<s>[INST] Proving that $f$ verifies $f(1−x) + f(x) = 1$ I have a function which is defined for $x \in (0,1)$ and for $p>1$ with the expression
\begin{align*} 
f(x) = \sum_{k=0}^{p-1} \frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\left(\prod_{i=k-p+1, i\neq0}^{k+p} (x+k-i)\right) 
\end{align*}
I would like to show that $f(1-x) + f(x) = 1$.
What I have done so far
With a little bit of algebra I have
\begin{align*} 
f(x) 
&= \frac{1}{(2p-1)!}\sum_{k=0}^{p-1} (-1)^{p+k} \binom{2p-1}{p+k} \left(\prod_{i=k-p+1, i\neq0}^{k+p} (x+k-i)\right)
&& \text{using }\binom{n}{m} = \frac{n!}{m!(n-m)!}
\\
&= \frac{1}{(2p-1)!}\sum_{k=0}^{p-1} (-1)^{p+k} \binom{2p-1}{p+k} \left(\prod_{j=1-p, j\neq-k}^{p} (x-j)\right) 
&& \text{using the change of indices j=i-k} 
\\
&= \frac{1}{(2p-1)!}\sum_{k=0}^{p-1} (-1)^{p+k} \binom{2p-1}{p+k} \left(\prod_{j=1-p}^{p} (x-j)\right) \frac{1}{x+k}
&& \text{completing the product} 
\\
&= \frac{1}{(2p-1)!} \left(\prod_{j=1-p}^{p} (x-j)\right) \left(\sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{x+k}\binom{2p-1}{p+k}\right)
&& \text{factoring the product from the sum} 
\end{align*}
Then I introduce the functions
\begin{align*} 
F(x) &= (2p-1)! f(x)  = G(x) H(x)\\
G(x) &= \prod_{j=1-p}^{p} (x-j) \\
H(x) &= \sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{x+k}\binom{2p-1}{p+k}
\end{align*}
So the initial question reduces to proving that $F(1-x) + F(x) = (2p-1)!$
Next, playing with the indices of the product I have been able to show that $G(1-x) = G(x)$.
So that now I only need to show $G(x) \left( H(1-x) + H(x)\right) = (2p-1)!$
EDIT:
I have just been able to simplify $H(1-x) + H(x)$ as follows :
\begin{align*} 
H(x) 
&=\sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{x+k}\binom{2p-1}{p+k} 
\\
&=\sum_{s=p}^{2p-1}\frac{(-1)^{s}}{x-p+s}\binom{2p-1}{s}
&& \text{using a change of index } s=p+k 
\end{align*}
And
\begin{align*} 
H(1-x) 
&=\sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{1-x+k}\binom{2p-1}{p+k} 
\\
&=\sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{1-x+k}\binom{2p-1}{p-1-k}
&& \text{using } \binom{2p-1}{p+k} = \binom{2p-1}{p-1-k} 
\\
&=\sum_{s=0}^{p-1}\frac{(-1)^{s}}{x-p+s}\binom{2p-1}{s}
&& \text{using a change of index } s=p-1-k 
\end{align*}
So finally
\begin{align*} 
H(1-x) + H(x) 
&=\sum_{s=0}^{2p-1}\frac{(-1)^{s}}{x-p+s}\binom{2p-1}{s}
\end{align*}
EDIT2:
I have just made the same work on $G(x)$ as follows
\begin{align*} 
G(x) 
&=\prod_{j=1-p}^{p} (x-j) \\
&=\prod_{s=0}^{2p-1} (x-p+s)
&& \text{using a change of index } s=p-i 
\end{align*}

Now I think the constant $(2p-1)!$ in the relation $F(1-x)+F(x)=(2p-1)!$ comes from the binomial factor in $H(1-x) + H(x)$ but I don't see how to prove that...
Any lead ?
 [/INST] Here we have a partial fraction decomposition in disguise.

