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数学

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Andrius Kulikauskas

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See: Quantum physics, Hermite polynomials, Associated Legendre polynomials

Investigation: What is the physical significance of the combinatorial interpretations of the orthogonal polynomials?


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

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

  • Rogers–Szegő polynomials

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.

Orthogonal polynomials

  • Empty space has unconnected coordinate system for each particle.
  • Nonempty space (the other four possibilities) is given by pairs of coordinate systems and they represent entanglements.
  • For orthogonal polynomials in quantum field theory, when particles go to another coordinate system and then return, it is because the original coordinate system is understood to be in a broader framework with presumptions about a second coordinate system.
  • What is the connection between growth in standard tableaux and growth in Zeng trees?
  • For Sheffer polynomials, loops are spatial representations, explaining how two coordinate systems are related by different steps they take there and back. But trees of links and kinks are temporal representations. But l=-a-b and k = (-a)(-b) how to interpret that temporally? How does that link time and space? Or causality and coordinates?
  • How are the orthogonal polynomial combinatorics related to the symmetric functions of eigenvalues?
  • In the combinatorics of Sheffer polynomials, {$\alpha$} describes the steps taken in our coordinate system away from our origin to the origin of another coordinate system, and {$\beta$} describes the steps taken from the origin of that coordinate system back to our origin. Thus the Sheffer polynomials ground the creation of space-time.
  • In the Meixner case, the general case, distinguish levels: singleton {$x$}, doubleton {$\gamma$}, outgoing {$\alpha$}, incoming {$\beta$}.
  • Study "Signals and Systems" to understand the Laplace transform, the Fourier transform, Shannon information.
  • Combinatorially, Jacobi configuration is a pair of Laguerre configurations (injections) back and forth. Is the Jacobi configuration like an adjunction?
  • Express the combinatorics of orthogonal polynomials in terms of symmetric functions of eigenvalues. For example, start with Hermite polynomials and define fixed points and involutions. In general, come up with a theory of species in terms of the symmetric functions of eigenvalues.
  • How do the combinatorial cycles that ground the Sheffer polynomials relate to my interpretations of the symmetric functions of eigenvalues? In particular, what may be the role of the forgotten symmetric function? And bricks of size 1 and 2? And having two alphabets?
  • Empty space (Hermite polynomials) has particle created and then annihilated.
  • Nonempty space has particle annihilated and then created (thereby moved).
  • Retaining identity in space and time - a principle grounded in the combinatorics of orthogonal polynomials - thus there exists identity as a long term entanglement
  • Wolfram's computational equivalence - simple program can be equivalent to the most complicated programs - and add to this the notion of degenerate programs which are less powerful. And an algebra of degeneracy which provides the desired content.
  • There is a symmetry of walking away and walking back, {$\alpha$} and {$\beta$}.
  • Moments - the numbers give the impact of {$x^k$}.
  • Hankel determinant - the numbers describe what momentum is.
  • In the combinatorial objects, note what gets weight {$x$} and what else is there.
  • How is the eccentricity of the conics related to the discriminant? And how is the equation {$(1-\alpha t)(1-\beta t)=1+lt+kt^2$} and the coefficients related to the conics?
  • The decision point (present, boundary) is the quantum {$x$} which commutes with either ground so that {$\int p_k x p_n$} can be interpreted ambiguously, and so relations with distant, nonproximate levels can be discarded.
  • Give combinatorial interpretation of {$\Delta_n$}, {$c_{nn}=\frac{\Delta_{n-1}}{\Delta_n}$}, {$\lambda_{n+1}=\frac{\Delta_{n-2}\Delta_{n}}{\Delta_{n-1}^2}$}.
  • Understand what function gives the zeroes that determine the range for the weight function.
  • Calculate the echo terms for {$A(t)$}, {$A'(t)$}, {$u(t)$}, {$u'(t)$} and compute {$\frac{A'(t)}{A(t)}$}.
  • Think of moments {$\mu_n$} in terms of walks that wander up to a distance {$n$}. These walks are the expressions of momentum, perhaps their building blocks.
  • How is the exponential world combinatorially different from the regular world? It is bounded from above by {$n!$}.
  • What are the transition matrices between the natural bases of the orthogonal polynomials? What are the combinatorial interpretations for their entries? They are indexed by integers rather than partitions.
  • What are natural bases for {$L^2$} ? The 5 Sheffer polynomials. What are natural bases for symmetric functions? There are 6. Are there four bases for the Yoneda lemma?
  • How does Viennot's "coherent transport" work as a functor?
  • The Sheffer polynomials are perhaps characterized by the operator {$x$} and its action on {$P_n(x)$} whereas the classical orthogonal polynomials are perhaps characterized by the operator {$\frac{\textrm{d}}{\textrm{dx}}$} as with the Rodrigues formula.
  • Interpret combinatorially the formula for an orthogonal polynomial as a determinant of moments for the given measure.
  • Wolfram's cellular automata can be thought of as acting on the hierarchy of levels, as with raising and lowering operators.
  • Is the Raudys classification some how related to the Pearson classification?
  • 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.
  • {$x$} 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 prganizes dimensions, higher and lower.
  • Krautchouk polynomials are a family that consists of only finitely many polynomials. The higher polynomials are zero. This introduces the notion of finiteness. It may be a sixth possibility that expresses the finite human situation, morality, the sixsome.

