# Mehler kernel

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Complex-valued function

The **Mehler kernel** is a complex-valued function found to be the [propagator](/source/Propagator) of the [quantum harmonic oscillator](/source/Quantum_harmonic_oscillator).

It was first discovered by [Mehler](/source/Gustav_Ferdinand_Mehler) in 1866, and since then, as [Einar Hille](/source/Einar_Hille) remarked in 1932, "has been rediscovered by almost everybody who has worked in this field".[1]

## Mehler's formula

[Mehler](/source/Gustav_Ferdinand_Mehler) ([1866](#CITEREFMehler1866)) defined a function[2]

E ( x , y ) = 1 1 − ρ 2 exp ⁡ ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y ( 1 − ρ 2 ) ) , {\displaystyle E(x,y)={\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}(x^{2}+y^{2})-2\rho xy}{(1-\rho ^{2})}}\right)~,}

and showed, in modernized notation,[3] that it can be expanded in terms of [Hermite polynomials](/source/Hermite_polynomials) H ( ⋅ ) {\displaystyle H(\cdot )} based on weight function exp ⁡ ( − x 2 ) {\displaystyle \exp(-x^{2})} as E ( x , y ) = ∑ n = 0 ∞ ( ρ / 2 ) n n ! H n ( x ) H n ( y ) . {\displaystyle E(x,y)=\sum _{n=0}^{\infty }{\frac {(\rho /2)^{n}}{n!}}~{\mathit {H}}_{n}(x){\mathit {H}}_{n}(y)~.}

This result is useful, in modified form, in quantum physics, [probability theory](/source/Probability_theory), and harmonic analysis. Equivalently, in the probabilist's Hermite polynomials: 1 1 − ρ 2 exp ⁡ ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) ) = ∑ n = 0 ∞ ρ n n ! He n ⁡ ( x ) He n ⁡ ( y ) {\displaystyle {\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}\left(x^{2}+y^{2}\right)-2\rho xy}{2\left(1-\rho ^{2}\right)}}\right)=\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)} Substituting ρ = e − t {\displaystyle \rho =e^{-t}} , and letting h n := He n ⁡ / n ! {\displaystyle h_{n}:=\operatorname {He} _{n}/{\sqrt {n!}}} , we have tanh ⁡ ( t / 2 ) exp ⁡ ( − e − t ( x 2 + y 2 ) − 2 x y 4 sinh ⁡ t ) = ∑ n = 0 ∞ ( 1 − e − t ) e − n t h n ( x ) h n ( y ) {\displaystyle {\sqrt {\tanh(t/2)}}\exp \left(-{\frac {e^{-t}(x^{2}+y^{2})-2xy}{4\sinh t}}\right)=\sum _{n=0}^{\infty }(1-e^{-t})e^{-nt}~h_{n}(x)h_{n}(y)}

## Physics version

In physics, the [fundamental solution](/source/Fundamental_solution), ([Green's function](/source/Green's_function)), or [propagator](/source/Propagator#Basic_Examples:_Propagator_of_Free_Particle_and_Harmonic_Oscillator) of the Hamiltonian for the [quantum harmonic oscillator](/source/Quantum_harmonic_oscillator) is called the **Mehler kernel**. It provides the [fundamental solution](/source/Fundamental_solution)[4] φ ( x , t ) {\displaystyle \varphi (x,t)} to ∂ φ ∂ t = ∂ 2 φ ∂ x 2 − x 2 φ ≡ D x φ . {\displaystyle {\frac {\partial \varphi }{\partial t}}={\frac {\partial ^{2}\varphi }{\partial x^{2}}}-x^{2}\varphi \equiv D_{x}\varphi ~.}

The orthonormal eigenfunctions of the operator D {\displaystyle D} are the [Hermite functions](/source/Hermite_polynomials#Hermite_functions), ψ n = H n ( x ) e − x 2 / 2 2 n n ! π , {\displaystyle \psi _{n}={\frac {H_{n}(x)\,e^{-x^{2}/2}}{\sqrt {2^{n}n!{\sqrt {\pi }}}}},} with corresponding eigenvalues ( − 2 n − 1 ) {\displaystyle (-2n-1)} , furnishing particular solutions φ n ( x , t ) = e − ( 2 n + 1 ) t H n ( x ) e − x 2 / 2 . {\displaystyle \varphi _{n}(x,t)=e^{-(2n+1)t}~H_{n}(x)e^{-x^{2}/2}.}

