Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods

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The Kibble-Slepian formula expresses the exponential of a quadratic form Q( x) = x t S( I + S) −1 x, S t = S, in n variables x = col( x 1,…, x n ) as a series of products of Hermite polynomials, thus generalizing Mehler's formula. This extension is restricted, however, to the case where the diagonal elements of the symmetric matrix S are all unity. We derive the general formula for an arbitrary symmetric matrix S, where I + S is positive definite, using techniques familiar from the boson operator treatment of the harmonic oscillator in quantum mechanics.