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The rational Heun operator and Wilson biorthogonal functions


We consider the rational Heun operator defined as the most general second-order q-difference operator which sends any rational function of type \([(n-1)/n]\) to a rational function of type \([n/(n+1)]\). We shall take the poles to be located on the Askey–Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars–van Diejen Hamiltonians. The Wilson biorthogonal functions of type \({_{10}}\Phi _9\) are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special “classical” kind.

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The authors are indebted to V. Spiridonov for drawing their attention to the reference [18].

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Correspondence to Luc Vinet.

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To the memory of Dick Askey, mathematician and pedagogue extraordinaire, with admiration and gratitude.

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The work of S.T. is partially supported by JSPS KAKENHI Grant Numbers 19H01792, 17K18725. The research of L.V. is funded in part by a discovery grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The work of A.Z. is supported by the National Science Foundation of China (Grant No. 11771015)

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Tsujimoto, S., Vinet, L. & Zhedanov, A. The rational Heun operator and Wilson biorthogonal functions. Ramanujan J (2021).

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  • Rational Heun operators
  • Ruijsenaars–van Diejen Hamiltonians
  • Wilson biorthogonal functions

Mathematics Subject Classification

  • 33D45
  • 47B36