This question is related to another question where we asked for evaluating a certain multi-dimensional integral. It turns out that that integral, from the question above, can be reduced to an action of a multivariate differential operator on a certain multivariate function. Now that very action can be further simplified by an appropriate change of variables which then leads to a quantity that we will be evaluating below. After this motivation let me formulate my question.
Let $n \ge 1$ be an integer and let $\vec{a}:= \left( a_i \right)_{i=1}^n \in {\mathbb N}_+^n$ and let $(y_i)_{i=1}^n$ by some symbolic parameters. Define $\vec{x}:= ( x_i)_{i=1}^n$ where $x_i := y_1+\cdots+y_i$ for $i=1,\cdots,n$. We define an action of a multivariate differential operator on a product of reciprocals of the quantities $x_{\cdot}$. We have:
\begin{eqnarray} {\mathfrak A}^{(\vec{a})}(\vec{x}) &:=& \prod\limits_{i=1}^n \left( \sum\limits_{j=i}^n \frac{\partial}{\partial x_j} \right)^{a_i} \cdot \prod\limits_{i=1}^n \frac{1}{x_i} \\ &=& \frac{\partial^{a_1}}{\partial y_1^{a_1}} \cdot \frac{\partial^{a_2}}{\partial y_2^{a_2}} \cdot \cdots \frac{\partial^{a_n}}{\partial y_n^{a_n}} \cdot \prod\limits_{i=1}^n \frac{1}{(y_1+\cdots+ y_i)} \tag{1} \end{eqnarray}
Now, by using iterated partial fraction decomposition in the variables $y_i$ starting from $i=n$ all the way down to $i=1$ we have obtained the following functional form of the quantity in $(1)$. We have:
\begin{eqnarray} &&\left.{\mathfrak A}^{(\vec{a})}(\vec{x}) = \right. \\ && \left. % {\mathfrak T}^{(0)}(\vec{a}) + \right. \\ % % && \left. \sum\limits_{i=0}^{n-2} {\mathfrak T}^{(1)}_i(\vec{y}) + \right. \\ && \left. \underbrace{\cdots}_{\mbox{additional $(2^{n-1}-2-2(n-1))$ terms}} + \right. \\ && \left. \sum\limits_{i=1}^{n-1} {\mathfrak T}^{(n-3)}_i(\vec{y}) + \right. \\ &&\left. {\mathfrak T}^{(n-2)}(\vec{a}) \right. \tag{2} \end{eqnarray}
where
\begin{eqnarray} {\mathfrak T}^{(0)}(\vec{a}) &:=& \left. % a_n! \cdot \sum\limits_{ \begin{array}{lll} \left( l_j \right. &=& 0, \cdots, (\sum\limits_{\eta=j}^n a_\eta ) - (\sum\limits_{\eta=j+1}^n l_\eta ) \left. \right)_{j=2}^n \end{array}} \frac{(-1)^{n-1}} {(\sum\limits_{\eta=1}^n y_\eta)^{1+ \sum\limits_{\eta=1}^n a_\eta - \sum\limits_{\eta=2}^n l_\eta}} \cdot \prod\limits_{\xi=2}^n \frac{(1+ \sum\limits_{\eta=\xi}^n (a_\eta-l_\eta))^{(a_{\xi-1})}}{( \sum\limits_{\eta=\xi}^n y_\eta )^{1+l_\xi}} + \right.\\ % {\mathfrak T}^{(1)}_i(\vec{a}) &:=& (-1)^{n-2} a_{n-i-1}! % \sum\limits_{ \begin{array}{lll} \left( l_j \right. &=& 0, \cdots, (\sum\limits_{\eta=j}^n a_\eta ) - (\sum\limits_{\eta=j+1}^n l_\eta ) \left. \right)_{j=n-i+1}^n \\ \left( l_j \right. &=& 0, \cdots, (\sum\limits_{\eta=j}^{n-i-1} a_\eta ) - (\sum\limits_{\eta=j+1}^{n-i-1} l_\eta ) \left. \right)_{j=2}^{n-i-1} \end{array} } \frac{1}{(\sum\limits_{\xi=1}^{n-i-1}y_\xi)^{1+(\sum\limits_{\xi=1}^{n-i-1} a_\xi) - (\sum\limits_{\xi=2}^{n-i-1} l_\xi) }} \cdot % \frac{1}{(\sum\limits_{\xi=n-i}^{n}y_\xi)^{1+(\sum\limits_{\xi=n-i}^{n} a_\xi) - (\sum\limits_{\xi=n-i+1}^{n} l_\xi) }} \cdot \prod\limits_{\xi=2}^{n-i-1} \frac{(1+ \sum\limits_{\eta=\xi}^{n-i-1} (a_\eta-l_\eta))^{(a_{\xi-1})}}{(\sum\limits_{\eta=\xi}^{n-i-1} y_\eta)^{1+l_\xi}} \cdot \prod\limits_{\xi=n-i+1}^{n} \frac{(1+ \sum\limits_{\eta=\xi}^{n} (a_\eta-l_\eta))^{(a_{\xi-1})}}{(\sum\limits_{\eta=\xi}^{n} y_\eta)^{1+l_\xi}} + % \\ % &\vdots& \\ {\mathfrak T}^{(n-3)}_i(\vec{a}) &:=& \prod\limits_{\begin{array}{c} \xi=1 \\ \xi \neq (i,i+1) \end{array}}^n \frac{a_\xi!}{y_{\xi}^{1+a_\xi}} \cdot \left( \sum\limits_{l_{i+1}=0}^{a_{i+1}} \frac{-(1+a_{i+1}-l_{i+1})^{(a_i)}}{(y_i+y_{i+1})^{1+a_i+a_{i+1}-l_{i+1}}} \cdot \frac{a_{i+1}!}{y_{i+1}^{1+l_{i+1}}} \right) \\ % {\mathfrak T}^{(n-2)}(\vec{a}) &:=& \prod\limits_{\xi=1}^n \frac{a_\xi!}{y_\xi^{1+a_\xi}} \end{eqnarray}
Now the Mathematica code snippet below verifies the formula $(2)$ for $n=1,\cdots,5$. We have:
[![(*Our objective is to compute the following expression \partial^a\[1\] \
\partial^a\[2\] .... \partial^a\[n\] \
Product\[1/Sum\[y\[xi\],{xi,1,i}\],{i,1,n}\]. We will be using iterated \
partial fraction decomposition for this purpose.*)
uPch\[a_, n_\] := Pochhammer\[a, n\];
NN = 8; Clear\[a\]; Clear\[y\];
(*i\[Equal\]NN,NN-1,NN-2,NN-3,NN-4*)
Do\[a\[xi\] = RandomInteger\[{1, 5}\], {xi, 1, NN}\];
res = Table\[(-1)^Sum\[a\[xi\], {xi, NN - i, NN}\] D\[1/
Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN}\],
Evaluate\[
Sequence @@ Table\[{y\[xi\], a\[xi\]}, {xi, NN - i, NN}\]\]\], {i, 0,
4}\];
(*i=NN*)
res1 = 1/Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN - 1}\] a\[NN\]!/
Sum\[y\[eta\], {eta, 1, NN}\]^(a\[NN\] + 1);
(*i=NN-1*)
res2 = Sum\[
uPch\[a\[NN\] + 1 - l\[NN\],
a\[NN - 1\]\]/(y\[NN - 1\] + Sum\[y\[xi\], {xi, 1, NN - 2}\] + y\[NN\])^(
a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\], 0,
a\[NN\]}\] +
uPch\[1, a\[NN - 1\]\]/(y\[NN - 1\] + Sum\[y\[xi\], {xi, 1, NN - 2}\])^(
1 + a\[NN - 1\]) (+1)/y\[NN\]^(a\[NN\] + 1);
res2 *= a\[NN\]!/Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN - 2}\];
(*i=NN-2*)
res3 =
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\],
a\[NN - 2\]\]/(y\[NN - 2\] + Sum\[y\[xi\], {xi, 1, NN - 3}\] +
y\[NN - 1\] + y\[NN\])^(
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] -
l\[NN - 1\]) (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
Sum\[(uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1,
a\[NN - 2\]\])/(y\[NN - 2\] + Sum\[y\[eta\], {eta, 1, NN - 3}\])^(
1 + a\[NN - 2\]) (+1)/(y\[NN - 1\] + y\[NN\])^(
a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\], 0,
a\[NN\]}\] +
Sum\[(uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\])/(y\[NN - 2\] + Sum\[y\[xi\], {xi, 1, NN - 3}\] +
y\[NN - 1\])^(
1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]) (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) + (