The general solution is then a [linear combination](/source/Linear_combination) of these; when fitted to the initial condition φ ( x , 0 ) {\displaystyle \varphi (x,0)} , the general solution reduces to φ ( x , t ) = ∫ K ( x , y ; t ) φ ( y , 0 ) d y , {\displaystyle \varphi (x,t)=\int K(x,y;t)\varphi (y,0)dy~,} where the kernel K {\displaystyle K} has the separable representation K ( x , y ; t ) ≡ ∑ n ≥ 0 e − ( 2 n + 1 ) t π 2 n n ! H n ( x ) H n ( y ) exp ⁡ ( − x 2 + y 2 2 ) . {\displaystyle K(x,y;t)\equiv \sum _{n\geq 0}{\frac {e^{-(2n+1)t}}{{\sqrt {\pi }}2^{n}n!}}~H_{n}(x)H_{n}(y)\exp \left(-{\frac {x^{2}+y^{2}}{2}}\right)~.}

Utilizing Mehler's formula then yields ∑ n ≥ 0 ( ρ / 2 ) n n ! H n ( x ) H n ( y ) exp ⁡ ( − x 2 + y 2 2 ) = 1 1 − ρ 2 exp ⁡ ( 4 x y ρ − ( 1 + ρ 2 ) ( x 2 + y 2 ) 2 ( 1 − ρ 2 ) ) . {\displaystyle {\sum _{n\geq 0}{\frac {(\rho /2)^{n}}{n!}}H_{n}(x)H_{n}(y)\exp \left(-{\frac {x^{2}+y^{2}}{2}}\right)={\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left({\frac {4xy\rho -\left(1+\rho ^{2}\right)\left(x^{2}+y^{2}\right)}{2\left(1-\rho ^{2}\right)}}\right)}~.}

On substituting this in the expression for K {\displaystyle K} with the value e − 2 t {\displaystyle e^{-2t}} for ρ {\displaystyle \rho } , Mehler's kernel finally reads

K ( x , y ; t ) = 1 2 π sinh ⁡ ( 2 t ) exp ⁡ ( − coth ⁡ ( 2 t ) ( x 2 + y 2 ) / 2 + csch ⁡ ( 2 t ) x y ) . {\displaystyle K(x,y;t)={\frac {1}{\sqrt {2\pi \sinh(2t)}}}~\exp \left(-\coth(2t)~(x^{2}+y^{2})/2+\operatorname {csch} (2t)~xy\right).}

When t = 0 {\displaystyle t=0} , variables x {\displaystyle x} and y {\displaystyle y} coincide, resulting in the limiting formula necessary by the [initial condition](/source/Initial_condition), K ( x , y ; 0 ) = δ ( x − y ) . {\displaystyle K(x,y;0)=\delta (x-y)~.}

As a fundamental solution, the kernel is additive, ∫ d y K ( x , y ; t ) K ( y , z ; t ′ ) = K ( x , z ; t + t ′ ) . {\displaystyle \int dy\,K(x,y;t)K(y,z;t')=K(x,z;t{+}t')~.}

This is further related to the symplectic rotation structure of the kernel K {\displaystyle K} .[5]

When using the usual physics conventions of defining the [quantum harmonic oscillator](/source/Quantum_harmonic_oscillator) instead via i ∂ φ ∂ t = 1 2 ( − ∂ 2 ∂ x 2 + x 2 ) φ ≡ H φ , {\displaystyle i{\frac {\partial \varphi }{\partial t}}={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}\right)\varphi \equiv H\varphi ,} and assuming [natural length and energy scales](/source/Quantum_harmonic_oscillator#Natural_length_and_energy_scales), then the Mehler kernel becomes the [Feynman propagator](/source/Propagator) K H {\displaystyle K_{H}} which reads ⟨ x | exp ⁡ ( − i t H ) | y ⟩ ≡ K H ( x , y ; t ) = 1 2 π i sin ⁡ t exp ⁡ ( i 2 sin ⁡ t ( ( x 2 + y 2 ) cos ⁡ t − 2 x y ) ) , t < π , {\displaystyle {\begin{aligned}\left\langle x\right|\exp(-itH)\left|y\right\rangle &\equiv K_{H}(x,y;t)\\&={\frac {1}{\sqrt {2\pi i\sin t}}}\exp \left({\frac {i}{2\sin t}}\left(\left(x^{2}+y^{2}\right)\cos t-2xy\right)\right),\quad t<\pi ,\end{aligned}}} i.e. K H ( x , y ; t ) = K ( x , y ; i t / 2 ) . {\displaystyle K_{H}(x,y;t)=K(x,y;it/2).}

