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Andrius Kulikauskas
 m a t h 4 w i s d o m  g m a i l
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正交多項式
How does the MacMahon Master theorem relate to orthogonal Sheffer polynomials? (in Zeng's 1997 paper) And what does that say about the role of symmetric functions of eigenvalues?
Jacobi polynomials
 Jacobi polynomials (such as the physicist Hermite polynomials) include a factor of 2x. These seem related to probability. But also for the geometry, as with the Legendre polynomials. Can we use the same combinatorics as Kim Zeng, but just alter by substitution for y=2x? and multiplying each polynomial by a fraction? Can we compare with Viennot's interpretation? Do they relate physics with probability?
 Combinatorially, Jacobi configuration is a pair of Laguerre configurations (injections) back and forth. Is the Jacobi configuration like an adjunction?
 How are the combinatorics of the Hermite (involution  one dimension), Laguerre (rook polynomials  two dimension) and Jacobi polynomials related? Are the Jacobi polynomials related to three dimensions?
Legendre polynomials
 How is the sequence that I used to get the Legendre polynomials related to approximating pi ?
 Interpret the Legendre and Laguerre polynomials and relate that to the Hermite polynomials.
Recurrence relations
 In orthogonal polynomials, understand {$x P_n(x)$} as the basis for the recurrence relation. It is the basis for the definition of {$P_n(x)$} in terms of {$x$}. {$x$} stands for the gap, encodes the gap between levels above and below. There is a hierarchy of gaps.
 Consider why the recurrence relations are based on second derivatives (slack as in symplectic geometry?) and go back only two values and how that restricts to objects (linear) and their relationships (quadratic). How does that relate to the Yoneda lemma?
Viennot
 Understand how Viennot's interpretation of orthogonal polynomials relates to taking steps S, E, EES.
 Relate Viennot's histoire with automata theory.
 Interpret Viennot's tableaux by rotating them 45 degrees so that they are like the Young tableaux and the binomial triangle.
Treelike tableaux
 What are treelike tableaux ? Aval, Boussicault, Nadeau 2011
读物
 维基百科: Orthogonal polynomials
 Kim, Zeng. A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials Walks on trees?
 Foata, Zeilberger. Laguerre polynomials, weighted derangements, and positivity.
 D.Stanton. Orthogonal polynomials and combinatorics.
 T.S.Chihara. 45 years of orthogonal polynomials: a view from the wings
 维基百科: Favard's theorem. A sequence of polynomials satisfying a suitable 3term recurrence relation is a sequence of orthogonal polynomials.
 On the unimodality and combinatorics of Bessel numbers. Ji Young Choi, Jonathan D.H. Smith The Bessel number {$B(n;k)=B_{k,2kn}$} is the numberof partitions of an nset into k nonempty subsets, each of size at most 2.
 Garsia, Remmel. qRook polynomials https://arxiv.org/pdf/math/9706219.pdf
 Chihara, Ismail. Orthogonal Polynomials Suggested by a Queueing Model.
 Gruenbaum. Random walks and orthogonal polynomials: Some challenges.
 Jacobi operator The case a(n) = 1 is known as the discrete onedimensional Schrödinger operator.
 C. Vignat1, P. W. Lamberti. A study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces No access.
 Arias. A Simple application of orthogonal polynomials to Physics. 2015.
 Alhaidari. Quantum mechanics with orthogonal polynomials. 2017. Abdulaziz D. Alhaidari website
 Alhaidari. Special orthogonal polynomials in quantum mechanics. 2019.
 Satoru Odake. Exactly solvable discrete quantum mechanical systems and multiindexed orthogonal polynomials of the continuous Hahn and Meixner–Pollaczek types
 Smirnov, Del Sol Mesa. Orthogonal Polynomials of Discrete Variable Associated with Quantum Algebras {$SU_q(2)$} and {$SU_q(1,1)$}.
 Van Diejen, Kirillov. A Combinatorial Formula for the Associated Legendre Functions of Integer Degree.
 Bob Griffiths. Stochastic processes with orthogonal polynomial eigenfunctions.
 Wim Schoutens. Stochastic Processes and Orthogonal Polynomials.
 Cotfas. Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics. 2004. No access.
 Iyela et al. Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials. 2013. No access.
