Is there a closed-form expression for the first k terms in a binomial series?

Question

This is kind of two questions in one.

Firstly, does the following expression have a closed form?

$$\sum_{i=0}^k \binom{n}{i}x^i$$ $$\text{(first $k$ terms in binomial series)}$$

where $k$ is some integer $0 < k < n$.

Secondly, this question got me thinking: is there a way to know if an expression does not have a closed form? If the above expression has no closed form, how would I know?

Answer 1

The direct answer: No, there exists no closed-form expression for the partial sum of binomial coefficients.

The extended answer: There does exist an expression that uses the Gaussian hypergeometric function $_2F_1$, but it isn't considered closed-form (shorter version of Claude Leibovici's answer):

$$\sum_{i=0}^k \binom{n}{i}x^i = (1+x)^n - x^{k+1} \binom{n}{k+1} \, _2F_1(1,k-n+1;k+2;-x)$$

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Thanks to mr_e_man who supplied me with these two links:

1. Calculating partial sum of binomial coefficients fast

2. Is there a simpler (preferably closed-form) way to compute this sum of binomial coefficients?

Answer 2

There is at least an expression.

Write $$\sum_{i=0}^k \binom{n}{i}x^i=\sum_{i=0}^n \binom{n}{i}x^i-\sum_{i=k+1}^n \binom{n}{i}x^i=(1+x)^n-\sum_{i=k+1}^n \binom{n}{i}x^i$$ $$\sum_{i=k+1}^n \binom{n}{i}x^i=x^{k+1} \binom{n}{k+1} \, _2F_1(1,k-n+1;k+2;-x)$$ where appears the Gaussian hypergeometric function.