When t > π {\displaystyle t>\pi } the i sin ⁡ t {\displaystyle i\sin t} in the inverse square-root should be replaced by | sin ⁡ t | {\displaystyle \left|\sin t\right|} and K H {\displaystyle K_{H}} should be multiplied by an extra [Maslov phase](/source/Lagrangian_Grassmannian#Maslov_Index) factor [6] exp ⁡ ( i θ Maslov ) = exp ⁡ ( − i π 2 ( 1 2 + ⌊ t π ⌋ ) ) . {\displaystyle \exp \left(i\theta _{\text{Maslov}}\right)=\exp \left(-i{\frac {\pi }{2}}\left({\frac {1}{2}}+\left\lfloor {\frac {t}{\pi }}\right\rfloor \right)\right).}

When t = π / 2 {\displaystyle t=\pi /2} the general solution is proportional to the [Fourier transform](/source/Fourier_transform) F {\displaystyle {\mathcal {F}}} of the initial conditions φ 0 ( y ) ≡ φ ( y , 0 ) {\displaystyle \varphi _{0}(y)\equiv \varphi (y,0)} since φ ( x , t = π 2 ) = ∫ K H ( x , y ; π 2 ) φ ( y , 0 ) d y = 1 2 π i ∫ e − i x y φ ( y , 0 ) d y = e − i π / 4 F [ φ 0 ] ( x ) , {\displaystyle {\begin{aligned}\varphi (x,\,t{=}{\tfrac {\pi }{2}})&=\int K_{H}(x,y;{\tfrac {\pi }{2}})\varphi (y,0)\,dy\\[1ex]&={\frac {1}{\sqrt {2\pi i}}}\int e^{-ixy}\varphi (y,0)\,dy\\[1ex]&=e^{-i\pi /4}{\mathcal {F}}[\varphi _{0}](x)~,\end{aligned}}} and the exact [Fourier transform](/source/Fourier_transform) is thus obtained from the quantum harmonic oscillator's [number operator](/source/Quantum_harmonic_oscillator#Ladder_operator_method) written as[7] N ≡ 1 2 ( x − ∂ ∂ x ) ( x + ∂ ∂ x ) = H − 1 2 = 1 2 ( − ∂ 2 ∂ x 2 + x 2 − 1 ) {\displaystyle {\begin{aligned}N&\equiv {\frac {1}{2}}\left(x-{\frac {\partial }{\partial x}}\right)\left(x+{\frac {\partial }{\partial x}}\right)\\&=H-{\frac {1}{2}}={\frac {1}{2}}\left(-{\frac {\partial ^{2}}{\partial x^{2}}}+x^{2}-1\right)~\end{aligned}}} since the resulting kernel ⟨ x | exp ⁡ ( − i t N ) | y ⟩ ≡ K N ( x , y ; t ) = e i t / 2 K H ( x , y ; t ) = e i t / 2 K ( x , y ; i t / 2 ) {\displaystyle {\begin{aligned}\left\langle x\right|\exp(-itN)\left|y\right\rangle &\equiv K_{N}(x,y;t)\\&=e^{it/2}K_{H}(x,y;t)\\&=e^{it/2}K(x,y;it/2)\end{aligned}}} also compensates for the [phase factor](/source/Phase_factor) still arising in K H {\displaystyle K_{H}} and K {\displaystyle K} , i.e. φ ( x , t = π 2 ) = ∫ K N ( x , y ; π / 2 ) φ ( y , 0 ) d y = F [ φ 0 ] ( x ) , {\displaystyle \varphi (x,\,t{=}{\tfrac {\pi }{2}})=\int K_{N}(x,y;\pi /2)\varphi (y,0)dy={\mathcal {F}}[\varphi _{0}](x)~,} which shows that the [number operator](/source/Quantum_harmonic_oscillator#Ladder_operator_method) can be interpreted via the Mehler kernel as the [generator](/source/Symmetry_in_quantum_mechanics) of [fractional Fourier transforms](#Fractional_Fourier_transform) for arbitrary values of t {\displaystyle t} , and of the conventional [Fourier transform](/source/Fourier_transform) F {\displaystyle {\mathcal {F}}} for the particular value t = π / 2 {\displaystyle t=\pi /2} , with the Mehler kernel providing an [active transform](/source/Active_and_passive_transformation#In_abstract_vector_spaces), while the corresponding passive transform is already embedded in the [basis change](/source/Bra%E2%80%93ket_notation#The_unit_operator) from position to [momentum space](/source/Momentum_operator#Fourier_transform). The eigenfunctions of N {\displaystyle N} are the usual [Hermite functions](/source/Hermite_polynomials#Hermite_functions) ψ n ( x ) {\displaystyle \psi _{n}(x)} which are therefore also [Eigenfunctions](/source/Fourier_transform#Eigenfunctions) of F {\displaystyle {\mathcal {F}}} .[8]

## Proofs

There are many proofs of the formula.

