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We present how explicit eigenfunctions of the principal Hamiltonian for the$BC_{m}$ relativistic Calogero-Moser-Sutherland model, due to van Diejen, canbe constructed using gauge and integral transformations. In particular, we findthat certain $BC$-type elliptic hypergeometric integrals, including ellipticSelberg integrals, of both Type I and Type II arise as eigenfunctions of thevan Diejen model under some parameter restrictions. Among these are also jointeigenfunctions of so-called modular pairs of van Diejen operators. Furthermore,these transformations are related to reflections of the $E_{8}$ Weyl groupacting on the space of model parameters.

Eigenfunctions of the van Diejen model generated by gauge and integral transformations