I have the following characteristic function $$\sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{m,k} \frac{\Gamma(\beta + m)}{\Gamma(\beta)},$$ where $i$ is the imaginary unit, $\beta>0$, $\Gamma(\cdot)$ is the Gamma function and $$\sigma_{m,k} \equiv \frac{\Gamma(\beta+\alpha)\Gamma(\beta+k)}{\Gamma(\beta + m)\Gamma(\beta + \alpha + k)},$$ where $k$ can be any non-negative integer and $\alpha > 0$ is another parameter. If there exists a $\sigma_k > 0$ (a positive constant that does not depend on $m$, but can depend on $k$), such that $$(\star) \qquad \qquad \sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{m,k} \frac{\Gamma(\beta + m)}{\Gamma(\beta)} = \sum_{m=0}^{\infty} \frac{(is)^m}{m!} \sigma_{k}^m \frac{\Gamma(\beta + m)}{\Gamma(\beta)}$$ then I know that the left hand side is the characteristic function of a Gamma distributed random variable with shape parameter $\beta$ and scale parameter $\sigma_k$, since the right hand-side is.

My Question: is it possible to show that there exists a $\sigma_k$ so that $(\star)$ is true? Is there some form of intermediate value theorem that I can appeal to here? Note that the sequence $\sigma_{0,k}, \sigma_{1,k}, ...$ is a decreasing sequence.


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