The formula is a special case of the [Hardy–Hille formula](/source/Laguerre_polynomials#Hardy–Hille_formula), using the fact that the Hermite polynomials are a special case of the [associated Laguerre polynomials](/source/Associated_Laguerre_polynomials): H 2 n ( x ) = ( − 1 ) n 2 2 n n ! L n ( − 1 / 2 ) ( x 2 ) H 2 n + 1 ( x ) = ( − 1 ) n 2 2 n + 1 n ! x L n ( 1 / 2 ) ( x 2 ) {\displaystyle {\begin{aligned}H_{2n}(x)&=\left(-1\right)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\\[4pt]H_{2n+1}(x)&=\left(-1\right)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\end{aligned}}} The formula is a special case of the [Kibble–Slepian formula](/source/Hermite_polynomials#Kibble–Slepian_formula), so any proof of it immediately yields of proof of the Mehler formula.[9]

Foata gave a [combinatorial proof](/source/Combinatorial_proof) of the formula.[10]

[Hardy](/source/G._H._Hardy) gave a simple proof by the Fourier integral representation of Hermite polynomials.[11] Using the Fourier transform of the Gaussian e − x 2 = 1 π ∫ e − t 2 + 2 i x t d t {\textstyle e^{-x^{2}}={\frac {1}{\sqrt {\pi }}}\int e^{-t^{2}+2ixt}dt} , we have H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2 = e x 2 π ∫ ( − 2 i t ) n e − t 2 + 2 i x t d t {\displaystyle {\begin{aligned}H_{n}(x)&=\left(-1\right)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}e^{-x^{2}}\\&={\frac {e^{x^{2}}}{\sqrt {\pi }}}\int \left(-2it\right)^{n}e^{-t^{2}+2ixt}dt\end{aligned}}} from which the summation ∑ n = 0 ∞ ( ρ / 2 ) n n ! H n ( x ) H n ( y ) {\displaystyle \sum _{n=0}^{\infty }{\frac {(\rho /2)^{n}}{n!}}{\mathit {H}}_{n}(x){\mathit {H}}_{n}(y)} converts to a double integral over a summation e x 2 + y 2 π ∬ R 2 e − ( t 2 + s 2 ) + 2 i ( x t + y s ) ∑ n = 0 ∞ ( − 2 t s ρ ) n n ! d t d s {\displaystyle {\frac {e^{x^{2}+y^{2}}}{\pi }}\iint _{\mathbb {R} ^{2}}e^{-\left(t^{2}+s^{2}\right)+2i(xt+ys)}\sum _{n=0}^{\infty }{\frac {\left(-2ts\rho \right)^{n}}{n!}}dt\,ds} which can be evaluated directly as two [Gaussian integrals](/source/Gaussian_integral).

## Probability version

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as x → x / 2 {\displaystyle x\to x/{\sqrt {2}}} , y → y / 2 {\displaystyle y\to y/{\sqrt {2}}} , so as to change from the "physicist's" Hermite polynomials H ( ⋅ ) {\displaystyle H(\cdot )} (with weight function e − x 2 {\displaystyle e^{-x^{2}}} ) to "probabilist's" Hermite polynomials He ⁡ ( ⋅ ) {\displaystyle \operatorname {He} (\cdot )} (with weight function exp ⁡ ( − x 2 / 2 ) {\displaystyle \exp(-x^{2}/2)} ). They satisfy H n ( x ) = 2 n 2 He n ⁡ ( 2 x ) , He n ⁡ ( x ) = 2 − n 2 H n ( x 2 ) . {\displaystyle {\begin{aligned}H_{n}(x)&=2^{\frac {n}{2}}\operatorname {He} _{n}\left({\sqrt {2}}\,x\right),&\operatorname {He} _{n}(x)&=2^{-{\frac {n}{2}}}H_{n}{\left({\frac {x}{\sqrt {2}}}\right)}.\end{aligned}}} Then, E {\displaystyle E} becomes 1 1 − ρ 2 exp ⁡ ( − ρ 2 ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) ) = ∑ n = 0 ∞ ρ n n ! He n ⁡ ( x ) He n ⁡ ( y ) . {\displaystyle {\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}\left(x^{2}+y^{2}\right)-2\rho xy}{2\left(1-\rho ^{2}\right)}}\right)=\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)~.}

