Higher order partial derivatives of a product of reciprocals of cumulative sums

Question

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?

Przemo · Answer 1 · 2023-05-17T13:51:18.593

Let us define $\vec{i}:=(i_\xi)_{\xi=1}^d$ subject to $i_0=-1$ and $i_{d+1}=n-1$. Then define $\chi_\xi(\vec{i}) := \sum\limits_{l=0}^d (n-i_l-1) \cdot 1_{n-i_l-1 \ge \xi \ge n-i_{l+1}+1 } $. Then the action of the multivariate operator on the product of reciprocals of partial sums reads as follows:

\begin{eqnarray} && \left[ \prod\limits_{j=1}^n (-1)^{a_j} \partial_{y_j}^{a_j} \right] \cdot \prod\limits_{j=1}^n \frac{1}{(y_1+\cdots+y_i)} =\\ && a_n! \sum\limits_{d=0}^{n-2} (-1)^{n-1-d} \sum\limits_{0 \le i_1 < i_2 < \cdots < i_d \le n-2} \sum\limits_{ \begin{array}{lll} \left( \right. l_j &=& 0,\cdots, a_j + \sum\limits_{\eta=j+1}^n (a_\eta-l_\eta) \left. \right)_{j=n-i_1+1}^n \\ \left( \right. l_j &=& 0,\cdots, a_j + \sum\limits_{\eta=j+1}^{n-i_1-1} (a_\eta-l_\eta) \left. \right)_{j=n-i_2+1}^{n-i_1-1} \\ \left( \right. l_j &=& 0,\cdots, a_j + \sum\limits_{\eta=j+1}^{n-i_2-1} (a_\eta-l_\eta) \left. \right)_{j=n-i_3+1}^{n-i_2-1} \\ &\vdots& \\ \left( \right. l_j &=& 0,\cdots, a_j + \sum\limits_{\eta=j+1}^{n-i_d-1} (a_\eta-l_\eta) \left. \right)_{j=2}^{n-i_d-1} \end{array} } \\ && \prod\limits_{\begin{array}{ccc} \xi=2 \\ \xi \neq n-i_1 \\ \xi \neq n-i_2 \\ \vdots \\ \xi \neq n_{i_d} \end{array}}^n \frac{\left( 1 + \sum\limits_{\eta=\xi}^{\chi_\xi(\vec{i})} (a_\eta-l_\eta)\right)^{(a_{\xi-1})}}{\left( \sum\limits_{\eta=\xi}^{\chi_\xi(\vec{i})} y_\eta \right)^{1+l_\xi}} \cdot \prod\limits_{l=1}^{d+1} \frac{(a_{n-1-i_l})!}{\left( \sum\limits_{\xi=n-i_l}^{n-i_{l-1}-1} y_\xi\right)^{1+a_{n-i_l} + \sum\limits_{\xi=n-i_l+1}^{n-i_{l-1}-1} (a_\xi-l_\xi)}} + \\ && \prod\limits_{\xi=1}^n \frac{a_\xi!}{(y_\xi)^{1+a_\xi}} \tag{1} \end{eqnarray}

Before we proceed let us make a comment.

The multivariate sum on the very right in the middle row runs over $(n-1-d)$ indices independently.Those indices are $(l_2,\cdots, l_{n-i_d-1}, l_{n-i_d+1}, \cdots, l_{n-i_1-1},l_{n-i_1+1}, \cdots, l_n )$. In other words the indices $l_{n-i_1}, l_{n-i_2}, \cdots, l_{n-i_d} $ are missing.It would be good to find a closed form expression for the total number of terms in that multivariate sum.

