Abstract
We prove an infinite family of identities satisfied by the Rankin–Cohen brackets involving the Racah polynomials. A natural interpretation in the representation theory of sl(2) is provided. From these identities and known properties of the Racah polynomials follows a short new proof of the associativity of the Eholzer product. Finally, we discuss, in the context of Rankin–Cohen algebras introduced by Zagier, how any algebraic identity satisfied by the Rankin–Cohen brackets can be seen as a consequence of the set of identities presented in this paper.
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Acknowledgements
The authors warmly thank M. Pevzner and G. Zhang for their interest in this work. The first author is supported by a research grant from the Villum Foundation (Grant No. 00025373). The second author is supported by Agence Nationale de la Recherche Projet AHA ANR-18-CE40-0001 and the international research project AAPT of the CNRS.
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Labriet, Q., Poulain d’Andecy, L. Identities for Rankin–Cohen brackets, Racah coefficients and associativity. Lett Math Phys 114, 20 (2024). https://doi.org/10.1007/s11005-023-01763-y
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DOI: https://doi.org/10.1007/s11005-023-01763-y