Analyticity criterion for suitable C-Summation sequences

Let $k_1$ and $k_2$ be the coefficients in the recurrence relating a sequence $(I_n)$ to the corresponding continued fraction, and let $n$ be fixed. The associated generating series is

$$\sum_{k=n}^\infty I_k z^k.$$

Analyticity criterion for C-Summation. If this series is analytic at $-\frac{k_2}{k_1}$, then $I_n$ is suitable for C-Summation.

This criterion is proposed to resolve the nonuniqueness of sequences corresponding to the same continued fraction and sum, particularly when the corresponding series is divergent. The source does not provide a resolution or status beyond stating the criterion.