Generating function of 1 over binomial

Question

Is there any known function for which it holds $$f(x)=\sum_{n\ge m}\frac{x^n}{\binom{n}{m}}?$$

I arrived to this question trying to bound a series and I have no experience with generating functions.

Answer 1

We have, if $\left|x\right|<1$ $$\sum_{n=m}^{\infty}\frac{x^{n}}{\dbinom{n}{m}}=\sum_{n=m}^{\infty}\frac{m!\left(n-m\right)!x^{n}}{n!}=x^{m}\sum_{n=0}^{\infty}\frac{m!n!}{\left(m+n\right)!}x^{n}=x^{m}\sum_{n=0}^{\infty}\frac{m!n!}{\left(m+n\right)!}x^{n}=x^{m}\sum_{n=0}^{\infty}\frac{n!n!}{\left(m+1\right)_{n}}\frac{x^{n}}{n!}=x^{m}\sum_{n=0}^{\infty}\frac{\left(1\right)_{n}\left(1\right)_{n}}{\left(m+1\right)_{n}}\frac{x^{n}}{n!}=x^{m}{}_{2}F_{1}\left(1,1;m+1;x\right)$$ where $\left(m+1\right)_{n}=\left(m+1\right)\cdots\left(m+n\right)$ is the Pochhammer symbol and $_{2}F_{1}\left(a,b;c;x\right)$ is the Hypergeometric function.