Sheffer classification gives 5 possibilities for synchronization. Consider {$\alpha$} and {$beta$} as vectors in the plane. Then we can have:

  • Two points at the origin.
  • A point at the origin and a vector.
  • The same vector twice.
  • Two nonzero vectors of different length on a line.
  • Complex conjugates in the plane

How are the natural orthogonal polynomials related to the natural bases of the symmetric functions?

  • Hermite - pairs - monomial?
  • Charlier - cylces - power?
  • Laguerre - strict ordering - elementary
  • Meixner-Pollaczek - alternatin permutation - Schur
  • Meixner - weak ordering - homogeneous

Leaving the forgotten as an extra basis.

  • https://en.wikipedia.org/wiki/Jacobi_operator
  • In what sense are the Krawtchouk polynomials possibly a sixth kind of orthgonal polynomial? They are a special case of the Mexiner polynomials of the first kind. They are only a finite sequence.
  • Fourier transform transforms the shift operator into the differentiation operator and vice versa. So how does it relate shift invariance (relativity, as with Sheffer polynomials) and invariance with regard to differentiation (as with weight functions like {$e^x$})?

In the classification of Sheffer polynomials, we have the choices:

  • {$\alpha = 0$}, {$\alpha \neq 0$}
  • {$\beta = 0$}, {$\beta = \alpha$}, {$\beta \neq \alpha$}, {$\beta = \overline{\alpha}$}

So there are eight possibilities, but some of them are related.

  • Wick's theorem? organizes creation and annihilation operators in very much the same way as the Hermite polynomials organize Zeng saplings. This suggests that momentum (the movement of a particle) can be thought of as the annihilation of a particle in one location and the creation of a particle in a different location. I am learning about Feynman diagrams to see how they might relate to Zeng saplings.
  • https://physics.stackexchange.com/questions/304508/why-are-all-my-separable-solutions-orthogonal-polynomials-in-l2
  • https://en.wikipedia.org/wiki/Spectral_theorem
  • 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?
  • The configurations that orthogonal polynomials code explain how phase space can be divided into distinct trees or cycles. The division of phase space into more compartments requires higher energy.
  • Orthogonal polynomial growth configurations (Zeng nurseries) can be thought of as a nondeterministic algorithm.
  • Consider Aristotle's four causes in terms of the two kinds of causality in decision making.
  • How does the generating function for {$P_n$} relate to the weight? They seem similar. Why?
  • Empty space is given by the weight. Differentiating it yields the same (almost) - a difference by a quantum. Differentiating {$x^k$} yields a lower degree, thus clearly different, and similarly with {$P_n(x)$}.

X.Viennot. The cellular ansatz: bijective combinatorics and quadratic algebra Robinson-Schensted-Knuth correspondence, trees and tableau, PASEP, quadratic algebra.

Razumov-Stroganov conjecture

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

Sturmian functions - harmonic oscillator

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
  • What is the meaning and significance of the conic sections "dot" and "X" (two lines crossed)?
  • For the conic sections, given the two cones, can we identify one with the inside and the other with the outside?
  • 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?

Kinks

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