I was doing the following sum: $$\sum_{i=2}^k \frac{(-1)^i}{i-1} \binom{2k-i-1}{k-1}x^i$$
First, Mathematica simplifies it to some DifferenceRoot
function:
Sum[(-1)^i/(i - 1) Binomial[2 k - i - 1, k - 1] x^i, {i, 2, k}]
(* DifferenceRoot[
Function[{\[FormalY], \[FormalN]}, {(-1 + \[FormalN]) (-\[FormalN] +
k) x \[FormalY][\[FormalN]] + (-\[FormalN] - \[FormalN]^2 +
2 \[FormalN] k - \[FormalN] x + \[FormalN]^2 x +
k x - \[FormalN] k x) \[FormalY][
1 + \[FormalN]] + \[FormalN] (1 + \[FormalN] -
2 k) \[FormalY][2 + \[FormalN]] == 0, \[FormalY][2] ==
0, \[FormalY][3] == x^2 Binomial[-3 + 2 k, -1 + k]}]][1 + k] *)
Then I tried to simplify it:
FullSimplify[%, k ∈ Integers && k >= 10]
And Mathematica throws a ComplexInfinity
error:
FullSimplify::infd: Expression
DifferenceRoot[Function[{\[FormalY],\[FormalN]},{(-1+\[FormalN]) (Times[<<2>>]+k) x \[FormalY][\[FormalN]]+(Times[<<2>>]+Times[<<2>>]+Times[<<3>>]+Times[<<3>>]+Times[<<2>>]+Times[<<2>>]+Times[<<4>>]) \[FormalY][Plus[<<2>>]]+\[FormalN] (1+\[FormalN]+Times[<<2>>]) \[FormalY][Plus[<<2>>]]==0,\[FormalY][2]==0,\[FormalY][3]==x^2 Binomial[-3+Times[<<2>>],-1+k]}]][1+k]
simplified toComplexInfinity
. >>
Just would like to understand what is going on here, what does Mathematica try to do? Perhaps it meets something like $\Gamma(z)$ for negative integer $z$?