  Kajihara obtained in 2004 a remarkable transformation formula connecting
multiple basic hypergeometric series associated with $A$-type root systems of
different ranks. By multiple principle 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.
