List of Orthogonal Polynomials

Introduction

This article lists the orthogonal polynomials the corpus introduces. An orthogonal polynomial sequence is a sequence $p_n$ of polynomials, of degree $n$, orthogonal with respect to a weight on an interval, and a row below names one family, records the weight, the recurrence and the differential equation that determine it, and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The families are grouped by the layer that introduces them: the four classical families of the real special functions, the generalised families of the spherical theory, and the zonal harmonics that make the addition theorem of a spherical harmonic expansion work. The rows that record a boundary — the monomials, which are not orthogonal; a weight with divergent moments, which generates no orthogonal sequence; the classical interval condition that fails for the Legendre weight on the whole line — stand beside the families as non-examples.

The Classical Families

Object	The weight, the recurrence and the equation	Introduced in
Legendre polynomials $P_n$	weight $1$ on $[-1,1]$; $(n+1)P_{n+1} = (2n+1)xP_n - nP_{n-1}$; $(1-x^2)y'' - 2xy' + n(n+1)y = 0$	Real Special Functions; Ordinary Differential Equations
Chebyshev polynomials of the first kind $T_n$	weight $(1-x^2)^{-1/2}$ on $[-1,1]$; $T_{n+1} = 2xT_n - T_{n-1}$; $(1-x^2)y'' - xy' + n^2y = 0$	Real Special Functions; Complex Special Functions
Hermite polynomials $H_n$	weight $e^{-x^2}$ on $\mathbb R$; $H_{n+1} = 2xH_n - 2nH_{n-1}$; $y'' - 2xy' + 2ny = 0$	Real Special Functions; Fourier Analysis on Euclidean Spaces
Laguerre polynomials $L_n$	weight $e^{-x}$ on $[0,\infty)$; $(n+1)L_{n+1} = (2n+1-x)L_n - nL_{n-1}$; $xy'' + (1-x)y' + ny = 0$	Real Special Functions; Complex Special Functions
the complex forms of the four families	the same weights and recurrences, with the orthogonality on the corresponding contour or curve	Complex Special Functions
the Hermite functions	$H_n(x)e^{-x^2/2}$; the eigenfunctions of the Fourier transform with eigenvalues $(-i)^n$	Fourier Analysis on Euclidean Spaces
the Hermite expansion	the completion of the Hermite functions in $L^2(\mathbb R)$; the basis in which the Fourier transform is diagonal	Fréchet Spaces

The Generalised Families

Object	The weight, the recurrence and the equation	Introduced in
Gegenbauer polynomials $C_n^{(\lambda)}$	weight $(1-x^2)^{\lambda-1/2}$ on $[-1,1]$; the three-term recurrence of the Gegenbauer equation	Symmetric Tensors and Spherical Harmonics
associated Legendre functions $P_n^m$	weight $1$ after the factor $(1-x^2)^{-m/2}$; the associated Legendre equation	Symmetric Tensors and Spherical Harmonics
spherical harmonics $Y_l^m$	the surface measure on $S^{n-1}$; orthogonal for distinct $l$ or $m$, with the addition theorem	Symmetric Tensors and Spherical Harmonics
zonal harmonics	the spherical harmonics depending only on one axis; the reproducing kernel of the addition theorem	Symmetric Tensors and Spherical Harmonics
the harmonic polynomials of degree $l$	the polynomials killed by the Laplacian; the decomposition of the polynomials into harmonic layers	Symmetric Tensors and Spherical Harmonics

