{"paper":{"title":"The conditional expectation of the product of the first $n-1$ Hermite polynomials in a multivariate normal distribution with respect to the $n$-th variable. A fresh perspective on the Kibble-Slepian formula","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pawe{\\l} J. Szab{\\l}owski","submitted_at":"2026-06-21T14:25:16Z","abstract_excerpt":"We calculate the conditional expectation of $\\prod_{j=1}^{n-1}H_{k_{j}}% (X_{j})$ given $X_{n}=z,$ if random vector $(X_{1},\\ldots,X_{n})^{T}$ has multivariate normal distribution and $H_{n}(x) $ denotes $n$-th Hermite polynomial. This expectation is a polynomial in $z$ of order $\\sum_{j=1}% ^{n-1}k_{j}$. Our formula has an iterative form with respect to $n$. We also present some auxiliary observations concerning the expansion of the density of the $n$-dimensional normal distribution in the series of the Hermite polynomials. Mostly concerning the properties of the coefficients of this expansio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22526","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.22526/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}