we have the sequence of positive real numbers ${a_{j}}$ , such that :
$$ \frac{1}{j+1} $$\frac{n+2}{n+1}=\prod_{j=1}^{\infty}\left(1-\frac{a_{j}^{2}}{(a_{j}n+a_{j}-1)^{2}}\right)$$
furthermore, the infinite product :
$$\prod_{j=1}^{\infty}(1+a_{j})e^{-a_{j}}$$
is convergent. in fact, there is an entire function defined as:
$$f(x)=C\prod_{j=1}^{\infty}(1+xa_{j})e^{-xa_{j}}$$
such that, at negative integers:
$$f(-n)=K(-1)^{n}n!$$
$C$ and $K$ being constants.
can we prove that such a sequence exists ? how can we solve for the numbers $a_{j}$ ?
an infinite system of infinite products
3
$\begingroup$
sequences-and-series
infinite-product
1 Answers
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If we define $b_j=1/{a_j}$ then we have $j