The left-hand side here is p ( x , y ) / p ( x ) p ( y ) {\displaystyle p(x,y)/p(x)p(y)} where p ( x , y ) {\displaystyle p(x,y)} is the [bivariate Gaussian probability density](/source/Multivariate_normal_distribution#Non-degenerate_case) function for variables x , y {\displaystyle x,y} having zero means and unit variances: p ( x , y ) = 1 2 π 1 − ρ 2 exp ⁡ ( − ( x 2 + y 2 ) − 2 ρ x y 2 ( 1 − ρ 2 ) ) , {\displaystyle p(x,y)={\frac {1}{2\pi {\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {(x^{2}+y^{2})-2\rho xy}{2(1-\rho ^{2})}}\right)~,} and p ( x ) , p ( y ) {\displaystyle p(x),p(y)} are the corresponding probability densities of x {\displaystyle x} and y {\displaystyle y} (both standard normal).

There follows the usually quoted form of the result (Kibble 1945)[12] p ( x , y ) = p ( x ) p ( y ) ∑ n = 0 ∞ ρ n n ! He n ⁡ ( x ) He n ⁡ ( y ) . {\displaystyle p(x,y)=p(x)p(y)\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)~.}

The exponent can be written in a more symmetric form: 1 1 − ρ 2 exp ⁡ ( ρ ( x + y ) 2 4 ( 1 + ρ ) − ρ ( x − y ) 2 4 ( 1 − ρ ) ) = ∑ n = 0 ∞ ρ n n ! He n ⁡ ( x ) He n ⁡ ( y ) . {\displaystyle {\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left({\frac {\rho (x+y)^{2}}{4(1+\rho )}}-{\frac {\rho (x-y)^{2}}{4(1-\rho )}}\right)=\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}~\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)~.} This expansion is most easily derived by using the two-dimensional Fourier transform of p ( x , y ) {\displaystyle p(x,y)} , which is c ( i u 1 , i u 2 ) = exp ⁡ ( − 1 2 ( u 1 2 + u 2 2 − 2 ρ u 1 u 2 ) ) . {\displaystyle c(iu_{1},iu_{2})=\exp \left(-{\tfrac {1}{2}}\left(u_{1}^{2}+u_{2}^{2}-2\rho u_{1}u_{2}\right)\right).}

This may be expanded as exp ⁡ ( − ( u 1 2 + u 2 2 ) / 2 ) ∑ n = 0 ∞ ρ n n ! ( u 1 u 2 ) n . {\displaystyle \exp(-(u_{1}^{2}+u_{2}^{2})/2)\sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}(u_{1}u_{2})^{n}~.} The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case.[12][13][14]

Erdélyi gave this as an integral over the [complex plane](/source/Complex_plane)[15] ∑ n = 0 ∞ ρ n n ! He n ⁡ ( x ) He n ⁡ ( y ) = 1 π t ∬ exp ⁡ [ − u 2 + v 2 ρ + ( u + i v ) x + ( u − i v ) y − 1 2 ( u + i v ) 2 − 1 2 ( u − i v ) 2 ] d u d v . {\displaystyle \sum _{n=0}^{\infty }{\frac {\rho ^{n}}{n!}}\operatorname {He} _{n}(x)\operatorname {He} _{n}(y)={\frac {1}{\pi t}}\iint \exp \left[-{\frac {u^{2}+v^{2}}{\rho }}+(u+iv)x+(u-iv)y-{\frac {1}{2}}(u+iv)^{2}-{\frac {1}{2}}(u-iv)^{2}\right]du\,dv.} which can be integrated with two Gaussian integrals, yielding the Mehler formula.

## Fractional Fourier transform

Main article: [Fractional Fourier transform](/source/Fractional_Fourier_transform)

Since Hermite functions ψ n {\displaystyle \psi _{n}} are orthonormal [eigenfunctions of the Fourier transform](/source/Fourier_transform#Eigenfunctions), F [ ψ n ] ( y ) = ( − i ) n ψ n ( y ) , {\displaystyle {\mathcal {F}}[\psi _{n}](y)=\left(-i\right)^{n}\psi _{n}(y)~,} in [harmonic analysis](/source/Harmonic_analysis) and [signal processing](/source/Signal_processing), they diagonalize the Fourier operator, F [ f ] ( y ) = ∫ d x f ( x ) ∑ n ≥ 0 ( − i ) n ψ n ( x ) ψ n ( y ) . {\displaystyle {\mathcal {F}}[f](y)=\int dxf(x)\sum _{n\geq 0}(-i)^{n}\psi _{n}(x)\psi _{n}(y)~.}

