LaguerreLGeneral

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed ν

For fixed z

General characteristics

Domain and analyticity

(Subscript[L, ν])(z) is an analytical function of ν, zm which is defined in (ℂ^2). For fixed ν, it is an entire function of z.  For positive integer ν, (Subscript[L, ν])(z) degenerates to a polynomial in z.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to ν

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

Branch points

With respect to z

For fixed ν, the function (Subscript[L, ν])(z) does not have branch points.

With respect to ν

For fixed z, the function (Subscript[L, ν])(z) does not have branch points.

Branch cuts

With respect to z

For fixed integer ν, the function (Subscript[L, ν])(z) does not have branch cuts.

With respect to ν

For fixed z, the function (Subscript[L, ν])(z) does not have branch cuts.

Series representations

Generalized power series

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

Expansions at z==0

Asymptotic series expansions

Integral representations

On the real axis

Of the direct function

Integral representations of negative integer order

Rodrigues-type formula.

Limit representations

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

Addition formulas

Multiple arguments

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

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 z

Symbolic differentiation

With respect to ν

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

Finite summation

Infinite summation

Operations

Limit operation

Orthogonality, completeness, and Fourier expansions

The set of functions (Subscript[L, n])(x), n=0,1,…, forms a complete, orthogonal (with weight (e^(-x))) system on the interval (0,∞).

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases for products of Laguerre LaguerreL

Classical cases involving  Exp and  products of LaguerreL

Classical cases involving Hypergeometric1F1

Classical cases involving  Exp and Hypergeometric1F1

Classical cases involving Hypergeometric1F1Regularized

Classical cases involving  Exp and Hypergeometric1F1Regularized

Classical cases involving hypergeometric HypergeometricU

Classical cases involving  Exp and hypergeometric HypergeometricU

Generalized cases for products of Laguerre LaguerreL

Generalized cases involving  Exp and  products of LaguerreL

Generalized cases involving Hypergeometric1F1

Generalized cases involving  Exp and Hypergeometric1F1

Generalized cases involving Hypergeometric1F1Regularized

Generalized cases involving  Exp and Hypergeometric1F1Regularized

Generalized cases involving hypergeometric HypergeometricU

Generalized cases involving  Exp and hypergeometric HypergeometricU

Through other functions

Involving some hypergeometric-type functions

Theorems

Expansions in generalized Fourier series

f(x)==(∑_(k=0)^∞)(Subscript[c, k])(Subscript[ψ, k])(x)/;   (Subscript[c, k ])==(∫_0^∞)f(t)(Subscript[ψ, k])(t)dt, (Subscript[ψ, k])(x)==(e^(-x/2))(Subscript[L, k])(x), k∈ℕ.

Excited coherent state

The (normalized) excited coherent state |α,m⟩==((((((a^(† ))))^m)|α⟩)/(⟨α|(((a^m)((a^(† ))))^m)|α⟩)) can be expressed through the number states |n⟩ as

|α,m⟩==(∑_(n=m)^∞)(((e^(-|α((|)^2)/2))(α^(n-m))(Sqrt[n!]))/((Subscript[L, m])(-|α((|)^2)m!)(n-m)!))|n⟩.

History

– J.-L. Lagrange
– R. Murphy (1833)
– E. N. Laguerre (1879)

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