thanks mr Gerhard for your hel, i need $p| C_{9.n}^{k-n} C_{8n+k}^8n $, $ \forall$ integer $ k$ satifying $ max(n,p)≤k≤10n.$ $S$ is not restrticted to $]8*n,9*n[$ because I have found that $\forall p$ prime $10.n \geq p > \sqrt{18n}, p \in ]4n=8*n/2,9*n/2[, $ we have $\forall k $ such that $ max(n,p)≤k≤10n$ one has $v_p( C_{9.n}^{k-n} C_{8n+k}^{8n}) \geq 1$

thanks mr Gerhard for your comment, S

thanks mister Gerarld for your help. for having 1) and 2), do i need , further more the condtion you wrote , $ \displaystyle \sum_{n \geq 0} \displaystyle \int_{0}^1 \int_{0}^1 \int_{0}^1 \int_{0}^1 | f_n(u,v,w,t)| \mathrm{d}u \mathrm{d}v \mathrm{d}w \mathrm{d}t $ converge? thanks

i'll check this formula, thanks for your help

thanks mr Fedor Petrov, i was expecting something more simple, ( didn't expect a complicate raison) but it's correct and thank you very much for your help..