Thus, the continuous generalization for [real](/source/Real_number) angle α {\displaystyle \alpha } can be readily defined ([Wiener](/source/Norbert_Wiener), 1929;[16] [Condon](/source/Edward_Condon), 1937[17]), the [fractional Fourier transform](/source/Fractional_Fourier_transform) (FrFT), with kernel F α = ∑ n ≥ 0 ( − i ) 2 α n / π ψ n ( x ) ψ n ( y ) . {\displaystyle {\mathcal {F}}_{\alpha }=\sum _{n\geq 0}(-i)^{2\alpha n/\pi }\psi _{n}(x)\psi _{n}(y)~.}

This is a *continuous family of linear transforms generalizing the [Fourier transform](/source/Fourier_transform)*, such that, for α = π / 2 {\displaystyle \alpha =\pi /2} , it reduces to the standard Fourier transform, and for α = − π / 2 {\displaystyle \alpha =-\pi /2} to the inverse Fourier transform.

The Mehler formula, for ρ = e − i α {\displaystyle \rho =e^{-i\alpha }} , thus directly provides F α [ f ] ( y ) = 1 − i cot ⁡ ( α ) 2 π e i 2 cot ⁡ ( α ) y 2 / 2 ∫ − ∞ ∞ e − i ( csc ⁡ ( α ) y x − 1 2 cot ⁡ ( α ) x 2 ) f ( x ) d x . {\displaystyle {\mathcal {F}}_{\alpha }[f](y)={\sqrt {\frac {1-i\cot(\alpha )}{2\pi }}}~e^{{\frac {i}{2}}\cot(\alpha )y^{2}/2}\int _{-\infty }^{\infty }e^{-i\left(\csc(\alpha )yx-{\frac {1}{2}}\!\cot(\alpha )x^{2}\right)}f(x)\,\mathrm {d} x\,.} The [square root](/source/Square_root) is defined such that the argument of the result lies in the interval [ − π / 2 , π / 2 ] {\displaystyle [-\pi /2,\pi /2]} .

If α {\displaystyle \alpha } is an integer multiple of π {\displaystyle \pi } , then the above [cotangent](/source/Cotangent) and [cosecant](/source/Cosecant) functions diverge. In the [limit](/source/Limit_of_a_function), the kernel goes to a [Dirac delta function](/source/Dirac_delta_function) in the integrand, δ ( x − y ) {\displaystyle \delta (x-y)} or δ ( x + y ) {\displaystyle \delta (x+y)} , for α {\displaystyle \alpha } an [even or odd](/source/Even_and_odd_numbers) multiple of π {\displaystyle \pi } , respectively. Since F 2 [ f ] = f ( − x ) {\displaystyle {\mathcal {F}}^{2}[f]=f(-x)} , F α [ f ] {\displaystyle {\mathcal {F}}_{\alpha }[f]} must be simply f ( x ) {\displaystyle f(x)} or f ( − x ) {\displaystyle f(-x)} for α {\displaystyle \alpha } an even or odd multiple of π {\displaystyle \pi } , respectively.

## See also

- [Oscillator representation § Harmonic oscillator and Hermite functions](/source/Oscillator_representation#Harmonic_oscillator_and_Hermite_functions)

- [Heat kernel](/source/Heat_kernel)

- [Hermite polynomials](/source/Hermite_polynomials)

- [Parabolic cylinder functions](/source/Parabolic_cylinder_function)

- [Laguerre polynomials § Hardy–Hille formula](/source/Laguerre_polynomials#Hardy–Hille_formula)

## References

1. **[^](#cite_ref-1)** Hardy, G. H. (1932-07-01). ["Addendum: Summation of a Series of Polynomials of Laguerre*"](https://academic.oup.com/jlms/article/s1-7/3/192/960574). *Journal of the London Mathematical Society*. **s1-7** (3): 192. [doi](/source/Doi_(identifier)):[10.1112/jlms/s1-7.3.192-s](https://doi.org/10.1112%2Fjlms%2Fs1-7.3.192-s). [ISSN](/source/ISSN_(identifier)) [0024-6107](https://search.worldcat.org/issn/0024-6107).

