Why is $\binom{n}{k}$ for fixed $n$ not Gosper summable? I am trying to understand the book A = B which has many much more complicated expressions which are Gosper-summable but then on p. 102 he states $\binom{n}{k}$ for fixed $n$ as a function of $k$ is not Gosper-summable. How can this be? Can someone show why it is not Gosper-summable?
 A: Gosper's algorithm can only find a formula for
$$
   S(k) = \sum_{i=0}^k \binom ni
$$
if it turns out that $S(k)$ is itself a hypergeometric term: if $\frac{S(k+1)}{S(k)}$ is a rational function of $k$.
However, for $k \ge n$, $\frac{S(k+1)}{S(k)} = \frac{2^n}{2^n} = 1$, so if it were a rational function, it would be equal to $1$ everywhere (aside possibly from places it's not defined). That's not what it does. Therefore it's not a rational function, $S(k)$ is not a hypergeometric term, and a series with partial sums $S(k)$ is not Gosper-summable.
In general, whenever we have a series with only finitely many nonzero terms (as we do here), it can only be Gosper-summable if the infinite sum over all terms is $0$. In that case, we can't carry out the argument above, because we can't say "our rational function is equal to $\frac{0}{0} = 1$ at infinitely many points".
A: I try an alternate proof and write the sum as $\sum_{n=0}^m \binom {p}{n}$ for $m,p$ arbitrary large
positive integers $p\!>\!m$. Per the book A=B denote the summand by $t_n=\binom{p}{n}$ and $r(n)=\frac{t_{n+1}}{t_n}=\frac{p-n}{n+1}$ here. On p75 eq.(5.2.3) is stated assume can write $$r(n)=\frac{a(n)c(n\!+\!1)}{b(n)c(n)}$$
where $a,b,c$ polynomials in n and $ gcd(a(n),b(n\!+\!h))\!=\!1$ for all integers $h\!>\!-1$ and I find $a\!=\!p\!-\!n,b\!=\!n\!+\!1,c\!=\!1$ as the lowest order or only? possibility. Anyway it goes on... and per p76 it requires a finite polynomial solution x to $$a(n)x(n\!+\!1)-b(n\!-\!1)x(n)\!=\!c(n)$$
which in this case is $$(p-n)x(n\!+\!1)-nx(n)\!=\!1$$  and if such a polynomial $x(n)$ cannot be found then Gosper's method fails.
Though I can't prove it rigorously it seems in this case such a polynomial does not exist for the only possibilities would be $x=cst$ which obviously does not satisfy or else of order >zero in which case the coeff. of the highest order term on lhs would have to be 0 which is not the case either so conclude Gosper's method fails so $\sum_{n=0}^m \binom {p}{n}$ is not Gosper summable ? For more detail the book is on the internet and can be found by searching 'book A=B' eg on google. By the way per Gosper's basic method it must
satisfy 1st order recurrence. It seems to me for most practical applications the cases where Gosper's method works are very limited.
