How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer Consider a truncated Taylor series of the exponential function to approximate $e$:
$$
E(n) = \sum_{k=0}^{n-1} \frac{1}{n!}
$$
I thought of computing this using the hypergeometric finite series $_1 F _1(a;c;z)$ for $a=-n+1$; $c=-n+1$; $z=1$. This is based on the fact that (see the Wikipedia link above)

The series is finite if the first argument, $a$, is a nonpositive integer, in which case the function reduces to a polynomial.

The series is finite in this case because the Pochhammer symbol in the numerator of $_1 F _1$ becomes $0$ from a given index onwards.
However, what if $b=a$? (as happens for the approximation of $e$: $_1 F _1(-n+1;-n+1;1$). In that case the Pochhammer symbol in the denominator of $_1 F _1$ becomes $0$ at the same index as that in the numerator, giving $0/0$. How is the function defined in that case?
According to Wolfram, that $0/0$ appears to be interpreted as $0$, and the series remains finite. See for example here: for arguments $-2$, $-2$, $1$ we get $2.5$, corresponding to $E(3) = 1 + 1 + 1/2$.
Matlab's hypergeometric function, on the other hand, seems to treat that $0/0$ as $1$: the series is infinite and always gives $e$:
>> hypergeom(-2,-2,1)
ans =
   2.718281828459045
>> hypergeom(-7,-7,1)
ans =
   2.718281828459045

The approximation to $e$ was just an example; my actual question is what is the correct definition for $_1 F _1(-n+1;-n+1;z)$? Or more generally, how is $_1F_1(a;c;z)$ defined for $c$ nonpositive integer? Or yet more generally if possible: how is $_mF_n$ defined when one of the denominator coefficients is a nonpositive integer?
 A: In regards to the specific example provided you can avoid the whole issue of negative bottom parameters by using DLMF 16.2.4 to formally write your truncated series as
$$
E(n)=\frac{1}{(n-1)!}{_2F}_0\left({1,1-n\atop -};-1\right),
$$
for $n=1,2,\dots$
You can confirm in MATLAB that the follow code will give you the correct answer for your truncated exponential Taylor series:
syms n
f(n)=hypergeom([1,1-n],[],-1)/factorial(n-1);

It is important to note that this representation for your series only holds for positive integer $n$ as the ${_2F_0}(1,1-n;;1)$ function diverges for $n\notin\Bbb N$. For a continuous generalization one could use
$$
E(n)=\sum_{k=0}^\infty\frac{1}{k!}-\sum_{k=n}^\infty\frac{1}{k!}=e-\frac{{_1 F}_1(1;n+1;1)}{\Gamma(n+1)},
$$
which works for noninteger values of $n$. In fact, according to the methods of Summability Calculus this representation is the unique generalization of your truncated series expansion to $n\in\Bbb C$.
But is $E(n)={_1F}_1(1-n;1-n;1)$ also a valid representation? It comes down to how you define such a quantity. As you have noticed, MATLAB's symbolic package does not know how to handle such expressions but this does not mean such a representation is invalid. In the literature, we typically interpret $(-n)_k/(-n)_k$ in the context of hypergeometric functions as being equal to one when $k\leq n$ and zero otherwise. In other words
$$
{_pF}_q\left({-n,a_2,\cdots,a_p\atop-n,b_2,\dots,b_q};z\right)=\sum_{k=0}^n\frac{(a_2)_k\cdots(a_p)_k}{(b_2)_k\cdots(b_q)_k}\frac{z^k}{k!}.
$$
An example of this is the Lerch sum. Indeed, Mathematica knows how to properly interpret hypergeometric functions for these parameters and will give you the numeric value of your truncated series. In your case, we would write
f[n_]:=Hypergeometric1F1[1 - n, 1 - n, 1]

