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