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Kajihara obtained in 2004 a remarkable transformation formula connectingmultiple basic hypergeometric series associated with $A$-type root systems ofdifferent ranks. By multiple principle specialisations of his formula, wededuce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano(NS) operators. The deformed MR operators were introduced by Sergeev andVeselov in the first order case and by Feigin and Silantyev in the higher ordercases. As applications of our kernel identities, we prove that all of theseoperators pairwise commute and are simultaneously diagonalised by thesuper-Macdonald polynomials. We also provide an explicit description of thealgebra generated by the deformed MR and/or NS operators by a Harish-Chandratype isomorphism and show that the deformed MR (NS) operators can be viewed asrestrictions of inverse limits of ordinary MR (NS) operators.

From Kajihara's transformation formula to deformed Macdonald-Ruijsenaars and Noumi-Sano operators