# Characteristic Function of Gamma Distributed Random Variables

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.