The Generating Functions and the Orthogonality Constants

Object	What it supplies	Introduced in
the Legendre generating function $(1-2xt+t^2)^{-1/2}=\sum_nP_n(x)t^n$	the generating datum of the Legendre family, from which the recurrence and the values $P_n(1)=1$ follow	Real Special Functions
the Chebyshev identity $T_n(\cos\theta)=\cos n\theta$	the trigonometric definition that makes $T_n$ a polynomial, and the extremal property of the family	Real Special Functions
the orthogonality constant $2/(2n+1)$	the norm of $P_n$ in $L^2([-1,1],dx)$, and the normalisation of the expansion	Real Special Functions
the recurrence constants $2n$ and $n+1$	the coefficients of the Hermite and Laguerre three-term recurrences	Real Special Functions
the normalisation of the Hermite functions	the constant $(\pi^{1/2}2^nn!)^{-1/2}$ that makes them an orthonormal basis of $L^2(\mathbb R)$	Fourier Analysis on Euclidean Spaces

The Orthogonal Polynomials of the Hypercomplex Systems

Object	The weight, the recurrence and the equation	Introduced in
the biquaternion Legendre, Chebyshev and Hermite polynomials $P_n(\tilde Q)$, $T_n(\tilde Q)$, $H_n(\tilde Q)$	the same recurrences in the biquaternion algebra; they reduce to the complex polynomials in the oscillatory regime and truncate in the nilpotent regime	Biquaternion Higher Special Functions
the split-biquaternion orthogonal polynomials	the same recurrences in the split-biquaternion algebra, with the two regimes of that algebra	Split-Biquaternion Higher Special Functions
the split-complex Legendre, Chebyshev and Hermite polynomials	the recurrences in the split-complex algebra, with the hyperbolic parametrisation	Split-Complex Special Functions

Warnings

Two families that a reader may expect are absent. The Jacobi polynomials $P_n^{(\alpha,\beta)}$, the two-parameter family with weight $(1-x)^\alpha(1+x)^\beta$ on $[-1,1]$ whose special cases are the Legendre, Chebyshev and Gegenbauer families, are not introduced anywhere in Parts I to III; the corpus treats their special cases directly and does not name the general family. The Chebyshev polynomials of the second kind $U_n$, orthogonal with weight $(1-x^2)^{1/2}$, are likewise not introduced. Both are therefore absent from the tables, and the record of their absence is this paragraph.

Boundary Cases

Object	The boundary it records	Introduced in
the monomials $1, x, x^2, \dots$	not orthogonal with respect to Lebesgue measure on $[-1,1]$: $\int_{-1}^1 x^m x^n\,dx \neq 0$ for $m+n$ even	Real Special Functions
the Legendre weight on the whole line	the integral $\int_{\mathbb R} P_mP_n\,dx$ diverges, so the Legendre polynomials are orthogonal only on the finite interval	Real Special Functions
a weight with divergent moments	no orthogonal sequence exists, the moments $\int x^n w(x)\,dx$ being the data of the three-term recurrence	Complex Special Functions
the second solution of the Legendre equation	$Q_n$ is not a polynomial; the polynomial solution is the one orthogonal with respect to the weight	Ordinary Differential Equations
the associated Legendre functions at $m > n$	vanish identically; the spherical harmonic $Y_l^m$ then carries no function	Symmetric Tensors and Spherical Harmonics

Summary

This list gathers the orthogonal polynomials of the corpus: the Legendre, Chebyshev of the first kind, Hermite and Laguerre families with their weights, recurrences and equations; their complex forms; the Hermite functions and the Hermite expansion; and the Gegenbauer, associated Legendre and spherical-harmonic families of the spherical theory. Each row records the weight that defines the orthogonality and the recurrence that generates the sequence, and the closing table records the families that are absent and the boundary cases.

Summary of Notation

The objects are named by their standard letters; the symbols in the tables are collected here.

Symbol	Meaning
$P_n$, $T_n$, $H_n$, $L_n$	Legendre, Chebyshev of the first kind, Hermite, Laguerre polynomials
$C_n^{(\lambda)}$, $P_n^m$	Gegenbauer polynomials, associated Legendre functions
$Y_l^m$	spherical harmonics
$w(x)$, $d\mu$	the weight, and the measure $w(x)\,dx$ it defines
$\langle p_m, p_n\rangle_w$	the inner product with respect to the weight