正交多項式
General picture
- 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?
- Does the local phase invariance deal with self-action?
Related math
- How is the sequence that I used to get the Legendre polynomials related to approximating pi ?
Combinatorial interpretations
- Understand Kim and Zeng's interpretation of the linearization of orthogonal polynomials in terms of derangements.
- What does it mean to choose 1/2 out of n/2 ? Or out of n?
- Interpret the Legendre and Laguerre polynomials and relate that to the Hermite polynomials.
- Interpret {$E^2=m^2c^4+p^2c^2$} in terms of cells.
- Interpret {$\sqrt{1-x^2}$} in terms of spinors.
- Interpret {$H\psi=E\psi$} in terms of cells.
- Interpret the Dirac equation, and the Klein-Gordon equation, in terms of cells.
- Understand how Viennot's interpretation of orthogonal polynomials relates to taking steps S, E, EES.
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.
Chebyshev polynomials, Wallis integrals
例子
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
- 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
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 non-reduced root system of rank 1.
General information
- Orthogonal polynomials {$P_n$} satisfy a recurrence relation of the form {$P_{n}(x)=(A_{n}x+B_{n})P_{n-1}(x)+C_{n}P_{n-2}(x)$}
- Orthogonal polynomials are related to continued fractions. Also related to the factorization of free monoids.
Combinatorial interpretation
- 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
{$$J_\alpha(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\;\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}$$}
{$$(2m+n)!2^{2m+n}J_{n,2m+n}=\binom{2m+n}{m}(-1)^m$$}
{$$(2m+n)!2^{2m+n}\binom{2m+2n}{m+n}J_{n-\frac{1}{2},2m+n}=\binom{2m+n}{m}(-1)^m4^{m+n}$$}
Hermite polynomials
- Gaussian beam and Hermite-Gaussian modes relate to the Gaussian distribution and the Hermite polynomials.
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.
Laguerre polynomials and Laguerre functions
- Laguerre function terms can be matched to the number of injective functions from subsets of [n] to [n].
- Number of terms in Laguerre polynomial.
- 1/n! on the outside of Laguerre polynomials means that the order (on one side) doesn't matter externally although it matters internally.