X.Viennot. The cellular ansatz: bijective combinatorics and quadratic algebra RobinsonSchenstedKnuth correspondence, trees and tableau, PASEP, quadratic algebra.
RazumovStroganov conjecture
F Colomo1, A G Pronko. The role of orthogonal polynomials in the sixvertex model and its combinatorial applications.
 The Hankel determinant representations for the partition function and boundary correlation functions of the sixvertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1, 2 and 3enumerations of alternating sign matrices (ASMs), these polynomials specialize to classical ones (continuous Hahn, Meixner–Pollaczek and continuous dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASM enumerations turns out to be possible, leading also to some new results such as the refined 3enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the onepoint boundary correlation functions of the homogeneous one.
 Vertex model
 Sixvertex model
 X.Viennot. Introduction to Enumerative, Algebraic, and Bijective Combinatorics
 X.Viennot. A combinatorial theory of orthogonal polynomials and continued fractions.
 X.Viennot. Combinatorics And Quadratic Algebras.
 X.Viennot. The birth of new domain: Combinatorial Physics
 Combinatorics and physics
 Christian Brouder. Quantum field theory meets Hopf algebra. Relates Feynman diagrams and semistandard Young tableaux.
 Gérard Henry Edmond Duchamp, H. Cheballah. Some Open Problems in Combinatorial Physics.
 Multiplicities in diag.
 Combinatorics of RiordanSheffer oneparameter groups.
 A corpus for combinatorial vector fields.
 Probabilistic study of approximate substitutions.
 Combinatorics, Physics and their Interactions Journal.
 Ismail. Classical and Quantum Orthogonal Polynomials in One Variable.
 Josef Meixner. Symmetric systems of orthogonal polynomials.
 Josef Meixner. Orthogonale polynomsysteme mit besonderen Gestalt der erzeugenden Funktion. Journal of the London Mathematical Society. January, 1934.
 Sheffer sequence Isador Sheffer
 Nima ArkaniHamed. Research Skills and Theoretical Physics.
 Gabor Hetyei. Shifted Jacobi polynomials and Delannoy numbers
 Reply to this question: https://math.stackexchange.com/questions/1161128/integralsoforthogonalpolynomialsandcombinatorics
 https://www.worldcat.org/title/orthogonalpolynomialstheoryandpracticeproceedingsofthenatoadvancedstudyinstitutecolumbusohio1989/oclc/472860183&referer=brief_results
 https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula used to establish the existence of Sheffer's polynomials.
 https://www.worldcat.org/title/classicalandquantumorthogonalpolynomialsinonevariable/oclc/928941543&referer=brief_results
 https://www.worldcat.org/title/orthogonalpolynomialstheoryandpracticeproceedingsofthenatoadvancedstudyinstitutecolumbusohio1989/oclc/472860183&referer=brief_results
 https://www.amazon.com/IntroductionOrthogonalPolynomialsDoverMathematics/dp/0486479293
 Wolfram Koepf, Dieter Schmersau. Recurrence equations and their classical orthogonal polynomial solutions.
 Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra
 Solving Special Function Equations Using Lie Symmetries
 http://wwwusers.math.umn.edu/~mille003/lietheoryspecialfunctions.html
 https://link.springer.com/book/10.1007/9789400905016 Orthogonal Polynomials, Theory and Practice, ed. Paul Nevai
 https://www.sciencedirect.com/science/article/pii/S0001870816310453?via%3Dihub Orthogonal polynomials through the invariant theory of binary forms, P.Petrullo, D.Senato, R.Simone
Nikiforov, Suslov, Uvarov. Classical Orthogonal Polynomials of a Discrete Variable.
Kim, Zeng. A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials.
Dennis Stanton. Publication list.
Jiang Zeng
Chebyshev polynomials, Wallis integrals
 https://arxiv.org/pdf/1004.2453.pdf Relates Wallis inequality, Schur functions, Feynman diagrams.
 http://www2.math.uu.se/~svante/papers/sj114.pdf Traveling Fly problem yields Wallis fraction.
 https://arxiv.org/pdf/1510.00399.pdf Recurrence relation for Wallis fraction: z_n+2 = 1/z_n+1 + z_n
 Mama Foupouagnigni. On characterization theorems for orthogonal polynomials on nonuniform lattices: the functional approach. Includes Pearson function.