In order to be absolutely sure that this is all correct we present two code snippets that evaluate expression $(1)$ for $N=6,7$. Here we go:

$N=6$ :

In[1]:= NN = 6; Clear[a, y];
uPch[a_, n_] := Pochhammer[a, n];
Do[a[xi] = RandomInteger[{1, 3}], {xi, 1, NN}];
(*From the defintion using Mathematiuca's D[..,] command.*)
res = (-1)^Sum[a[xi], {xi, 1, NN}] D[1/
    Product[Sum[y[eta], {eta, 1, xi}], {xi, 1, NN}], 
    Evaluate[Sequence @@ Table[{y[xi], a[xi]}, {xi, 1, NN}]]];
(From the closed form expression.)
Clear[ii, ll, i, j, k, xi, jt];
res6 =
  (One five-fold sum )
  (-1)^-1 Sum[
     1/(0 + Sum[y[eta], {eta, NN - 5, NN}])^(
      1 + a[NN - 5] + Sum[a[eta] - ll[eta], {eta, NN - 4, NN}])
       Product[
       Pochhammer[1 + Sum[a[eta] - ll[eta], {eta, xi, NN}], 
        a[xi - 1]]/Sum[y[eta], {eta, xi, NN}]^(
       1 + ll[xi]), {xi, NN - 4, NN}], 
     Evaluate[
      Sequence @@ 
       Table[{ll[jt], 0, 
         a[jt] + Sum[a[eta] - ll[eta], {eta, jt + 1, NN}]}, {jt, NN, 
         NN - 4, -1}]]] +
   (Binomial[5,d] multivariate sums of depth 5-d.)
   Sum[(-1)^(-d - 1) 1/a[NN]! Sum[Sum[
       (Product[
           a[NN - ii[eta - 1] - 1]!/(If[eta == d + 1, 0, 0] + 0 + 
              Sum[y[xi], {xi, NN - ii[eta], NN - ii[eta - 1] - 1}])^(
            1 + a[NN - ii[eta]] + 
             Sum[a[xi] - ll[xi], {xi, NN - ii[eta] + 1, 
               NN - ii[eta - 1] - 1}]), {eta, 1, 
            d + 1}] /. {ii[0] :> -1, ii[d + 1] :> 5})
        (Times @@ 
          Delete[Table[
            With[{upLim = (Sum[(NN - ii[eta] - 1) If[
                    NN - ii[eta] - 1 >= xi >= NN - ii[eta + 1] + 1, 1,
                     0], {eta, 0, d}] /. {ii[0] :> -1, 
                  ii[d + 1] :> 5})}, 
             uPch[1 + Sum[a[eta] - ll[eta], {eta, xi, upLim}], 
               a[xi - 1]]/Sum[y[eta], {eta, xi, upLim}]^(
              1 + ll[xi])], {xi, NN, NN - 4, -1}], 
           Table[{ii[eta] + 1}, {eta, 1, d}]])
       , Evaluate[Sequence @@
     Delete[Table[
       With[{upLim = (Sum[(NN - ii[eta] - 1) If[
               NN - ii[eta] - 1 &gt;= jt &gt;= NN - ii[eta + 1] + 1, 1, 
               0], {eta, 0, d}] /. {ii[0] :&gt; -1, 
             ii[d + 1] :&gt; 5})}, {ll[jt], 0, 
         a[jt] + 
          Sum[a[eta] - ll[eta], {eta, jt + 1, upLim}]}], {jt, NN, 
        NN - 4, -1}], Table[{ii[eta] + 1}, {eta, 1, d}]]
                 ]
   ], 
  Evaluate[
   Sequence @@ 
    Table[{ii[eta], If[eta == 1, 0, ii[eta - 1] + 1], 
      Min[NN - 2, 4]}, {eta, 1, d}]]], {d, 1, Min[NN - 2, 4]}] +

(One Free term)
   (-1)^-6 a[NN - 5]!/(0 + y[NN - 5])^(1 + a[NN - 5])
     Product[a[xi]!/(y[xi])^(1 + a[xi]), {xi, NN - 4, NN}]/a[NN]!;
res6 *= a[NN]!;
(Compare.)
(res - res6) /. y[j_] -> RandomInteger[{1, 5}]
Out[8]= 0

$N=7$ :

