 81     1:09:01
interpretation of Meixner-Pollaczek polynomials   84     1:12:09
interpretation of Krawtchouk polynomials   87     1:14:28
interpretation of Hahn polynomials   89     1:15:19
the tableau of limit formulae   93     1:18:10
Sheffer polynlomials and delta operators  (summary of Ch 5c)  98     1:20:10
The end 105     1:25:56

Chapter 5c

 March 13 , 2019
slides of Ch5c   (pdf,  19 Mo)                
video Ch5c  link to Ekalavya  (IMSc Media Center)
video Ch5c  link to YouTube  (from IMSc Matsciencechanel Playlist)
video Ch5c link to bilibili
Orthogonal Sheffer polynomials   3     0:28
definition: Sheffer polynomials, binomial type polynomials   4     0:32
Meixner'd theorem: characterization of orthogonal Sheffer polynomials   5-6     1:14
Delta operators and umbral operators   8     3:14
an example of umbral calculus: Bernoulli polynomials   10-11     4:34
discussion   6:43
Gian-Carlo Rota, Stanley and Garsia  13-14     8:20
Sheffer polynlomials: definition with delta operators   16     12:07
binomial type polynomials: definition  17     12:14
shift-invariant operators  18     13:37
delta operators: definition   19     15:47
basic sequence for Q delta operator   20     17:44
isomorphism shift-invariant operators -- formal powers series   21     19:31
discussion     20:59
exponential generating function for binomial type polynomials   22     22:55
Sheffer polynomials: definition with delta operator Q   22     24:05
characterization of Sheffer polynomials with two delta operators S and Q   23     25:34
exponential generating function for Sheffer polynomials   24     28:26
definition: inverse polynomials   25     30:31
inverse sequence of a Sheffer sequence   26     30:53
Riordan arrays   28     32:46
Appell sequence  30     35:03
Inverse sequence of orthogonal polynlomials (from Ch 1d)   32     36:54
Reversing the paths interpreting mu_n,i   39     38:49
Laguerre histories and restricted Laguerre histories  (Ch2b, 19-23)   44     42:30
Delta operators Q and S for Laguerre polynomials   49     44:14
Delta operators Q and S for general Sheffer polynomials   54     49:50
Delta operators Q and S for the 5 Sheffer orthogonal polynomials   62     54:10
in conclusion: delta operators S and Q interpreted with left and right subtrees   68-71 (slides not in the video)
The end 73     1:01:36