Deformed Schur Indices of BCD-type for N=4 Super Yang-Mills and Symmetric Functions

We investigate the deformed Schur index in four dimensional N=4 super Yang-Mills theories with $SO$ and $Sp$ gauge groups, generalizing Hatsuda's recent calculations. We express the deformed Schur index as integrals of Koornwinder polynomials and Macdonald polynomials, then perform the integrals in terms of the normalization constants of Macdonald polynomials. We provide explicit results for some low rank gauge groups and for expansion in a $u$ parameter. We discuss various special limits and the tests of S-duality.