概论
数学
发现
Andrius Kulikauskas
 ms@ms.lt
 +370 607 27 665
 My work is in the Public Domain for all to share freely.
用中文
Software

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 selfaction?
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 {$\sqrt{1x^2}$} in terms of spinors.
 Interpret {$H\psi=E\psi$} in terms of cells.
 Interpret the Dirac equation, and the KleinGordon 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 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.
Chebyshev polynomials, Wallis integrals
例子
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
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.
General information
 Orthogonal polynomials {$P_n$} satisfy a recurrence relation of the form {$P_{n}(x)=(A_{n}x+B_{n})P_{n1}(x)+C_{n}P_{n2}(x)$}
 Orthogonal polynomials are related to continued fractions. Also related to the factorization of free monoids.
Combinatorial interpretation
 Laguerre {$(n+1)L_{n+1}(x) + nL_{n1}(x) = (2n+1x)L_n(x)$}
 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
{$$J_\alpha(x)=\sum_{m=0}^{\infty}\frac{(1)^m}{m!\;\Gamma(m+\alpha+1)}(\frac{x}{2})^{2m+\alpha}$$}
{$$(2m+n)!2^{2m+n}J_{n,2m+n}=\binom{2m+n}{m}(1)^m$$}
{$$(2m+n)!2^{2m+n}\binom{2m+2n}{m+n}J_{n\frac{1}{2},2m+n}=\binom{2m+n}{m}(1)^m4^{m+n}$$}
Hermite polynomials
 Gaussian beam and HermiteGaussian 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=ab 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 spacetime.
 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 {$\Delta_n$}, {$c_{nn}=\frac{\Delta_{n1}}{\Delta_n}$}, {$\lambda_{n+1}=\frac{\Delta_{n2}\Delta_{n}}{\Delta_{n1}^2}$}.
 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 {$p_{n+1}$}, {$p_n$}, {$p_{n1}$}. {$\frac{\textrm{d}}{\textrm{dx}}$} 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 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
 MeixnerPollaczek  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/whyareallmyseparablesolutionsorthogonalpolynomialsinl2
 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 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.
 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?
 https://en.m.wikipedia.org/wiki/Sheffer_sequence is what i am interested in.
Sturmian functions  harmonic oscillator
 Sturmian words  billiard ball bouncing between horizontal and vertical sides of table
 Sturmian functions https://arxiv.org/pdf/quantph/9809062.pdf
 This paper relates Sturmian words, Sturmian functions and Sturmian numbers.
 Relativistic quantum harmonic oscillator
 A.Poszwa. Relativistic Generalizations of the Quantum Harmonic Oscillator
 Spherical harmonics
 R. P. Martı́nezyRomero. Relativistic quantum mechanics of a Dirac oscillator 1999. Free article. Refers to spinor spherical harmonics.
 The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential.
 See: Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 9783642880827.
 Relativistic hydrogen atom
 R. P. Martı́nezyRomero. Relativistic hydrogen atom revisited American Journal of Physics 68, 1050 (2000). The hydrogen atom is solved using a simple method. We show that this system has an exact solution that can be written in terms of Laguerre polynomials of noninteger index, instead of the hypergeometric series. This point is important because Laguerre polynomials of integer index appear in the solution of the nonrelativistic hydrogen atom, giving students a more unified point of view for this system.
 A. D. Alhaidari, H. Bahlouli, M. E. H. Ismail. The DiracCoulomb Problem: a mathematical revisit Alhaidari researches orthogonal polynomials. Describes the relativistic hydrogen atom in terms of Pollaczek polynomials.
 Guillermo Palma, Ulrich Raff. The One Dimensional Hydrogen Atom Revisited
 Spacetime is made of events (the coordinates). But in quantum mechanics, an event is only when a wave function collapses. So time and space combine only when the wave function collapses. So relativity is only important for the actual collapse of the wave function. The wrapper {$e^{x^2}$} may explain the quadratic nature of gravity. But how?
 Orthogonality measures selfduality. Two pyramids give the perspectives of position and momentum.
 Ismail. Classical and Quantum Orthogonal Polynomials in One Variable.
 Based on the case of the harmonic oscillator, setting units like h to 1, energy is simply the frequency. In general, is energy the frequency of conversion of potential energy to kinetic energy and back again, as with the harmonic oscillator?
 The combinatorial configurations into cells and relationships are presumably the same regardless of which is assigned to position and which to momentum... and other, possible hybrid, physical qualities? Or does the Fourier transform introduce some subtle difference? So what is the meaning of these deeper configurations? And what does it mean when they signify the result of a measurement?
 Measurement is a trivial translation of coordinate system from x to xa and thus relates to the trivial selfentanglement of a tree.
 Delayed signals? (Kinks?) Attiyah: Retarded Dirac operator. Attiyah, Moore. A Shifted View of Fundamental Physics.
 Timedelayed Dirac delta, Group delay and phase delay
 Connes noncommutative geometry is based on every point being actually a pair of points (on two sheets), on which 2x2 matrices operate, and yields the Standard Model. Are these pairs of points expressing the dual causality of the fivesome, links and kinks?
 How is KimZeng linearization (generalized derangements) related to the representation of the symmetric group and general linear group?
 https://www.amazon.com/IntroductionOrthogonalPolynomialsDoverMathematics/dp/0486479293
 Physicists know how to copy equations, how to manipulate them, how to solve them, how to set them up, but not how to read them mathematically, conceptually. Good physicists can read the physics and yet not the math.
 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
 Consider the weights of the various families of orthogonal polynomials and their properties. For example, 1/2cosh(pi x/2) is, up to a dilation, its own Fourier transform. Are the other weights similarly related to geometric transformations?
 Wolfram Koepf, Dieter Schmersau. Recurrence equations and their classical orthogonal polynomial solutions.
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 ́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?
Kinks
 There are 5 conic sections plus 3 more: the point C (where the cones meet), two lines that cross in an X, and the line segment (which arises in the limiting case as the hyperbola passes over to an ellipse). Given any other point P on the lower cone, P not C, then we get conic figures by rotating the plane that contains C along the axis that is tangent to the cone. We can start with a circle, then it becomes an ellipse ever longer, until it reaches infinity and becomes a parabola, which then intersects the upper cone as a hyperbola, which then becomes a line as the hyperbola gets squeezed and infinitely narrow, but then the line drops down to a line segment from P to C, from where it opens up as an ellipse, which grows in width until it becomes the circle. So the line segment is a moment of ambiguity where it equals the line and yet the limit cases from either side are different. The line segment seems like a string or loop in string theory.
 Nurseries, Zeng saplings
 What would a kink look like for a tableaux?
 An observer carves up spacetime into kinks.
 Kinks express the need that a question be answered, that there be resolution, that tension be resolved, which is important in all conceptual languages.
 Kinks express the yearning for justice, that everything be set right, that there be no loose ends.
 Kinks express the obligations met by a pushdown automata, as with balanced parentheses.
 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
 A kink is a contraction (a line) as defined for Wick's theorem. What is a normal ordering? An abnormal ordering? A link?
 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
 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.
 The 5 natural bases for {$L^2$} or the 6 natural bases for the symmetric functions are perhaps related to Mtheory.