uPch\[1, a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\])/(y\[NN - 2\] +
Sum\[y\[eta\], {eta, 1, NN - 3}\])^(
1 + a\[NN - 2\]) (+1)/(y\[NN - 1\])^(a\[NN - 1\] + 1) (+1)/y\[NN\]^(
a\[NN\] + 1);
res3 *= a\[NN\]!/Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN - 3}\];
(*i=NN-3*)
res4 =
(*One Triple sum (0,1,2)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\] -
l\[NN - 2\],
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\] +
y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
a\[NN - 3\] + a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] -
l\[NN - 1\] -
l\[NN - 2\]) (-1)/(y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0,
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 0 - l\[NN\] -
l\[NN - 1\]}\] (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
(*Three Double sums: (0,1),(0,2),(1,2)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] uPch\[
1, a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\])^(
1 + a\[NN - 3\]) (+1)/(y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] -
l\[NN - 1\]) (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + 1 - l\[NN - 2\],
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\] +
y\[NN - 2\])^(
a\[NN - 3\] + a\[NN - 2\] + 1 - l\[NN - 2\]) (-1)/(y\[NN - 2\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0, a\[NN - 2\]}\] (+1)/(y\[NN - 1\] +
y\[NN\])^(a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(
l\[NN\] + 1), {l\[NN\], 0, a\[NN\]}\] +
Sum\[uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\] Sum\[
uPch\[1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\] - l\[NN - 2\],
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\] +
y\[NN - 2\] + y\[NN - 1\])^(
1 + a\[NN - 3\] + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\] -
l\[NN - 2\]) (-1)/(y\[NN - 2\] + y\[NN - 1\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0,
0 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]}\] (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
(*Three Single sums: (0),(1),(2)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\])^(
1 + a\[NN - 3\]) (+1)/(y\[NN - 2\])^(
a\[NN - 2\] + 1) (+1)/(y\[NN - 1\] + y\[NN\])^(
a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\], 0,
a\[NN\]}\] +
Sum\[uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\])^(
1 + a\[NN - 3\]) (+1)/(y\[NN - 2\] + y\[NN - 1\])^(
1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]) (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
uPch\[1, a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] Sum\[
uPch\[1 + a\[NN - 2\] - l\[NN - 2\],
a\[NN - 3\]\]/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\] +
y\[NN - 2\])^(
1 + a\[NN - 3\] + a\[NN - 2\] - l\[NN - 2\]) (-1)/(y\[NN - 2\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0, 0 + a\[NN - 2\]}\] (+1)/(y\[
NN - 1\])^(a\[NN - 1\] + 1) (+1)/y\[NN\]^(a\[NN\] + 1) +
(*One Free term*)
(uPch\[1, a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\])/(y\[NN - 3\] + Sum\[y\[xi\], {xi, 1, NN - 4}\])^(
