Formula for a geometric series weighted by binomial coefficients (sum over the upper index):$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$

Question

The binomial sum is $$\sum\limits_{i=0}^n {n\choose i}\ x^i = (1+x)^n,$$ where $\displaystyle{n\choose i}=\frac{n!}{(n-i)!i!}.$

Is there a corresponding formula when you sum over the upper index of the binomial coefficients, not the lower index: $$\sum_{i=0}^L {n+i\choose n}\ x^i =\ ?$$ Equivalently, I am looking for the generating function of the sequence $$a_i={r+i\choose k}\qquad i\ge0,$$ where $r\ge k\ge0$ are fixed parameters.

Answer 1

From a CAS

$$\sum_{i=0}^L {n+i\choose n}\ x^i =\frac{1-(L+1) \binom{L+n+1}{n} B_x(L+1,n+1) }{(1-x)^{n+1}}$$ where appears the incomplete beta function.