 Orthogonal Polynomials: Theory and Practice Various articles available online.
 Dennis Stanton. An Introduction to Group Representations and Orthogonal Polynomials.
 Ian Macdonald. Orthogonal polynomials associated with root systems.
 Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of Winvariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q,t1,t2,...,tr, where r (=1,2 or 3) is the number of Worbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and padic symmetric spaces. Also when R=S is of type An, they conincide with the symmetric polynomials.
 https://en.wikipedia.org/wiki/Jacobi_operator
 https://en.wikipedia.org/wiki/Spectral_theorem
 Jang Soo Kim. Combinatorics of orthogonal polynomials. There is a video.
 Jang Soo Kim
 Renato ́AlvarezNodarse. On some applications of Orthogonal Polynomials in MathematicalPhysics Alberto Grunbaum one time said (in a summer school in 2004):“Special functions are to mathematics what pipes are to a house: nobodywants to exhibit them openly but nothing works without them”.
 Danilo Latini, D.Riglioni. From ordinary to discrete quantum mechanics: The Charlier oscillator and its coalgebra symmetry The coalgebraic structure of the harmonic oscillator is used to underline possible connections between continuous and discrete superintegrable models which can be described in terms of SUSY discrete quantum mechanics. A set of 1parameter algebraic transformations is introduced in order to generate a discrete representation for the coalgebraic harmonic oscillator. This set of transformations is shown to play a role in the generalization of classical orthogonal polynomials to the realm of discrete orthogonal polynomials in the Askey scheme. As an explicit example the connection between Hermite and Charlier oscillators, that share the same coalgebraic structure, is presented and a twodimensional maximally superintegrable version of the Charlier oscillator is constructed.
 Richard Askey, Sergei K. Suslov. The qharmonic oscillator and an analogue of the Charlier polynomials.
 Richard Askey explained why hypergeometric functions appear so frequently in mathematical applications: "Riemann showed that the requirement that a differential equation have regular singular points at three given points and every other complex point is a regular point is so strong a restriction that the differential equation is the hypergeometric equation with the three singularities moved to the three given points. Differential equations with four or more singular points only infrequently have a solution which can be given explicitly as a series whose coefficients are known, or have an explicit integral representation. This partly explains why the classical hypergeometric function arises in many settings that seem to have nothing to do with each other. The differential equation they satisfy is the most general one of its kind that has solutions with many nice properties. Special functions: group theoretical aspects and applications.
 How do regular singular points and Riemann's differential equation relate to the eccentricities of the conics (0,1,infinity)? and to the Mobius transformations?
例子
Various kinds of orthogonal polynomials
Askey scheme for organizing orthogonal polynomials and also their qanalogues
Classical orthogonal polynomials
 Jacobi polynomials  orthogonal on [1,1]
 Gegenbauer polynomials
 Legendre polynomials
 Chebyshev polynomials  trigonometric
 Laguerre polynomials (and associated Laguerre polynomials)  orthogonal on [0,infinity)  Hydrogen atom
 Hermite polynomials  orthogonal on (infinity, infinity)  harmonic oscillator
Wilson polynomials, which generalize the Jacobi polynomials. They include many orthogonal polynomials as special cases:
 Meixner–Pollaczek polynomials
 continuous Hahn polynomials
 continuous dual Hahn polynomials
 classical polynomials described by the Askey scheme
Askey–Wilson polynomials introduce an extra parameter q into the Wilson polynomials
 The Chebyshev polynomials of the second kind are orthogonal polynomials with respect to the Wigner semicircle distribution.
Orthogonal polynomials on the unit circle
On the cylinder?
And sphere?
Macdonald polynomials are orthogonal polynomials in several variables, depending on the choice of an affine root system.
 Jack polynomials
 Hall–Littlewood polynomials
 Heckman–Opdam polynomials
 Koornwinder polynomials
 Askey–Wilson polynomials are the special case of Macdonald polynomials for a certain nonreduced root system of rank 1.
