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
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