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 6 q-analogues of some orthogonal polynomials
Chapter 6a
March 4 , 2019
slides of Ch6a (pdf 20 Mo )
video Ch6a link to Ekalavya (IMSc Media Center)
video Ch6a link to YouTube (from IMSc Matsciencechanel Playlist)
video Ch6a link to bilibili
q-analogue, n! and binomials coefficients 3 0:31
6 q-analogues of orthogonal polynomials 11 9:12
continuous and discrete Hermite, Charlier and Laguerre polynomials
scheme of basic hypergeometric orthogonal polynomials 13 12:55
discussion: what makes a "good" q-analogue 14:05
Continuous q-Hermite polynomials 14 16:48
continuous q-Hermite polynomials (Hermite I) 15
recalling Hermite histories 16-37 17:06
q-analogue of Hermite histories 38 21:20
crossing number 42 22:19
proposition: interpretation of the moments of continuous q-Hermite I with crossings 44 23:23
q-analogue of Hermite histories with nestings 45 24:07
moments of continous q-Hermite 60 27:16
formula of these moments 62 27:55
the philosophy of histoires and its q-analogues 63 28:57
exercise: the double distribution (crossings, nestings) 70 33:25
q-analogue of continous Hermite polynomials 71 33:48
proposition: interpretation of the coefficients of continous Hermite 83 38:48
Discrete q-Hermite (Hermite II) 86 42:25
proposition: formula for the moments of discrete q-Hermite 88 44:13
number of "inversions" Inv(I) of a chord diagram I 95 45:25
relation relating Inv(I) and number of crossings and nestings 97 46:16
proof of the formula for the moments of discrete q-Hermite 98-102 47:50
a funny relation between continuous and discrete q-Hermite 103 50:36
q-Charlier polynomials 104 51:42
Discrete q-Charlier (Charlier II) (de Médicis, Stanton, White) 108 54:42
formula expressing discrete q-Charlier polynomials 111 56:27
interpretation of the discrete q-Charlier polynomials 112 58:01
formula for the moments of discrete q-Charlier polynomials with q-Stirling numbers 113 58:48
restricted growth functions for set partitions 114 59:47
the parameter rs for restricted growth functions (Wachs, White) 116 1:02:05
0-1 tableaux (Leroux) 119 1:04:35
proposition: interpretation of the moments of discre q-Charlier with the parameter rs 120 1:05:26
classical q-Charlier polynomials (Zeng) 122-123 1:08:26
Continuous q-Charlier (Charlier I) (Kim, Stanton, Zeng) 124 1:09:40
formula for expressing the q-Charlier polynomials 126 1:10:42
definition of the index w(k) for a permutation (Simion, Stanton) 127-128 1:11:20
proposition: combinatorial interpretation of the continuous q-Chalier polynomials (with Simion-Stanton) 128 1:12:34
formula for the moments mu_n(a;q) 129 1:13:38
Chapter 6b
March 11 , 2019
slides of Ch6b (v2, pdf 29Mo)
video Ch6b link to Ekalavya (IMSc Media Center)
video Ch6b link to YouTube (from IMSc Matsciencechanel Playlist)
video Ch6b link to bilibili
Reminding Ch 6a 3 0:24
Basic hypergeometric series 14 6:58
Continuous q-Laguerre polynomials (q-Laguerre I) 18 12:13
Al-Salam - Chihara polynomials 21 14:08
formula for the q-Laguerre polynomials (Kasraoui, Stanton, Zeng ; Simion, Stanton) 22-23 15:06
Moments of the continous q-Laguerre polynomials 24 17:01
weighted q-Laguerre histories 30 22:30
lemma: relation with the patterns 31-2 31 23:27
proposition: interpretation of the moments with the patterns 31-2 35 31:29
formula for the moments of the continous q-Laguerre polynomials (Corteel, Josuat-Vergès, Prellberg, Rubey) 36 32:22
Continous q-Laguerre polynomials with parameter beta 38 33:37
proposition: interpretation of the moments of the continuous q-Laguerre with parameter beta 42 38:58
Subdivided Laguerre histories (and its q-analogues) 43 39:57
q-analogue of Euler's (Stieljes) continued fraction for n! 48-49 41:33
Bijection subdivided Laguerre histories H -- restricted Laguerre histories h (Ch 3b, 82-91) 51 44:19
Bijection permutations sigma -- subdivided Laguerre histories H (Ch 3b, 43-72) 58 47:35
parameter q and nestings of the associated pairs of Hermite histories 64 51:58
parameter q and crossings of the associated pairs of Hermite histories 65 53:08
relation with the number of crossings of a permutation (Corteel) 66-67 54:10
corollary: interpretation of the moments of continuous q-Laguerre polynomials with numbers of crossing of permutations 68 55:52
the commutative diagram: permutations sigma - pairs of Hermite histories - subdived Laguerre histories H - restricted Laguerre histories h 70 56:18
q-moments with the eulerian parameter y 71 58:02
Interpretation with Laguerre heaps of segments
bijection restricted Laguerre histories h - Laguerre heaps of segments E (Ch2c, p95-104, p113-133) 72 58:48
number of crossings of a Laguerre heap of segments 85 1:02:36
intepretation with Laguerre heaps of the moments of the continuous q-Laguerre polynomials with parameter beta 86 1:03:34
Bijection restricted Laguerre histories h -- permutations tau (Ch 3b, 127-129) 88 1:07:09
commutative diagram h, E, tau, inverse of tau 91 1:07:58
q-analogue of Euler continued fraction with parameter beta 93 1:09:50
interpretation of the beta-q parameters on the (A,K) chord diagrams related to the subdivided Laguerre history 98 1:11:50
discrete q-Laguerre polynomials (Laguerre II) 99 1:13:13
definition of discrete q-Laguerre polynomials with the 3 terms recurrence relation 100 1:13:18
relation between the number of inversions of a permutation and
the number of crossing and nesting of the associated pair of Hermite histories 103 1:14:28
proposition: expresssion for the moments of discrete q-Laguerre polynomials (Heine, Biane) 105 1:17:12
proposition: expresssion for the moments of discrete beta-q-Laguerre polynomials 106 1:18:22
Bijective proof for the Askey-Wilson integral (Ismail, Stanton, X.V.) 108 1:18:53
Askey-Wilson polynomials (definition) 110 1:19:07
orthogonality of Askey-Wilson polynomials 111 1:19:30
the Askey-Wilson integral 112 1:19:54
bijective proof: the Askey-Wilson integral as a product of four continuous q-Hermite polynomials 115 1:21:05
PASEP and orthogonal polynomials 117 1:22:50
The end 126 1:26:15