{
  "abstract": "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 expansion. To perform these calculations, we give a few auxiliary formulas concerning Hermite polynomials and multivariate normal distributions. We apply this result to obtain exact, simple forms of these expansions for $n=2$ and $3$, thus looking at known results from a different perspective. We believe that the results can be very useful in Probability Theory, in particular in the Theory of distributions, Statistics, and the Theory of Stochastic Processes, in particular Gaussian ones.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.22526",
  "author_total": 1,
  "authors": [
    {
      "name": "Paweł J. Szabłowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ezw159zmmtjh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.22526",
  "fulltext_url": null,
  "id": "8ezw159zmmtjh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.22526",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7165623170",
  "pdf_url": "https://arxiv.org/pdf/2606.22526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ezw159zmmtjh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.22526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.22526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "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",
  "title_text": "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",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8ezw159zmmtjh",
  "venue": "arXiv",
  "versions": []
}