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正交多項式

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 3-term 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 {$B(n;k)=B_{k,2k-n}$} is the numberof partitions of an n-set into k nonempty subsets, each of size at most 2.
• Garsia, Remmel. q-Rook 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 one-dimensional 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 multi-indexed 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 Robinson-Schensted-Knuth correspondence, trees and tableau, PASEP, quadratic algebra.

• Chapter 5a: Tableaux and Orthogonal Polynomials
• Chapter 5b: Tableaux and Orthogonal Polynomials
• Chapter 5c: Tableaux and Orthogonal Polynomials

Razumov-Stroganov conjecture

• Combinatorial proof by Cantini, Sportielo
• Alternating Sign Matrices
• Motivated by study of {$\lambda$}-determinants.
• Schur function

F Colomo1, A G Pronko. The role of orthogonal polynomials in the six-vertex model and its combinatorial applications.

• The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex 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 3-enumerations 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 3-enumerations 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 one-point boundary correlation functions of the homogeneous one.
• Vertex model
• Six-vertex 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 Riordan-Sheffer one-parameter 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 Arkani-Hamed. Research Skills and Theoretical Physics.
• Gabor Hetyei. Shifted Jacobi polynomials and Delannoy numbers
• Reply to this question: https://math.stackexchange.com/questions/1161128/integrals-of-orthogonal-polynomials-and-combinatorics
• https://www.worldcat.org/title/orthogonal-polynomials-theory-and-practice-proceedings-of-the-nato-advanced-study-institute-columbus-ohio-1989/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/classical-and-quantum-orthogonal-polynomials-in-one-variable/oclc/928941543&referer=brief_results
• https://www.worldcat.org/title/orthogonal-polynomials-theory-and-practice-proceedings-of-the-nato-advanced-study-institute-columbus-ohio-1989/oclc/472860183&referer=brief_results
• https://www.amazon.com/Introduction-Orthogonal-Polynomials-Dover-Mathematics/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://www-users.math.umn.edu/~mille003/lietheoryspecialfunctions.html
• https://link.springer.com/book/10.1007/978-94-009-0501-6 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.

• Stanton. Orthogonal polynomials and combinatorics.
• Corteel, Kim, Stanton. Moments of Orthogonal Polynomials and Combinatorics.
• Karlin, McGregor. The Differential Equations of Birth-and-Death Processes, and the Stieltjes Moment Problem
• Karlin, McGregor. The Classification of Birth and Death Processes
• Karlin, McGregor. Many server queueing processes with Poisson input and exponential service time.
• M/M/1 Queue
• We can write a probability mass function dependent on t to describe the probability that the M/M/1 queue is in a particular state at a given time. We assume that the queue is initially in state i and write p_k(t) for the probability of being in state k at time t. Then<ref>{{cite book | title = Queueing Systems Volume 1: Theory | first1=Leonard | last1=Kleinrock | author-link = Leonard Kleinrock | isbn = 0471491101 | year=1975 | page=77}}</ref>{$p_k(t)=e^{-(\lambda+\mu)t} \left[ \rho^{\frac{k-i}{2}} I_{k-i}(at) + \rho^{\frac{k-i-1}{2}} I_{k+i+1}(at) + (1-\rho) \rho^{k} \sum_{j=k+i+2}^{\infty} \rho^{-j/2}I_j(at) \right]$} where {$i$} is the initial number of customers in the station at time {$t=0$}, {$\rho=\lambda/\mu$}, {$a=2\sqrt{\lambda\mu}$} and {$I_k$} is the modified Bessel function of the first kind. Moments for the transient solution can be expressed as the sum of two monotone functions.