1 + a\[NN - 3\]) (+1)/(y\[NN - 2\])^(1 + a\[NN - 2\]) (+1)/(y\[NN - 1\])^(
a\[NN - 1\] + 1) (+1)/y\[NN\]^(a\[NN\] + 1);
res4 *= a\[NN\]!/Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN - 4}\];
(*i=NN-4*)
S5 = Sum\[y\[xi\], {xi, 1, NN - 5}\];
res5 =
(*One Quadruple sum (0,1,2,3)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\] -
l\[NN - 2\], a\[NN - 3\]\] Sum\[
uPch\[1 + a\[-3 + NN\] + a\[-2 + NN\] + a\[-1 + NN\] + a\[NN\] -
l\[-3 + NN\] - l\[-2 + NN\] - l\[-1 + NN\] - l\[NN\],
a\[-4 + NN\]\] (S5 + y\[-4 + NN\] + y\[-3 + NN\] + y\[-2 + NN\] +
y\[-1 + NN\] + y\[NN\])^(-1 - a\[-4 + NN\] - a\[-3 + NN\] -
a\[-2 + NN\] - a\[-1 + NN\] - a\[NN\] + l\[-3 + NN\] +
l\[-2 + NN\] + l\[-1 + NN\] +
l\[NN\]) (-1)/((y\[-3 + NN\] + y\[-2 + NN\] + y\[-1 + NN\] +
y\[NN\])^(1 + l\[-3 + NN\])) , {l\[-3 + NN\], 0,
a\[-3 + NN\] + a\[-2 + NN\] + a\[-1 + NN\] + a\[NN\] - l\[-2 + NN\] -
l\[-1 + NN\] - l\[NN\]}\] (-1)/(y\[NN - 2\] + y\[NN - 1\] +
y\[NN\])^(l\[NN - 2\] + 1), {l\[NN - 2\], 0,
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 0 - l\[NN\] -
l\[NN - 1\]}\] (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
(*Four triple sums: (0,1,2),(0,1,3),(0,2,3),(1,2,3)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\] -
l\[NN - 2\], a\[NN - 3\]\] uPch\[1,
a\[-4 + NN\]\] (S5 + y\[-4 + NN\])^(-1 -
a\[-4 + NN\]) (y\[-3 + NN\] + y\[-2 + NN\] + y\[-1 + NN\] +
y\[NN\])^(-1 - a\[-3 + NN\] - a\[-2 + NN\] - a\[-1 + NN\] - a\[NN\] +
l\[-2 + NN\] + l\[-1 + NN\] +
l\[NN\]) (-1)/(y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0,
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 0 - l\[NN\] -
l\[NN - 1\]}\] (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] uPch\[
1, a\[NN -
3\]\] Sum\[-uPch\[1 + a\[-3 + NN\] - l\[-3 + NN\],
a\[-4 + NN\]\] y\[-3 + NN\]^(-1 -
l\[-3 + NN\]) (S5 + y\[-4 + NN\] + y\[-3 + NN\])^(-1 -
a\[-4 + NN\] - a\[-3 + NN\] + l\[-3 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\]}\] (+1)/(y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] -
l\[NN - 1\]) (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + 1 - l\[NN - 2\],
a\[NN - 3\]\] (Sum\[-uPch\[
1 + a\[-3 + NN\] + a\[-2 + NN\] - l\[-3 + NN\] - l\[-2 + NN\],
a\[-4 + NN\]\] (y\[-3 + NN\] + y\[-2 + NN\])^(-1 -
l\[-3 + NN\]) (S5 + y\[-4 + NN\] + y\[-3 + NN\] +
y\[-2 + NN\])^(-1 - a\[-4 + NN\] - a\[-3 + NN\] - a\[-2 + NN\] +
l\[-3 + NN\] + l\[-2 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\] + a\[-2 + NN\] - l\[-2 + NN\]}\]) (-1)/(y\[NN - 2\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0, a\[NN - 2\]}\] (+1)/(y\[NN - 1\] +
y\[NN\])^(a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(
l\[NN\] + 1), {l\[NN\], 0, a\[NN\]}\] +
Sum\[uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\] Sum\[
uPch\[1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\] - l\[NN - 2\],
a\[NN - 3\]\] (Sum\[-uPch\[
1 + a\[-3 + NN\] + a\[-2 + NN\] + a\[-1 + NN\] - l\[-3 + NN\] -
