Mittag-Leffler for non-compact Riemann surfaces

Question

Quote from Grauert & Remmert's Theory of Stein spaces: 'Behnke and Stein showed in 1948 that the Mittag-Leffier Partial Fraction Theorem and the Weierstrass Product Theorem (i.e. the Cousin Theorems) are valid on non-compact Riemann surfaces. The following lemma appears at the end of their paper:

Hilfssatz C: Let $D$ be a discrete set in a non-compact Riemann surface $X$. For every $p\in D$ let $z_p$ be a local coordinate at p. Suppose that at all $p \in D$ there is prescribed a finite Laurent-Taylor series $h_p = \sum_{\nu=-m_p}^{n_p}a_\nu z_p^{\nu}$, $0\leq m_p,n_p<\infty$. Then there exists a function $H$ which is meromorphic on $X$, holomorphic on $X\setminus D$, and whose Laurent development at $p$ with respect to $z_p$ agrees with $h_p$ up to the $n_p$-th term.'

Does $H$ still exist if we ask it to have no zeroes in $X\setminus D$? That is, I want to $H$ to realize exactly a prescribed divisor. References are welcomed.

Answer 1

Considering the reciprocal function, it is sufficient to construct a holomorphic function with prescribed zeros, and prescribed finite portions of Taylor series at those zeros. For the plane and the unit disk this is an interpolation problem whose positive solution follows from Weierstrass's theorem (and its generalization to the unit disk).