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$q$-Racah probability distribution

Masahito Hayashi, Akihito Hora, Shintarou Yanagida

TL;DR

This work constructs a multi-parameter discrete probability distribution $P_{n,m,k,l;q}$ whose pmf is governed by a $q$-Racah polynomial and whose cdf is a terminating ${}_{4}\phi_{3}$-series, arising from a zonal spherical function on the Grassmannian over a finite field. Positivity of the pmf is established in three regimes—$q$ a prime power, $q=1$, and $q$ near 1—via representation-theoretic realisations using $\mathrm{GL}(n,\mathbb{F}_q)$ and Schur-Weyl duality, with the normalization and cdf derived from two terminating hypergeometric summations. The $q\to1$ Racah limit yields a Racah-polynomial form, enabling detailed asymptotic analysis and potential applications in quantum information theory. Overall, the paper positions the $q$-Racah family as a master-class for a new master-distribution in discrete probability, connecting hypergeometric orthogonal polynomials, Gelfand pairs, and Grassmannian geometry in a unified framework.

Abstract

We introduce a certain discrete probability distribution $P_{n,m,k,l;q}$ having non-negative integer parameters $n,m,k,l$ and quantum parameter $q$ which arises from a zonal spherical function of the Grassmannian over the finite field $\mathbb{F}_q$ with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single $q$-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating ${}_4 φ_3$-hypergeometric series.

$q$-Racah probability distribution

TL;DR

This work constructs a multi-parameter discrete probability distribution whose pmf is governed by a -Racah polynomial and whose cdf is a terminating -series, arising from a zonal spherical function on the Grassmannian over a finite field. Positivity of the pmf is established in three regimes— a prime power, , and near 1—via representation-theoretic realisations using and Schur-Weyl duality, with the normalization and cdf derived from two terminating hypergeometric summations. The Racah limit yields a Racah-polynomial form, enabling detailed asymptotic analysis and potential applications in quantum information theory. Overall, the paper positions the -Racah family as a master-class for a new master-distribution in discrete probability, connecting hypergeometric orthogonal polynomials, Gelfand pairs, and Grassmannian geometry in a unified framework.

Abstract

We introduce a certain discrete probability distribution having non-negative integer parameters and quantum parameter which arises from a zonal spherical function of the Grassmannian over the finite field with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single -Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating -hypergeometric series.
Paper Structure (16 sections, 15 theorems, 94 equations)

This paper contains 16 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Racah and $q$-Racah polynomials
  3. Features of this article
  4. Summary of results
  5. Organization and notation
  6. Summation formulas
  7. The first summation formula
  8. The second summation formula
  9. The case $q$ is a prime power
  10. Gelfand pair (GL(n,Fq),P(m,n-m,Fq) and q-Hahn polynomials
  11. The positivity from GL(n,Fq)-action
  12. The case q=1
  13. The limit form under q to 1
  14. The positivity from Sn-action
  15. The case 0<|q-1|<<1
  16. ...and 1 more sections

Key Result

Theorem 1.2

For $(n,m,k,l) \in \mathfrak{N}$ and $x\in\{0,1,\dotsc,m\}$, we have the following identity in $\mathbb{Q}(q)$.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2: $q$-Hahn presentation, \ref{['thm:sum1']}
  • Theorem 1.3: \ref{['thm:q:sum=1']}
  • Theorem 1.4: $q$-Racah probability distribution
  • Remark 1.5: added after the referees' comments
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • ...and 17 more