Editing Hermite polynomials (section)

=== Kibble–Slepian formula ===

Let <math display="inline">M</math> be a real <math display="inline">n\times n</math> symmetric matrix, then the '''Kibble–Slepian formula''' states that<math display="block">\det(I+M)^{-\frac 12 } e^{x^T M (I+M)^{-1}x} = \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \cdots H_{k_n}(x_n)</math> where <math display="inline">\sum_K</math> is the <math>\frac{n(n+1)}{2}</math>-fold summation over all <math display="inline">n \times n</math> symmetric matrices with non-negative integer entries, <math>tr(K)</math> is the [[Trace (linear algebra)|trace]] of <math>K</math>, and <math display="inline">k_i</math> is defined as <math display="inline">k_{ii} + \sum_{j=1}^n k_{ij}</math>. This gives [[Mehler kernel|Mehler's formula]] when <math>M = \begin{bmatrix} 0 & u \\ u & 0\end{bmatrix}</math>.

Equivalently stated, if <math display="inline">T</math> is a [[Positive semidefinite matrices|positive semidefinite matrix]], then set <math display="inline">M = -T(I+T)^{-1}</math>, we have <math display="inline">M(I+M)^{-1} = -T</math>, so <math display="block">
e^{-x^T T x} = \det(I+T)^{-\frac 12} \sum_K \left[\prod_{1\leq i \leq j \leq n} \frac{(M_{ij}/2)^{k_{ij}}}{k_{ij}!}\right] 2^{-tr(K)} H_{k_1}(x_1) \dots H_{k_n}(x_n)
</math>Equivalently stated in a form closer to the [[boson]] [[quantum mechanics]] of the [[harmonic oscillator]]:<ref name=":0">{{Cite journal |last=Louck |first=J. D |date=1981-09-01 |title=Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods |url=https://dx.doi.org/10.1016/0196-8858%2881%2990005-1 |journal=Advances in Applied Mathematics |volume=2 |issue=3 |pages=239–249 |doi=10.1016/0196-8858(81)90005-1 |issn=0196-8858}}</ref><math display="block">
\pi^{-n/4}\det(I+M)^{-\frac 12 }e^{- \frac 12  x^T(I-M)(I+M)^{-1} x}= \sum_K\left[\prod_{1 \leq i \leq j \leq n} M_{i j}^{k_{i j}} / k_{i j}!\right]\left[\prod_{1 \leq i \leq n} k_{i}!\right]^{1 / 2}  2^{-\operatorname{tr} K} \psi_{k_1}\left(x_1\right) \cdots \psi_{k_n}\left(x_n\right) .
</math> where each <math display="inline">\psi_n(x)</math> is the <math display="inline">n</math>-th eigenfunction of the harmonic oscillator, defined as <math display="block">\psi_n(x) := \frac{1}{\sqrt{2^n n!}}\left(\frac{1}{\pi}\right)^{\frac{1}{4}} e^{-\frac{1}{2} x^2} H_n(x)  </math>The Kibble–Slepian formula was proposed by Kibble in 1945<ref>{{Cite journal |last=Kibble |first=W. F. |date=June 1945 |title=An extension of a theorem of Mehler's on Hermite polynomials |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/an-extension-of-a-theorem-of-mehlers-on-hermite-polynomials/6CD265E3054D1595062F1CA83D492AC2 |journal=Mathematical Proceedings of the Cambridge Philosophical Society |language=en |volume=41 |issue=1 |pages=12–15 |doi=10.1017/S0305004100022313 |bibcode=1945PCPS...41...12K |issn=1469-8064}}</ref> and proven by Slepian in 1972 using Fourier analysis.<ref>{{Cite journal |last=Slepian |first=David |date=November 1972 |title=On the Symmetrized Kronecker Power of a Matrix and Extensions of Mehler's Formula for Hermite Polynomials |url=https://epubs.siam.org/doi/abs/10.1137/0503060 |journal=SIAM Journal on Mathematical Analysis |volume=3 |issue=4 |pages=606–616 |doi=10.1137/0503060 |issn=0036-1410}}</ref> Foata gave a combinatorial proof<ref>{{Cite journal |last=Foata |first=Dominique |date=1981-09-01 |title=Some Hermite polynomial identities and their combinatorics |url=https://dx.doi.org/10.1016/0196-8858%2881%2990006-3 |journal=Advances in Applied Mathematics |volume=2 |issue=3 |pages=250–259 |doi=10.1016/0196-8858(81)90006-3 |issn=0196-8858}}</ref> while Louck gave a proof via boson quantum mechanics.<ref name=":0" /> It has a generalization for complex-argument Hermite polynomials.<ref>{{Cite journal |last1=Ismail |first1=Mourad E.H. |last2=Zhang |first2=Ruiming |date=September 2016 |title=Kibble–Slepian formula and generating functions for 2D polynomials |url=https://doi.org/10.1016/j.aam.2016.05.003 |journal=Advances in Applied Mathematics |volume=80 |pages=70–92 |doi=10.1016/j.aam.2016.05.003 |issn=0196-8858|arxiv=1508.01816 }}</ref><ref>{{Cite journal |last1=Ismail |first1=Mourad E. H. |last2=Zhang |first2=Ruiming |date=2017-04-01 |title=A review of multivariate orthogonal polynomials |journal=Journal of the Egyptian Mathematical Society |volume=25 |issue=2 |pages=91–110 |doi=10.1016/j.joems.2016.11.001 |issn=1110-256X|doi-access=free }}</ref>