1. **[^](#cite_ref-2)** Mehler, F. G. (1866), ["Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung"](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975), *Journal für die Reine und Angewandte Mathematik* (in German) (66): 161–176, [ISSN](/source/ISSN_(identifier)) [0075-4102](https://search.worldcat.org/issn/0075-4102), [ERAM](/source/ERAM_(identifier)) [066.1720cj](https://zbmath.org/?format=complete&q=an:066.1720cj) (cf. p 174, eqn (18) & p 173, eqn (13) )

1. **[^](#cite_ref-3)** [Erdélyi, Arthur](/source/Arthur_Erd%C3%A9lyi); [Magnus, Wilhelm](/source/Wilhelm_Magnus); Oberhettinger, Fritz; [Tricomi, Francesco G.](/source/Francesco_Tricomi) (1955), *Higher transcendental functions. Vol. II*, McGraw-Hill ([scan](http://www.nr.com/legacybooks): [p.194 10.13 (22)](http://apps.nrbook.com/bateman/Vol2.pdf))

1. **[^](#cite_ref-4)** [Pauli, W.](/source/Wolfgang_Pauli), *Wave Mechanics: Volume 5 of Pauli Lectures on Physics* (Dover Books on Physics, 2000) [ISBN](/source/ISBN_(identifier)) [0486414620](https://en.wikipedia.org/wiki/Special:BookSources/0486414620) ; See section 44.

1. **[^](#cite_ref-5)** The [quadratic form](/source/Quadratic_form) in its exponent, up to a factor of −1/2, involves the simplest (unimodular, symmetric) [symplectic matrix](/source/Symplectic_matrix) in Sp(2,**R**). That is, ( x , y ) M ( x y ) , {\displaystyle (x,y){\mathbf {M} }{\begin{pmatrix}x\\y\end{pmatrix}}~,~} where M ≡ csch ⁡ ( 2 t ) ( cosh ⁡ ( 2 t ) − 1 − 1 cosh ⁡ ( 2 t ) ) , {\displaystyle {\mathbf {M} }\equiv \operatorname {csch} (2t){\begin{pmatrix}\cosh(2t)&-1\\-1&\cosh(2t)\end{pmatrix}}~,} so it preserves the symplectic metric, M T ( 0 1 − 1 0 ) M = ( 0 1 − 1 0 ) . {\displaystyle {\mathbf {M} }^{\text{T}}~{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~{\mathbf {M} }={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}~.}

1. **[^](#cite_ref-6)** Horvathy, Peter (1979). "Extended Feynman Formula for Harmonic Oscillator". *International Journal of Theoretical Physics*. **18** (4): 245–250. [Bibcode](/source/Bibcode_(identifier)):[1979IJTP...18..245H](https://ui.adsabs.harvard.edu/abs/1979IJTP...18..245H). [doi](/source/Doi_(identifier)):[10.1007/BF00671761](https://doi.org/10.1007%2FBF00671761). [S2CID](/source/S2CID_(identifier)) [117363885](https://api.semanticscholar.org/CorpusID:117363885).

1. **[^](#cite_ref-7)** Wolf, Kurt B. (1979), *Integral Transforms in Science and Engineering*, Springer ([\[1\]](https://doi.org/10.1007/978-1-4757-0872-1) and [\[2\]](https://www.fis.unam.mx/~bwolf/integraleng.html)); see section 7.5.10.

1. **[^](#cite_ref-8)** Celeghini, Enrico; Gadella, Manuel; del Olmo, Mariano A. (2021). ["Hermite Functions and Fourier Series"](https://doi.org/10.3390%2Fsym13050853). *Symmetry*. **13** (5): 853. [arXiv](/source/ArXiv_(identifier)):[2007.10406](https://arxiv.org/abs/2007.10406). [Bibcode](/source/Bibcode_(identifier)):[2021Symm...13..853C](https://ui.adsabs.harvard.edu/abs/2021Symm...13..853C). [doi](/source/Doi_(identifier)):[10.3390/sym13050853](https://doi.org/10.3390%2Fsym13050853).

1. **[^](#cite_ref-9)** Ismail, Mourad E. H.; Zhang, Ruiming (2017-04-01). ["A review of multivariate orthogonal polynomials"](https://www.sciencedirect.com/science/article/pii/S1110256X16300761#bib0028). *Journal of the Egyptian Mathematical Society*. **25** (2): 91–110. [doi](/source/Doi_(identifier)):[10.1016/j.joems.2016.11.001](https://doi.org/10.1016%2Fj.joems.2016.11.001). [ISSN](/source/ISSN_(identifier)) [1110-256X](https://search.worldcat.org/issn/1110-256X).