Jiang Zeng

• Jiang Zeng. Chapter 5 - Combinatorics of Orthogonal Polynomials and their Moments in Lectures on Orthogonal Polynomials and Special Functions, 2020.
• Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials. 2013.
• Ira M. Gessel, Jiang Zeng. Moments of Orthogonal Polynomials and Exponential Generating Functions. 2021.
• Qiongqiong Pan, Jiang Zeng. Combinatorics of {$(q,y)$}-Laguerre polynomials and their moments.
• Jiang Zeng at Arxiv.org
• Viennot. PASEP.
• https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-8/issue-1/Many-server-queueing-processes-with-Poisson-input-and-exponential-service/pjm/1103040247.full
• https://en.m.wikipedia.org/wiki/Jacobi_operator. It arises in many areas of mathematics and physics. The case a(n) = 1 is known as the discrete one-dimensional Schrödinger operator.

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 W-invariant 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 W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic 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 ́Alvarez-Nodarse. On some applications of Orthogonal Polynomials in Mathematical-Physics 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 1-parameter 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 two-dimensional maximally superintegrable version of the Charlier oscillator is constructed.
• Richard Askey, Sergei K. Suslov. The q-harmonic 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 q-analogues

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
  • Rook polynomials - No two rooks may be in the same row or column.
  • Tricomi–Carlitz polynomials - related to random walks on positive integers - {$l_n(x)=(-1)^nL_n^{(x-n)}(x)$}
  • Chihara–Ismail polynomials - generalize the Tricomi-Carlitz polynomials
• 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

• Rogers–Szegő polynomials

On the cylinder?

• Bessel functions of the first kind and second kind

And sphere?

• Spherical Bessel functions

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 non-reduced root system of rank 1.
• Laguerre {$(n+1)L_{n+1}(x) + nL_{n-1}(x) = (2n+1-x)L_n(x)$}
• probabilist Hermite {$\mathit{He}_{n+1}(x) + n\mathit{He}_{n-1}(x) = x\mathit{He}_n(x)$}
• physicist Hermite {$H_{n+1}(x) + 2nH_{n-1}(x) = 2xH_n(x)$}
• Legendre {$(n+1)P_{n+1}(x) + n P_{n-1}(x)= (2n+1) x P_n(x)$}
• Bessel polynomials {$xZ_{\alpha+1}(x) + xZ_{\alpha - 1}(x) = 2\alpha Z_{\alpha}(x)$}
• Jacobi polynomials {$2n (n + \alpha + \beta) (2n + \alpha + \beta - 2) P_n^{(\alpha,\beta)}(z)$}

{$+ 2 (n+\alpha - 1) (n + \beta-1) (2n+\alpha + \beta) P_{n-2}^{(\alpha, \beta)}(z)$} {$= (2n+\alpha + \beta-1) \Big\{ (2n+\alpha + \beta)(2n+\alpha+\beta-2) z + \alpha^2 - \beta^2 \Big\} P_{n-1}^{(\alpha,\beta)}(z) $}

• Bessel functions of the first kind

{$$J_\alpha(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\;\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}$$}

{$$(2m+n)!2^{2m+n}J_{n,2m+n}=\binom{2m+n}{m}(-1)^m$$}

{$$(2m+n)!2^{2m+n}\binom{2m+2n}{m+n}J_{n-\frac{1}{2},2m+n}=\binom{2m+n}{m}(-1)^m4^{m+n}$$}

• Askey-Wilson 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 {$P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)$}
• {$x$} is the raising operator which defines {$x p_n(x)$} in terms of {$p_{n+1}$}, {$p_n$}, {$p_{n-1}$}. {$\frac{\textrm{d}}{\textrm{dx}}$} 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 {$P_{n-1}$}. 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.

What is the combinatorics of convex spaces and how does that relate to orthogonal polynomials, which give different ways of looking at the geometry?

• Vilmos Totik. Orthogonal polynomials.
• Christopher Boyd, Raymond A. Ryan, Nina Snigireva. Geometry of Spaces of Orthogonally Additive Polynomials on C(K).

Other topics

• Orthogonal polynomials are related to continued fractions.
• Also related to the factorization of free monoids.