NN = 7; Clear[a, y];
uPch[a_, n_] := Pochhammer[a, n];
Do[a[xi] = RandomInteger[{1, 3}], {xi, 1, NN}];
(*From the defintion using Mathematiuca's D[..,] command.*)
res = (-1)^Sum[a[xi], {xi, 1, NN}] D[1/
    Product[Sum[y[eta], {eta, 1, xi}], {xi, 1, NN}], 
    Evaluate[Sequence @@ Table[{y[xi], a[xi]}, {xi, 1, NN}]]];
(From the closed form expression.)
Clear[ii, ll, i, j, k, xi, jt];
res7 =
  (One six-fold sum )
  (-1)^-0 Sum[( 1/(0 + Sum[y[eta], {eta, -5 + NN, NN}] + y[-6 + NN])^(
       1 + a[-6 + NN] + 
        Sum[a[eta] - ll[eta], {eta, -5 + NN, NN}])) Product[
       Pochhammer[1 + Sum[a[eta] - ll[eta], {eta, xi, NN}], 
        a[xi - 1]]/Sum[y[eta], {eta, xi, NN}]^(
       1 + ll[xi]), {xi, NN - 5, NN}], 
     Evaluate[
      Sequence @@ 
       Table[{ll[jt], 0, 
         a[jt] + Sum[a[eta] - ll[eta], {eta, jt + 1, NN}]}, {jt, NN, 
         NN - 5, -1}]]] +
   (Binomial[6,d] multivariate sums of depth 6-d.)
   Sum[(-1)^-d 1/a[NN]! Sum[Sum[
       (Product[
           a[NN - ii[eta - 1] - 1]!/(If[eta == d + 1, 0, 0] + 0 + 
              Sum[y[xi], {xi, NN - ii[eta], NN - ii[eta - 1] - 1}])^(
            1 + a[NN - ii[eta]] + 
             Sum[a[xi] - ll[xi], {xi, NN - ii[eta] + 1, 
               NN - ii[eta - 1] - 1}]), {eta, 1, 
            d + 1}] /. {ii[0] :> -1, ii[d + 1] :> 6})
        (Times @@ 
          Delete[Table[
            With[{upLim = (Sum[(NN - ii[eta] - 1) If[
                    NN - ii[eta] - 1 >= xi >= NN - ii[eta + 1] + 1, 1,
                     0], {eta, 0, d}] /. {ii[0] :> -1, 
                  ii[d + 1] :> 6})}, 
             uPch[1 + Sum[a[eta] - ll[eta], {eta, xi, upLim}], 
               a[xi - 1]]/Sum[y[eta], {eta, xi, upLim}]^(
              1 + ll[xi])], {xi, NN, NN - 5, -1}], 
           Table[{ii[eta] + 1}, {eta, 1, d}]])
       , Evaluate[Sequence @@
     Delete[Table[
       With[{upLim = (Sum[(NN - ii[eta] - 1) If[
               NN - ii[eta] - 1 &gt;= jt &gt;= NN - ii[eta + 1] + 1, 1, 
               0], {eta, 0, d}] /. {ii[0] :&gt; -1, 
             ii[d + 1] :&gt; 6})}, {ll[jt], 0, 
         a[jt] + 
          Sum[a[eta] - ll[eta], {eta, jt + 1, upLim}]}], {jt, NN, 
        NN - 5, -1}], Table[{ii[eta] + 1}, {eta, 1, d}]]
                 ]
   ], 
  Evaluate[
   Sequence @@ 
    Table[{ii[eta], If[eta == 1, 0, ii[eta - 1] + 1], 
      Min[NN - 2, 5]}, {eta, 1, d}]]], {d, 1, Min[NN - 2, 5]}] +

(One free term)
   (-1)^-6 (uPch[1, a[-6 + NN]] (0 + y[-6 + NN])^(-1 - 
       a[-6 + NN]) ) Product[a[xi]!/(y[xi])^(
      1 + a[xi]), {xi, NN - 5, NN}]/a[NN]!;
res7 *= a[NN]!;
(Compare.)
(res - res7) /. y[j_] -> RandomInteger[{1, 5}]
Out[16]= 0

Higher order partial derivatives of a product of reciprocals of cumulative sums

1 Answers1

$N=6$ :

$N=7$ :