GegenbauerC3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n,λ

For fixed n,z

For fixed λ,z

Values at infinities

General characteristics

Domain and analyticity

The function (Subscript[C, n]^λ)(z) is defined over ℕ⊗ℂ⊗ℂ. For fixed n,λ, the function (Subscript[C, n]^λ)(z) is a polynomial in z of degree n. For fixed n,z, the function (Subscript[C, n]^λ)(z) is a polynomial in λ of degree n.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

For fixed λ the function (Subscript[C, n]^λ)(z) is polynomial and has pole of order n at z =(OverTilde[∞]).

With respect to λ

For fixed n, z, the function (Subscript[C, n]^λ)(z) has an infinite set of singular points:
a)  λ==-((n+j)/2)/;j∈ℕ, are the simple poles with residues (((((-1))^j)(2^(j+ν)) (Sqrt[π]) )/(ν! j! Γ(-((ν+j)/2))))(Subscript[, 2])(Subscript[(OverTilde[F]), 1])(-j,-ν;((1-ν-j)/2);((1-z)/2));
b) λ==(OverTilde[∞]) is an essential singular point.

Branch points

With respect to z

For fixed n,λ, the function (Subscript[C, n]^λ)(z) does not have branch points.

With respect to λ

For fixed n,z, the function (Subscript[C, n]^λ)(z) does not have branch points.

Branch cuts

With respect to z

For fixed  n,λ, the function (Subscript[C, n]^λ)(z) does not have branch cuts.

With respect to λ

For fixed  n,z, the function (Subscript[C, n]^λ)(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point λ==(Subscript[λ, 0])

Expansions at generic point z==(Subscript[z, 0])

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

Expansions at λ==0

Expansions at λ==∞

Expansions at n==∞

Integral representations

On the real axis

Of the direct function

Integral representations of negative integer order

Rodrigues-type formula.

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products of the direct function

Addition formulas

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Relations of special kind

Complex characteristics

Real part

Imaginary part

Argument

Conjugate value

Differentiation

Low-order differentiation

With respect to λ

With respect to z

Symbolic differentiation

With respect to λ

With respect to z

Fractional integro-differentiation

With respect to λ

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Definite integration

Involving the direct function

Summation

Finite summation

Infinite summation

Operations

Limit operation

Orthogonality, completeness, and Fourier expansions

The set of functions (Subscript[C, n]^λ)(x), n=0,1,…, forms a complete, orthogonal (with weight ((n! (n+λ) ((Γ(λ))^2))/(π (2^(1-2 λ)) Γ(n+2 λ)))(((1-(x^2)))^(λ-(1/2)))) system on the interval (-1,1).

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric2F1Regularized

Involving Hypergeometric2F1

Through hypergeometric functions of two variables

Through Meijer G

Classical cases for the direct function itself

Classical cases involving algebraic functions

Classical cases involving unit step UnitStep

Generalized cases involving algebraic functions

Generalized cases involving unit step UnitStep

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Theorems

Expansions in generalized Fourier series

f(x)==(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(x)/;   (Subscript[c, k ])==(∫_(-1)^1)f(t)(Subscript[ψ, k])(t)dt, (Subscript[ψ, k])(x)==(((2^(1/2-λ))(Sqrt[π Γ(2 λ+n)]))/(Γ(λ)(Sqrt[n! (λ+n)])))(((1-(x^2)))^((2λ-1)/4)) (Subscript[C, k]^λ)(x), k∈ℕ.

Eigenfunctions of the angular part of a d-dimensional Laplace operator

The eigenfunctions of the angular part (L^2)==-(∑_(i>j)^d)((((Subscript[x, i])(∂/(∂(Subscript[x, j])))-(Subscript[x, j])(∂/(∂(Subscript[x, i])))))^2) of a d-dimensional Laplace operator Δ==(1/(r^(d-1)))(∂/(∂r))(r^(d-1))(∂/(∂r))+((L^2)/(r^2)),  where u,u' are two unit vectors in  (ℝ^d), has the representation
(L^2)(Subscript[C, n]^(d/2-1))(u.u') ==n(n+d-2)(Subscript[C, n]^(d/2-1))(u.u').

Removing Gibbs oscillations from Fourier series

Let f(x) be a doubly periodic function with  f(-1)≠f(1).  Let (OverHat[(Subscript[f, k])]) be its Fourier components (OverHat[(Subscript[f, k])])=(1/2) (∫_(-1)^1)f(x) (e^(-i k π x))dx. Then the Fourier sum (∑_(k=-n)^n)(OverHat[(Subscript[f, k])]) (e^(i k π x)) exhibits Gibbs oscillations. 
It is possible to recover the original function f(x) as a sum without such Gibbs oscillations in the following
 manner: (∑_(k=0)^n)(Subscript[g, k]^n)(Subscript[C, k]^λ)(x), where (Subscript[g, k]^n)=(Subscript[δ, 0 k])(OverHat[(Subscript[f, 0])])+Γ(λ) (i^k) (k+λ) (∑_(l=-n)^n)(1-(Subscript[δ, 0 l]))(Subscript[J, k+λ])(π l) ((((2/(l π))))^λ) (OverHat[(Subscript[f, l ])])/; λ=⌊((2 π e n)/27)⌋.
This sum converges pointwise to f(x).

Quantum mechanical eigenfunctions of the hydrogen atom

The quantum mechanical eigenfunctions (Subscript[ψ, n m l])(p,θ,ϕ) of the hydrogen atom in the momentum representation are:

(Subscript[ψ, n m l])(p,θ,ϕ)==16π (Sqrt[m])(Subscript[κ, n]^2)(2^l)l!(Sqrt[(((n-l-1)!)/((n+l)!))])(1/((((Subscript[κ, n]^2)+(p^2)))^2))((((((Subscript[κ, n])p)/((Subscript[κ, n]^2)+(p^2)))))^l)(Subscript[C, n-l-1]^(l+1))((((Subscript[κ, n]^2)-(p^2))/((Subscript[κ, n]^2)+(p^2))))(Subscript[Y, l]^m)(θ,ϕ)/;(Subscript[κ, n])=(τ/n)/;τ>0, n,l∈ℕ,l≤n,m∈ℤ,|m|≤l.

History

–  L. Gegenbauer (1893)

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