JacobiPGeneral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν,a,b

For fixed ν,a,z

For fixed ν,b,z

For fixed a,b,z

For fixed ν,z

General characteristics

Some abbreviations

Domain and analyticity

(Subscript[P, ν]^((a,b)))(z) is an analytical function of ν, a, b, z which is defined in (ℂ^4). For fixed ν, a, z, it is an entire function of b. For positive integer ν, the function (Subscript[P, ν]^((a,b)))(z) degenerates to a polynomial in z of order ν.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to b

For fixed ν, a, z, the function (Subscript[P, ν]^((a,b)))(z) has only one singular point at b= (OverTilde[∞]).  It is an essential singular point.

With respect to a

For fixed ν, b, z, the function (Subscript[P, ν]^((a,b)))(z) has an infinite set of singular points:
a) a==-ν- k/;k∈(SuperPlus[ℕ]), are the simple poles with residues (((((-1))^(k-1)) Γ(k-b))/((k-1)! Γ(-b-ν) Γ(ν+1)))(Subscript[P, -b+k-1]^((-k-ν,b)))(z);
b) a==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

With respect to ν

For fixed a, b, z, the function (Subscript[P, ν]^((a,b)))(z) has an infinite set of singular points:
a) ν==-a- k/;k∈(SuperPlus[ℕ]), are the simple poles with residues (((((-1))^(k-1)) Γ(k-b))/((k-1)! Γ(-b-ν) Γ(ν+1)))(Subscript[P, -b+k-1]^((-k-ν,b)))(z);
b) ν==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

Branch points

With respect to z

For fixed a,b and integer ν, the function (Subscript[P, ν]^((a,b)))(z) does not have branch points.

With respect to b

For fixed ν, a, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch points.

With respect to a

For fixed ν, b, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch points.

With respect to ν

For fixed a, b, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch points.

Branch cuts

With respect to z

For fixed a,b and integer ν, the function (Subscript[P, ν]^((a,b)))(z) does not have branch cuts.

With respect to b

For fixed ν, a, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch cuts.

With respect to a

For fixed ν, b, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch cuts.

With respect to ν

For fixed a, b, z, the function (Subscript[P, ν]^((a,b)))(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

Expansions at z==1

Expansions at z==-1

Expansions at z==∞

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

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Additional relations between contiguous functions

Relations of special kind

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

With respect to ν

With respect to a

With respect to b

With respect to z

Symbolic differentiation

With respect to z

Fractional integro-differentiation

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

Infinite summation

Operations

Limit operation

Orthogonality, completeness, and Fourier expansions

The set of functions (Subscript[P, n]^((a,b)))(x), n=0,1,…, forms a complete, orthogonal (with weight ((n! (a+b+2 n+1) Γ(a+b+n+1))/((2^(a+b+1)) Γ(a+n+1) Γ(b+n+1))) (((1-x))^a) (((1+x))^b)) 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

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)==(Sqrt[((n! (a+b+2 n+1) Γ(a+b+n+1))/((2^(a+b+1)) Γ(a+n+1) Γ(b+n+1)))])(((1-x))^(a/2)) (((1+x))^(b/2)) (Subscript[P, k]^((a,b)))(x), k∈ℕ.

The quantum mechanical representation matrices of angular momentum

The quantum mechanical representation matrices (Subscript[D, m m']^L)(α,β,γ) of angular momentum L are given by

(Subscript[D, m m']^L)(α,β,γ)==(e^(i(m α+m'γ)))(Sqrt[(((L+m')!(L-m')!)/((L+m)!(L-m)!))])(((cos((β/2))))^(m+m'))(((sin((β/2))))^(m-m'))(Subscript[P, L-m']^((m-m',m+m')))(cos(β))

where α, β, γ are the Euler angles and L,m,m'∈ℕ,-L≤m,m'≤L .

The expected value of the number of real eigenvalues of a one matrix

The expected value (Subscript[r, n]) of the number of real eigenvalues of a n×n matrix whose matrix elements are random variables with Gaussian distribution (mean = 0, variance = 1) is

(Subscript[r, n])==(1-(((-1))^n))/2+(Sqrt[2])(Subscript[P, n-2]^((1-n,3/2)))(3).

The equilibrium positions of n unit charges

The equilibrium positions of n unit charges, a charge q at -1, and a charge p at +1 interacting with potential -log(x) are the zeros of (Subscript[P, n]^((2 p-1,2 q-1)))(x).

History

– C. J. Jacobi (1859)
– P.  L. Chebyshev (1870).

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