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For integers $n\ge2$ and $k\ge2$, fix the notation $$ [m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}. $$ Consider the following trigonometric numbers: $$ a_i=\frac{[i+k-2]![n-i+k-2]!}{[k-2]![n+k-2]!}, \qquad i=1,2,\dots,n-1. $$ Is it possible for any $n$ to express the quantities $$ A_j=1-\frac{[k-1]\cdot [n+k-1]}{[j+k-1]\cdot [n-j+k-1]}, \qquad j=1,2,\dots,n-1, $$ as a product/quotient of terms of the form $(1-\text{product of some }a_i)$? If not (for $n\ge4$), is it possible to prove that?

The affirmative answer is known for $n=2$ and $n=3$. Namely, if $n=2$ so that we have only one $a_1$ and one $A_1$, then $$ A_1=1-a_1. $$ If $n=3$, then $$ A_j=(1-a_j)(1-a_1a_2) \quad\text{for } j=1,2. $$ (The last formula is a nice trigonometric identity, by the way.)

My question is motivated (in a very sophisticated way) by a recent question on Rogers--Ramanujan identities. The latter one reminded me about the problem of possible $\mathfrak{sl}_n$ generalizations of RRs in their classical form "a $q$-sum"="a $q$-product". The only cases $n=2$ and $n=3$ are known; these are the Andrews--Gordon identities and the Andrews--Schilling--Warnaar identities (see [S. Ole Warnaar, Adv. in Math. 200 (2006) 403--434]). An indirect implication of such identities is the family of (highly nontrivial) numerical identities for the dilogarithm function; these come as the limit $q\to1$ specialisation and some multivariate asymptotics. The trigonometric identities above come into play from these considerations for $n=2$ and $n=3$; any answer for $n>3$ can shed some light on the existence of new RRs.

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Wadim, do you mean that in the "product of some $a_i$" and in the "product/quoient" each term appears at most once? If so, the answer is NO with a counterexample $n=4$, $k=3$, $j=1$. Proof is done by brute-force of all such possible products/quotients. – Max Alekseyev Jul 17 '10 at 5:31
Max, a natural requirement is to have all the multiples in the product/quotients of the form $1\pm a_{j_1}\dots a_{j_k}$ where $j_1,\dots,j_k$ are not necessarily distinct. – Wadim Zudilin Jul 17 '10 at 8:14
do these have to hold for arbitrary $k$, or is $k$ fixed? If $k=1$, things become much nicer but that is disallowed here :-\ – Suvrit Sep 25 '13 at 20:15

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