Mittag-Leffler Theorem Correction Factor

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From what I learned and also in http://math.iisc.ac.in/~vvdatar/courses/2020_Jan/Lecture_Notes/Lecture-17.pdf, it is said if $\Omega=\mathbb{C}$, then $q_k$ can be polynomial. I wonder how if $\Omega \subsetneq \mathbb{C}$? Can $q_k$ be not a polynomial?