From Kajihara’s transformation formula to deformed Macdonald–Ruijsenaars and Noumi–Sano operators
Journal article, 2022

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald–Ruijsenaars (MR) and Noumi–Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.

Macdonald–Ruijsenaars operators

Multiple basic hypergeometric series

Noumi–Sano operators

Kernel identities


Martin Hallnäs

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

Edwin Langmann

Royal Institute of Technology (KTH)

Masatoshi Noumi

Kobe University

Royal Institute of Technology (KTH)

Hjalmar Rosengren

University of Gothenburg

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Selecta Mathematica, New Series

1022-1824 (ISSN) 14209020 (eISSN)

Vol. 28 2 24

Quasi-invariants of finite Coxeter groups and integrable systems

Swedish Research Council (VR) (2018-04291), 2019-01-01 -- 2022-12-31.

Subject Categories


Mathematical Analysis



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