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  "body": "\n## Statement\n\nLet $H_n$ be the (physicists') [[Hermite polynomial|MGACY98E9H]] of degree $n$.\nLet $\\rho \\in \\C$ with $\\abs{\\rho} \u003c 1$, and let $x, y \\in \\R$.\nThen (Mehler's formula)\n$$\\sum_{n=0}^{\\infty} \\frac{H_n(x) H_n(y)}{n!} \\left( \\frac{\\rho}{2} \\right)^n = \\frac{1}{\\sqrt{1 - \\rho^2}} \\exp\\left( \\frac{2\\rho x y - \\rho^2 (x^2 + y^2)}{1 - \\rho^2} \\right),$$\nwhere the square root is the principal one.\nThe right-hand side, as a function of $(x, y)$, is the *Mehler kernel*.\n\n## Remarks\n\nProof idea: insert the integral formula $H_n(x) = \\frac{2^n}{\\sqrt{\\pi}} \\int_{\\R} (x + it)^n e^{-t^2} \\, dt$ for both factors.\nThe sum over $n$ becomes an exponential, and the remaining double integral is Gaussian and can be computed in closed form.\nCheck at $x = y = 0$: $H_0(0) = 1$, $H_2(0) = -2$ and $H_4(0) = 12$ give $1 + \\frac{\\rho^2}{2} + \\frac{3\\rho^4}{8} + \\cdots$, which is the expansion of $(1 - \\rho^2)^{-1/2}$.\nIn terms of the [[Hermite functions|52VS07H6GK]] $\\psi_n$ the formula reads\n$$\\sum_{n=0}^{\\infty} \\rho^n \\psi_n(x) \\psi_n(y) = \\frac{1}{\\sqrt{\\pi (1 - \\rho^2)}} \\exp\\left( -\\frac{(1 + \\rho^2)(x^2 + y^2) - 4\\rho x y}{2 (1 - \\rho^2)} \\right).$$\nSo the Mehler kernel is the integral kernel of the operator that multiplies the $n$-th Hermite function by $\\rho^n$.\nThe hypothesis $\\abs{\\rho} \u003c 1$ is needed: at $\\rho = 1$ the right side is undefined and the series does not converge to a function.\n\n## Explanations\n\n### L1 human\n\nMehler's formula adds up an infinite series of products of Hermite polynomials and gets a single Gaussian expression.\nIt is the Hermite analogue of summing a geometric series.\nThe resulting kernel describes how a quantum harmonic oscillator evolves in time.\nIt also describes a particle pulled back to the origin by a spring and shaken by noise.\n\n### L2 human\n\nWrite $N$ for the operator with $N \\psi_n = n \\psi_n$, the number operator of the harmonic oscillator.\nThen the Mehler kernel with $\\rho = e^{-t}$, $t \u003e 0$, is the kernel of $e^{-tN}$, a smoothing semigroup on $L^2(\\R)$.\nIn probability, the probabilists' version of the same formula gives the transition density of the Ornstein–Uhlenbeck process.\nLetting $\\rho$ tend to a point $e^{-i\\alpha}$ of the unit circle leads to the kernel of the fractional Fourier transform.\nThe formula is a model for bilinear generating functions of other orthogonal polynomials, such as the Hille–Hardy formula for Laguerre polynomials.\n",
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