Is there are a term for a generalized exponential function? The series expansions of sine and cosine look very similar to the exponential function's series expansion
\begin{align} & \sum_{n = 0}^{\infty} \frac{x^n}{n!} \\ & \sum_{n = 0}^{\infty} \frac{(-x)^n}{n!} \\ & \sum_{n = 0}^{\infty} \frac{x^{2n}}{(2n)!} & & \sum_{n = 0}^{\infty} \frac{x^{2n + 1}}{(2n + 1)!} & \\ & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!} \\ & \sum_{n = 0}^{\infty} \frac{x^{3n}}{(3n)!} & & \sum_{n = 0}^{\infty} \frac{x^{3n + 1}}{(3n + 1)!} & & \sum_{n = 0}^{\infty} \frac{x^{3n + 2}}{(3n + 2)!} \\ & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n}}{(3n)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n + 1}}{(3n + 1)!} & & \sum_{n = 0}^{\infty} \frac{(-1)^nx^{3n + 2}}{(3n + 2)!} \\ & \vdots && \vdots && \vdots \end{align}
and so I was wondering as to whether or not there is a name for all of these types of infinite sums and what properties they all share.