l\[-2 + NN\] - l\[-1 + NN\], a\[-4 + NN\]\] (y\[-3 + NN\] +
y\[-2 + NN\] + y\[-1 + NN\])^(-1 -
l\[-3 + NN\]) (S5 + y\[-4 + NN\] + y\[-3 + NN\] + y\[-2 + NN\] +
y\[-1 + NN\])^(-1 - a\[-4 + NN\] - a\[-3 + NN\] - a\[-2 + NN\] -
a\[-1 + NN\] + l\[-3 + NN\] + l\[-2 + NN\] +
l\[-1 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\] + a\[-2 + NN\] + a\[-1 + NN\] - l\[-2 + NN\] -
l\[-1 + NN\]}\]) (-1)/(y\[NN - 2\] + y\[NN - 1\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0,
0 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]}\] (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
(*Six double sums: (0,1),(0,2),(0,3),(1,2),(1,3),(2,3)*)
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] Sum\[
uPch\[a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] - l\[NN - 1\], a\[NN - 2\]\] uPch\[
1, a\[NN - 3\]\] uPch\[1, a\[-4 + NN\]\] (S5 + y\[-4 + NN\])^(-1 -
a\[-4 + NN\])
y\[-3 + NN\]^(-1 -
a\[-3 + NN\]) (+1)/(y\[NN - 2\] + y\[NN - 1\] + y\[NN\])^(
a\[NN - 2\] + a\[NN - 1\] + a\[NN\] + 1 - l\[NN\] -
l\[NN - 1\]) (-1)/(y\[NN - 1\] + y\[NN\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0,
a\[NN - 1\] + a\[NN\] + 0 - l\[NN\]}\] (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] Sum\[
uPch\[a\[NN - 2\] + 1 - l\[NN - 2\],
a\[NN - 3\]\] (uPch\[1, a\[-4 + NN\]\] (S5 + y\[-4 + NN\])^(-1 -
a\[-4 + NN\]) (y\[-3 + NN\] + y\[-2 + NN\])^(-1 - a\[-3 + NN\] -
a\[-2 + NN\] + l\[-2 + NN\])) (-1)/(y\[NN - 2\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0, a\[NN - 2\]}\] (+1)/(y\[NN - 1\] +
y\[NN\])^(a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(
l\[NN\] + 1), {l\[NN\], 0, a\[NN\]}\] +
Sum\[uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\] (Sum\[-uPch\[1 + a\[-3 + NN\] - l\[-3 + NN\],
a\[-4 + NN\]\] y\[-3 + NN\]^(-1 -
l\[-3 + NN\]) (S5 + y\[-4 + NN\] + y\[-3 + NN\])^(-1 - a\[-4 + NN\] -
a\[-3 + NN\] + l\[-3 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\]}\]) (+1)/(y\[NN - 2\])^(
a\[NN - 2\] + 1) (+1)/(y\[NN - 1\] + y\[NN\])^(
a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (-1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\], 0,
a\[NN\]}\] +
Sum\[uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\] Sum\[
uPch\[1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\] - l\[NN - 2\],
a\[NN - 3\]\] (uPch\[1, a\[-4 + NN\]\] (S5 + y\[-4 + NN\])^(-1 -
a\[-4 + NN\]) (y\[-3 + NN\] + y\[-2 + NN\] + y\[-1 + NN\])^(-1 -
a\[-3 + NN\] - a\[-2 + NN\] - a\[-1 + NN\] + l\[-2 + NN\] +
l\[-1 + NN\])) (-1)/(y\[NN - 2\] + y\[NN - 1\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0,
0 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]}\] (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
Sum\[uPch\[1, a\[NN - 1\]\] uPch\[1 + a\[NN - 1\] - l\[NN - 1\],
a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\] (Sum\[-uPch\[1 + a\[-3 + NN\] - l\[-3 + NN\],
a\[-4 + NN\]\] y\[-3 + NN\]^(-1 -
l\[-3 + NN\]) (S5 + y\[-4 + NN\] + y\[-3 + NN\])^(-1 -
