The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e.,

$$_2F_1(-n;-r;1;2) = \mbox{#} \lbrace x \in \mathbb{Z}^n : |x_1|^2 + \cdots + |x_n|^2 \leq r \rbrace $$

and another formula for $_2F_1(-n;-r;1;2)$ is given by

$$_2F_1(-n;-r;1;2) = \sum_{i=1}^{\min\lbrace{n,r}\rbrace}{n \choose i}{r \choose i}2^i$$

Does anyone know an asymptotic formula for this function when $n$ is large?

I know there are some closed formulas for the Gaussian hypergeometric (i.e. $_2F_1(a,b;2,1)$) and in some other cases, but haven't find any clue in this case.

The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e.,

$$_2F_1(-n;-r;1;2) = \mbox{#} \lbrace x \in \mathbb{Z}^n : |x_1| + \cdots + |x_n| \leq r \rbrace $$

and another formula for $_2F_1(-n;-r;1;2)$ is given by

$$_2F_1(-n;-r;1;2) = \sum_{i=1}^{\min\lbrace{n,r}\rbrace}{n \choose i}{r \choose i}2^i$$

Does anyone know an asymptotic formula for this function when $n$ is large?

I know there are some closed formulas for the Gaussian hypergeometric (i.e. $_2F_1(a,b;2,1)$) and in some other cases, but haven't find any clue in this case.