  We consider dual polynomials of the multi-indexed ($q$-)Racah orthogonal
polynomials. The $M$-indexed ($q$-)Racah polynomials satisfy the second order
difference equations and various $1+2L$ ($L\geq M+1$) term recurrence relations
with constant coefficients. Therefore their dual polynomials satisfy the three
term recurrence relations and various $2L$-th order difference equations. This
means that the dual multi-indexed ($q$-)Racah polynomials are ordinary
orthogonal polynomials and the Krall-type. We obtain new exactly solvable
discrete quantum mechanics with real shifts, whose eigenvectors are described
by the dual multi-indexed ($q$-)Racah polynomials. These quantum systems
satisfy the closure relations, from which the creation/annihilation operators
are obtained, but they are not shape invariant.
