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It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$. $F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...

Define the sequence of functions $$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$ Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...