$$e^{tx+h} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{tx+h-\ln({\frac{t+1}{t}})(m-1)}{mt\ln({\frac{t+1}{t}})}$$ This is my attempt at parameterizing the function $ e^{tx+h} $ in terms of a power series using a repeated product. I would very much appreciate help in actually proving it. I don't know how to find the radius of convergence either, so that might be a good start.
Some of my favourite math hobbies are Maclaurin and Taylor series. Recently I was interested to see what would happen when they are parameterized in terms of a repeated product. The following relationship is pretty straightforward: $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = \sum_{n=0}^\infty \prod_{m=1}^n \frac{x}{m}$$
The original motivators to find a parameterization were the following relationships, which change the exponential base from e to 2.
$$\sum_{n=0}^\infty \frac{\ln(2)^nx^n}{n!} = 2^x = \sum_{n=0}^\infty \prod_{m=1}^n \frac{x-m+1}{m}$$ and similarly, $$2^x = \sum_{n=0}^\infty \prod_{m=1}^n \frac{x+m-1}{2m}$$