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I am looking for an asymptotic expansion of J(n)

$J(n)=\frac {2} {\pi} \int_{0}^{\pi/n} \prod_{k=1}^n \frac {\sin kx} {\sin x} dx$, $n=2,3,4,\dots$ The first approximation is managed to get $F_1(n)=\frac {n!} {\sqrt{\pi A}}$, $A=n(n-1)(2n+5)/36$

Is a general expansion known for this?

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This is probably an overkill way to a solution, but applying Euler reflection formula $$ \sin(z)=\frac{\pi}{\Gamma(1-\frac{z}{\pi})\Gamma(\frac{z}{\pi})}, $$ one gets products of gamma functions both in the numerator and denominator. From this point on, I think, your integral is computable/representable via a generalized Hypergeometric Function of Fox type: (I am mentioning the wiki link mainly because of the references there), i.e. Barnes-Integral with a kernel of fraction of products of Gamma functions. If my memory serves me right, this type of integral has been treated also in the book of Bleistein and Handelsman. If it does not, then the relevant paper in this case is "Asymptotic expansions and analytic continuations for a class of Barnes-integrals" by Braaksma, Compositio Math. 15,1964, p. 239–341.

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