For $a_1,a_2,\ldots,a_n < 1$ let $$f(a_1,a_2,\ldots,a_n) = \sum_{k_1,k_2,...,k_n = 1}^\infty \frac{(k_1 + k_2)!}{k_1! k_2!} \frac{(k_2 + k_3)!}{k_2! k_3!}\ldots \frac{(k_{n-1} + k_{n})!}{k_{n-1}! k_{n}!} a_1^{k_1} a_2^{k_2} \ldots a_n^{k_n}$$ Any idea how to attack this problem? At least for $n=3$...
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4$\begingroup$ what is the problem? $\endgroup$– Fedor PetrovCommented Jul 20, 2020 at 13:29
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$\begingroup$ If you are interested in getting simple rational expressions for $f(a_1,\ldots,a_n)$, I would change the lower bound in the sum to 0 first. $\endgroup$– Timothy BuddCommented Jul 20, 2020 at 16:37
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$\begingroup$ Fedor: finding explicit rational expressions. $\endgroup$– tomateCommented Jul 21, 2020 at 6:42
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