HermiteHGeneral

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

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

With respect to ν

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

With respect to ν

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

Branch cuts

With respect to z

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

With respect to ν

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

Series representations

Generalized power series

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

Expansions at z==0

Expansions at z==∞

Asymptotic series expansions

Integral representations

On the real axis

Integral representations of negative integer order

Rodrigues-type formula.

Limit representations

Generating functions

Differential equations

Ordinary linear differential equations and wronskians

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

Multiple arguments

Products, sums, and powers of the direct function

Products of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Complex characteristics

Real part

Imaginary part

Differentiation

Low-order differentiation

With respect to ν

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

Brychkov Yu.A. (2006)

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

Involving functions of the direct function and elementary functions

Involving direct function and Gamma-, Beta-, Erf-type 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[H, n])(x), n=0,1,…, forms a complete, orthogonal (with weight (1/((2^n) n! (Sqrt[π])))(e^(-(x^2)))) system on the interval (-∞,∞).

Representations through more general functions

Through hypergeometric functions

Involving Hypergeometric1F1Regularized

Involving Hypergeometric1F1

Involving HypergeometricPFQ

Involving hypergeometric HypergeometricU

Through Meijer G

Classical cases for the direct function itself

Classical cases involving Exp

Classical cases involving Exp and Cosh

Classical cases involving Exp and Sinh

Classical cases for products of HermiteH

Classical cases involving  Exp and products of HermiteH

Classical cases involving  Exp and ParabolicCylinderD

Generalized cases for the direct function itself

Generalized cases involving Exp

Generalized cases involving Exp and Cosh

Generalized cases involving Exp and Sinh

Generalized cases for products of HermiteH

Generalized cases involving  Exp and products of HermiteH

Generalized cases involving  Exp and ParabolicCylinderD

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Zeros

Theorems

Expansions in generalized Fourier series

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

Fourier transform eigenfunctions

Hermite polynomials together with their weighting function are eigenfunctions of the Fourier and inverse Fourier transforms:

(1/(Sqrt[2π]))(∫_(-∞)^∞)(e^(i t x))(e^(-(x^2)/2)) (Subscript[H, n])(x)dx==-(i^n) (e^(-(t^2)/2)) (Subscript[H, n])(t)/;n∈ℕ

Zeros of Hermite polynomials

For any given interval (a,b), there exists some n∈ℕ such that (Subscript[H, n])(x) has a zero in this interval.

The number of simple graphs

The number of simple graphs with no cycles and n vertices is (Subscript[H, n])(n+1)-n (Subscript[H, n-1])(n+1).

History

– P. S. Laplace (1810)
– Ch. Hermite (1864)
– N. J. Sonine (1880)

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