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i have a quite complicated formula in d dimensions and i am trying to find a recursion rule by $n$ for it.

$S^d_n = \displaystyle \sum_{j_1,\dotsc,j_{d-1} \in \mathbb{N}_0\\ \sum_{i=1}^{d-1}j_i \leq n} \sum_{i_1,\dotsc,i_{d-1} \in \{0,1\}} (-1)^{i_1+\dotsc,i_{d-1}} f\left(j_1-i_1,\dotsc,j_{d-1}-i_{d-1},n-\sum_{i=1}^{d-1}j_i\right)$

* Edit For $f$ it holds that $f(j_1,\dotsc,j_d) = 0$ if any $j_1,\dotsc,j_d = 0$ Edit *

For $d=2$ I found the following rule:

$S^2_n = \displaystyle \sum_{j=0}^n f(j,n-j) - S^2_{n-1-j}$

I would really appreciate your help.

amWhy
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matosch
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  • Would it be possible to provide some context regarding the meaning of $S_n^d$? It seems to relate (at least tangentially) to a counting problem of some sort, which could be useful in informing the properties of the resulting series. – Kajelad Aug 12 '18 at 15:01
  • Hello, it is related to convolution of d random variables which are not independent. f is the Copula and it takes as input the discrete marginal distributions. – matosch Aug 12 '18 at 16:33
  • In that case, would we define $f$ to be zero if any of its arguments are negative? Also, would the result $S_n^d=0$ be sensible? – Kajelad Aug 13 '18 at 15:34

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