1. **[^](#cite_ref-10)** Foata, Dominique (1978-05-01). ["A combinatorial proof of the Mehler formula"](https://www.sciencedirect.com/science/article/pii/0097316578900663). *Journal of Combinatorial Theory, Series A*. **24** (3): 367–376. [doi](/source/Doi_(identifier)):[10.1016/0097-3165(78)90066-3](https://doi.org/10.1016%2F0097-3165%2878%2990066-3). [ISSN](/source/ISSN_(identifier)) [0097-3165](https://search.worldcat.org/issn/0097-3165).

1. **[^](#cite_ref-11)** Watson, G. N. (July 1933). ["Notes on Generating Functions of Polynomials: (2) Hermite Polynomials"](http://doi.wiley.com/10.1112/jlms/s1-8.3.194). *Journal of the London Mathematical Society*. **s1-8** (3): 194–199. [doi](/source/Doi_(identifier)):[10.1112/jlms/s1-8.3.194](https://doi.org/10.1112%2Fjlms%2Fs1-8.3.194).

1. ^ [***a***](#cite_ref-Kibble_12-0) [***b***](#cite_ref-Kibble_12-1) Kibble, W. F. (1945). "An extension of a theorem of Mehler's on Hermite polynomials". *[Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society)*. **41** (1): 12–15. [Bibcode](/source/Bibcode_(identifier)):[1945PCPS...41...12K](https://ui.adsabs.harvard.edu/abs/1945PCPS...41...12K). [doi](/source/Doi_(identifier)):[10.1017/S0305004100022313](https://doi.org/10.1017%2FS0305004100022313). [MR](/source/MR_(identifier)) [0012728](https://mathscinet.ams.org/mathscinet-getitem?mr=0012728). [S2CID](/source/S2CID_(identifier)) [121931906](https://api.semanticscholar.org/CorpusID:121931906).

1. **[^](#cite_ref-13)** Slepian, David (1972), "On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials", *SIAM Journal on Mathematical Analysis*, **3** (4): 606–616, [doi](/source/Doi_(identifier)):[10.1137/0503060](https://doi.org/10.1137%2F0503060), [ISSN](/source/ISSN_(identifier)) [0036-1410](https://search.worldcat.org/issn/0036-1410), [MR](/source/MR_(identifier)) [0315173](https://mathscinet.ams.org/mathscinet-getitem?mr=0315173)

1. **[^](#cite_ref-14)** [Hörmander, Lars](/source/Lars_H%C3%B6rmander) (1995). "Symplectic classification of quadratic forms, and general Mehler formulas". *Mathematische Zeitschrift*. **219**: 413–449. [doi](/source/Doi_(identifier)):[10.1007/BF02572374](https://doi.org/10.1007%2FBF02572374). [S2CID](/source/S2CID_(identifier)) [122233884](https://api.semanticscholar.org/CorpusID:122233884).

1. **[^](#cite_ref-15)** Erdélyi, Artur (1939-12-01). ["Über eine erzeugende Funktion von Produkten Hermitescher Polynome"](https://link.springer.com/article/10.1007/BF01210650). *Mathematische Zeitschrift* (in German). **44** (1): 201–211. [doi](/source/Doi_(identifier)):[10.1007/BF01210650](https://doi.org/10.1007%2FBF01210650). [ISSN](/source/ISSN_(identifier)) [1432-1823](https://search.worldcat.org/issn/1432-1823).

1. **[^](#cite_ref-16)** [Wiener](/source/Norbert_Wiener), N (1929), "Hermitian Polynomials and Fourier Analysis", *Journal of Mathematics and Physics* **8**: 70–73.

1. **[^](#cite_ref-17)** [Condon, E. U.](/source/Edward_Condon) (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", *Proc. Natl. Acad. Sci. USA* **23**, 158–164. [online](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf)

- Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). *Heat Kernels and Dirac Operators*, (Springer: Grundlehren Text Editions) Paperback [ISBN](/source/ISBN_(identifier)) [3540200622](https://en.wikipedia.org/wiki/Special:BookSources/3540200622)

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- Srivastava, H. M.; Singhal, J. P. (1972). ["Some extensions of the Mehler formula"](https://doi.org/10.1090%2FS0002-9939-1972-0285738-4). *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **31**: 135–141. [doi](/source/Doi_(identifier)):[10.1090/S0002-9939-1972-0285738-4](https://doi.org/10.1090%2FS0002-9939-1972-0285738-4).

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Adapted from the Wikipedia article [Mehler kernel](https://en.wikipedia.org/wiki/Mehler_kernel) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mehler_kernel?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.