We obtain
\begin{align*}
\color{blue}{f(x)}&=\sum_{k=0}^{p-1}\frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}
\prod_{\substack{j=k-p+1\\j\neq 0}}^{k+p}(x+k-j)\\
&=\prod_{j=-p+1}^{p}(x-j)\sum_{k=0}^{p-1}
\frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\,\frac{1}{x+k}\tag{1}\\
&\color{blue}{=\prod_{j=-p}^{p-1}(x+j)\sum_{k=0}^{p-1}
\frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\,\frac{1}{x+k}}\tag{2}\\
\end{align*}

Comment:

*

*In (1) we expand the summands with $(x+k)/(x+k)$ and shift the product by $k$ to start with $j=-p+1$. This way the product can be factored out from the sum.


*In (2) we substitute $j$ with $-j$ in the product.

We start with (2) and consider $f(1-x)$. We obtain
\begin{align*}
\color{blue}{f(1-x)}&=\prod_{j=-p}^{p-1}(1-x+j)\sum_{k=0}^{p-1}
\frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\,\frac{1}{1-x+k}\\
&=\prod_{j=-p}^{p-1}(x-j-1)\sum_{k=0}^{p-1}
\frac{(-1)^{k-1}}{k!(2p-1-k)!}\,\frac{1}{-x+p-k}\tag{3}\\
&=\prod_{j=-p+1}^{p}(x+j-1)\sum_{k=0}^{p-1}
\frac{(-1)^{k}}{k!(2p-1-k)!}\,\frac{1}{x+k-p}\tag{4}\\
&\,\,\color{blue}{=\prod_{j=-p}^{p-1}(x+j)\sum_{k=-p}^{-1}
\frac{(-1)^{p+k}}{(p+k)!(p-1-k)!}\,\frac{1}{x+k}}\tag{5}\\
\end{align*}

Comment:

*

*In (3) we factor out $(-1)^{2p}=1$ from the product. We also change the order of summation $k\to p-1-k$.


*In (4) we substitute $j$ with $-j$ in the product and we expand numerator and denominator of the summands with $-1$.


*In (5) we shift the index $j$ by one to start with $j=-p$. We also shift the index $k$ by $p$ to start with $k=-p$.

We can now sum up (2) and (5) and get
\begin{align*}
\color{blue}{f(x)+f(1-x)}&\color{blue}{=\prod_{j=-p}^{p-1}(x+j)\sum_{k=-p}^{p-1}
\frac{(-1)^{p+k}}{(p+k)!(p-1-k)!}\,\frac{1}{x+k}}\tag{6}
\end{align*}

The right-hand side of (6) looks like a partial fraction decomposition. Indeed, we consider a function
\begin{align*}
g(x)=\frac{1}{\prod_{j=-p}^{p-1}(x+j)}
\end{align*}
The function $g(x)$ is a rational function with $2p$ pairwise different simple poles. Therefore we can find a representation
\begin{align*}
g(x)=\frac{1}{\prod_{j=-p}^{p-1}(x+j)}=\sum_{k=-p}^{p-1}\frac{a_k}{x+k}\qquad\qquad a_k\in\mathbb{R}
\end{align*}
Multiplication with the denominator of $g(x)$ gives
\begin{align*}
\color{blue}{1=\prod_{j=-p}^{p-1}(x+j)\sum_{k=-p}^{p-1}\frac{a_k}{x+k}}\tag{7}
\end{align*}

We can now evaluate (7) at $x=-k_0, -p\leq k_0\leq p-1$. We obtain
\begin{align*}
1&=\left.\prod_{j=-p}^{p-1}(x+j)\sum_{k=-p}^{p-1}\frac{a_k}{x+k}\right|_{x=-k_0}\\
&=\prod_{\substack{j=-p\\j\neq k_0}}^{p-1}\left(-k_0+j\right)a_{k_0}\\
a_{k_0}&=\frac{1}{\prod_{\substack{j=-p\\j\neq k_0}}^{p-1}(j-k_0)}\\
&=\frac{1}{\prod_{j=-p}^{k_0-1}(j-k_0)}\,\frac{1}{\prod_{j=k_0+1}^{p-1}(j-k_0)}\\
&=\frac{(-1)^{p+k_0}}{(p+k_0)!}\,\frac{1}{\left(p-1-k_0\right)!}
\end{align*}
The coefficients $a_{k_0}$ can now be put into (7) and we obtain expression (6). We conclude
\begin{align*}
\color{blue}{f(x)+f(1-x)=1}
\end{align*}
and the claim follows.

</s>