 Laguerre {$(n+1)L_{n+1}(x) + nL_{n1}(x) = (2n+1x)L_n(x)$}
 probabilist Hermite {$\mathit{He}_{n+1}(x) + n\mathit{He}_{n1}(x) = x\mathit{He}_n(x)$}
 physicist Hermite {$H_{n+1}(x) + 2nH_{n1}(x) = 2xH_n(x)$}
 Legendre {$(n+1)P_{n+1}(x) + n P_{n1}(x)= (2n+1) x P_n(x)$}
 Bessel polynomials {$xZ_{\alpha+1}(x) + xZ_{\alpha  1}(x) = 2\alpha Z_{\alpha}(x)$}
 Jacobi polynomials {$2n (n + \alpha + \beta) (2n + \alpha + \beta  2) P_n^{(\alpha,\beta)}(z)$}
{$+ 2 (n+\alpha  1) (n + \beta1) (2n+\alpha + \beta) P_{n2}^{(\alpha, \beta)}(z)$}
{$= (2n+\alpha + \beta1) \Big\{ (2n+\alpha + \beta)(2n+\alpha+\beta2) z + \alpha^2  \beta^2 \Big\} P_{n1}^{(\alpha,\beta)}(z) $}
 Bessel functions of the first kind
{$$J_\alpha(x)=\sum_{m=0}^{\infty}\frac{(1)^m}{m!\;\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}$$}
{$$(2m+n)!2^{2m+n}J_{n,2m+n}=\binom{2m+n}{m}(1)^m$$}
{$$(2m+n)!2^{2m+n}\binom{2m+2n}{m+n}J_{n\frac{1}{2},2m+n}=\binom{2m+n}{m}(1)^m4^{m+n}$$}
 AskeyWilson polynomials subsume the various orthogonal polynomials. They are defined by five parameters. They have a combinatorial interpretation in terms of PASEP and their moments are given by staircase tableaux.
Chebyshev polynomials
 The Wallis integrals are helpful for interpreting the orthogonality of the Chebyshev polynomials of the second kind. I am trying to interpret their values, the fractions of the double factorials. I need to understand Euler's Beta function and Gamma function.
Theoretical Physics Reference: Hypergeometric functions
 {${}_0F_1$} confluent hypergeometric limit function
 {${}_1F_1$} Kummer’s confluent hypergeometric function of the first kind
 {${}_2F_1$} Gauss’ hypergeometric function
General information about orthogonal polynoials
Recurrence relation
 Orthogonal polynomials {$P_n$} satisfy a recurrence relation of the form {$P_{n}(x)=(A_{n}x+B_{n})P_{n1}(x)+C_{n}P_{n2}(x)$}
 {$x$} is the raising operator which defines {$x p_n(x)$} in terms of {$p_{n+1}$}, {$p_n$}, {$p_{n1}$}. {$\frac{\textrm{d}}{\textrm{dx}}$} is the lowering operator which defines {$y'=\frac{\textrm{d}y}{\textrm{dx}}$} in terms of {$y''$}, {$y'$}, {$y$}.
 We can think of {$x$} as moving from {$xP_n(x)$} to {$P_{n1}$}. Which is why it does not vanish, why it does not go to zero by linearization.
 The middle term is defined with regard to the terms above and below. This is like Wildberger's mutation game defined for Dynkin diagrams that are a chain. The chain organizes dimensions, higher and lower.
 Viennot video. Combinatorial interpretation of Favard theorem (about the recurrence formula for orthogonal polynomials). The duality between the combinatorics of moments and the combinatorics of the coefficients of orthogonal polynomials is the duality between elementary symmetric functions and homogeneous symmetric functions. (But which way?)
 Tridiagonal matrices are related to PASEP, random matrix theory, orthogonal polynomials.
Raising and lowering operators
Polya urn, AS  SA = I, PASAP, related to raising and lowering operators, and the twelvefold way
Note that q = 1 is bosonic (commutator), q = 1 is fermionic (anticommutator), and also there is q = 0.
quadratic algebra Q  Qtableaux  bijection  representation of Q by combinatorial operators 
UD = qDU + I  permutations, towers placement  Robinson–Schensted–Knuth correspondence  pairs of Young tableaux 
DE = qED + E + D  alternative tableaux  Exchangefusion algorithm  Laguerre histories  permutations, orthogonal polynomials, data structures "histoires" 
What is the combinatorics of convex spaces and how does that relate to orthogonal polynomials, which give different ways of looking at the geometry?
Other topics
 Orthogonal polynomials are related to continued fractions.
 Also related to the factorization of free monoids.
