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q-Racah probability distribution

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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 \({}_4 \phi _3\)-hypergeometric series.

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Acknowledgements

They would also like to thank the referees for valuable comments and suggestions on improvements. In particular, as explained in Remarks 1.5, 3.2 and 3.4, the statement of Theorem 1.4 1 and some of the arguments in Sect. 3.2 are added after their suggestions. Finally, S.Y. thanks Professor Masatoshi Noumi for valuable comments on spherical functions and hypergeometric functions.

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Authors and Affiliations

  1. School of Data Science, The Chinese University of Hong Kong, Shenzhen, Longgang District, Shenzhen, 518172, China

    Masahito Hayashi

  2. International Quantum Academy (SIQA), Shenzhen, 518048, China

    Masahito Hayashi

  3. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Masahito Hayashi & Shintarou Yanagida

  4. Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

    Akihito Hora

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  1. Masahito Hayashi
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  2. Akihito Hora
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  3. Shintarou Yanagida
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Correspondence to Shintarou Yanagida.

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M. H. was supported in part by the National Natural Science Foundation of China under Grant 62171212. A. H. was supported in part by JSPS Grant-in-Aids for Scientific Research Grant Number 19K03532. S. Y. was supported in part by JSPS Grant-in-Aids for Scientific Research Grant Number 19K03399. A part of this article is presented in the 16th OPFSA (International Symposium on Orthogonal Polynomials, Special Functions and Application). The authors thank the organizers for the opportunity.

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Hayashi, M., Hora, A. & Yanagida, S. q-Racah probability distribution. Ramanujan J (2024). https://doi.org/10.1007/s11139-024-00859-w

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