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We present new exactly solvable systems of the discrete quantum mechanicswith pure imaginary shifts, whose physical range of the coordinate is the wholereal line. These systems are shape invariant and their eigenfunctions aredescribed by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonalpolynomials. The set of degrees of these multi-indexed polynomials are$\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$,where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : amulti-index set), but they form a complete set of orthogonal basis in theweighted Hilbert space.

Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types