a\[-4 + NN\] - a\[-3 + NN\] + l\[-3 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\]}\]) (+1)/(y\[NN - 2\] + y\[NN - 1\])^(
1 + a\[NN - 2\] + a\[NN - 1\] - l\[NN - 1\]) (-1)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
Sum\[(Sum\[(-uPch\[
1 + a\[-3 + NN\] + a\[-2 + NN\] - l\[-3 + NN\] - l\[-2 + NN\],
a\[-4 + NN\]\])/(S5 + y\[-4 + NN\] + y\[-3 + NN\] + y\[-2 + NN\])^(
1 + a\[-4 + NN\] + a\[-3 + NN\] + a\[-2 + NN\] - l\[-3 + NN\] -
l\[-2 + NN\])
uPch\[1 + a\[NN - 2\] - l\[NN - 2\],
a\[NN - 3\]\]/(y\[-3 + NN\] + y\[-2 + NN\])^(
1 + l\[-3 + NN\]), {l\[-3 + NN\], 0,
a\[-3 + NN\] + a\[-2 + NN\] - l\[-2 + NN\]}\]) (-a\[NN - 2\]!)/(y\[
NN - 2\])^(l\[NN - 2\] + 1), {l\[NN - 2\], 0,
0 + a\[NN - 2\]}\] (+a\[NN - 1\]!)/(y\[NN - 1\])^(a\[NN - 1\] + 1) (+1)/
y\[NN\]^(a\[NN\] + 1) +
(*Four single sums (0),(1),(2),(3)*)
a\[NN - 4\]!/(S5 + y\[-4 + NN\])^(1 + a\[-4 + NN\]) a\[NN - 3\]!/
y\[-3 + NN\]^(1 + a\[-3 + NN\]) (+a\[NN - 2\]!)/(y\[NN - 2\])^(
a\[NN - 2\] + 1)
Sum\[(-uPch\[a\[NN\] + 1 - l\[NN\], a\[NN - 1\]\])/(y\[NN - 1\] + y\[NN\])^(
a\[NN - 1\] + a\[NN\] + 1 - l\[NN\]) (+1)/y\[NN\]^(l\[NN\] + 1), {l\[NN\],
0, a\[NN\]}\] +
a\[NN - 4\]!/(S5 + y\[-4 + NN\])^(1 + a\[-4 + NN\]) a\[NN - 3\]!/
y\[-3 + NN\]^(1 + a\[-3 + NN\])
Sum\[(-uPch\[1 + a\[NN - 1\] - l\[NN - 1\], a\[NN - 2\]\])/(y\[NN - 2\] +
y\[NN - 1\])^(
1 + a\[NN - 2\] + a\[NN - 1\] -
l\[NN - 1\]) (+a\[NN - 1\]!)/(y\[NN - 1\])^(
l\[NN - 1\] + 1), {l\[NN - 1\], 0, a\[NN - 1\]}\] (+1)/y\[NN\]^(
a\[NN\] + 1) +
a\[NN - 4\]!/(S5 + y\[-4 + NN\])^(1 + a\[-4 + NN\])
Sum\[(-uPch\[1 + a\[NN - 2\] - l\[NN - 2\], a\[NN - 3\]\])/(y\[-3 + NN\] +
y\[-2 + NN\])^(
1 + a\[-3 + NN\] + a\[-2 + NN\] -
l\[-2 + NN\]) (a\[NN - 2\]!)/(y\[NN - 2\])^(
l\[NN - 2\] + 1), {l\[NN - 2\], 0, 0 + a\[NN - 2\]}\] (+a\[NN - 1\]!)/(y\[
NN - 1\])^(a\[NN - 1\] + 1) (+1)/y\[NN\]^(a\[NN\] + 1) +
Sum\[ (-uPch\[1 + a\[-3 + NN\] - l\[-3 + NN\], a\[-4 + NN\]\])/(S5 +
y\[-4 + NN\] + y\[-3 + NN\])^(
1 + a\[-4 + NN\] + a\[-3 + NN\] - l\[-3 + NN\]) (+a\[NN - 3\]!)/ (
y\[-3 + NN\]^(1 + l\[-3 + NN\])), {l\[-3 + NN\], 0,
a\[-3 + NN\]}\] (+a\[NN - 2\]!)/(y\[NN - 2\])^(
1 + a\[NN - 2\]) (+a\[NN - 1\]!)/(y\[NN - 1\])^(a\[NN - 1\] + 1) (+1)/
y\[NN\]^(a\[NN\] + 1) +
(*One Free term*)
uPch\[1, a\[NN - 1\]\] uPch\[1, a\[NN - 2\]\] uPch\[1,
a\[NN - 3\]\] (uPch\[1, a\[-4 + NN\]\] (S5 + y\[-4 + NN\])^(-1 -
a\[-4 + NN\]) y\[-3 + NN\]^(-1 - a\[-3 + NN\])) (+1)/(y\[NN - 2\])^(
1 + a\[NN - 2\]) (+1)/(y\[NN - 1\])^(a\[NN - 1\] + 1) (+1)/y\[NN\]^(
a\[NN\] + 1);
res5 *= a\[NN\]!/Product\[Sum\[y\[eta\], {eta, 1, xi}\], {xi, 1, NN - 5}\];
(res\[\[1\]\] - res1) /. y\[j_\] -> RandomInteger\[{1, 5}\]
(res\[\[2\]\] - res2) /. y\[j_\] -> RandomInteger\[{1, 5}\]
(res\[\[3\]\] - res3) /. y\[j_\] -> RandomInteger\[{1, 5}\]
(res\[\[4\]\] - res4) /. y\[j_\] -> RandomInteger\[{1, 5}\]
(res\[\[5\]\] - res5) /. y\[j_\] -> RandomInteger\[{1, 5}\]
As you can see some terms are missing in formula $(2)$. Can we find those terms in closed form?