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In his proof of the Rogers-Ramanujan identities I. Schur introduced two $q-$analogues of the Fibonacci numbers ${F_n}({q})$ and ${G_n}({q})$, which satisfy

${F_0}({q})=0$, ${F_1}({q})=1$ and $${F_n}({q})={F_{n-1}}({q})+q^{n-2}{F_{n-2}}({q})$$ and ${G_0}({q})=0$, ${G_1}({q})=1$ and $${G_n}({q})={G_{n-1}}({q})+q^{n-1}{G_{n-2}}({q}).$$ Computer experiments suggest that if $k$ is not a multiple of $5$ and ${\zeta _k }$ a primitive $k-$th root of unity then for each positive integer $n$ $${F_{kn}}({\zeta _k }) {G_{kn}}({\zeta _k })=F_{n}^2$$ holds. For example $${F_{4n}}({i}) {G_{4n}}({i})=F_{n}^2$$ The special case $${F_{k}}({\zeta _k }) {G_{k}}({\zeta _k })=1$$ for $k \equiv \pm 2\bmod 5$ follows from a result by L. Carlitz, Solution of Problem H- 138, Fibonacci Quart. 8(1), 1970, 76-81.

Is there a proof in the literature for the general identity or at least for $n=1$ and $k \equiv \pm 1\bmod 5$?

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