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New relations between certain monomials of the fundamental generators - indexed by the integer $r\\in\\mathbb{Z}^{+}$ - are conjectured. These relations can be seen as deformed analogues of Lusztig's $r-$th higher order $q-$Serre relations associated with ${\\cal U}_q({\\widehat g})$, which are recovered as special cases. The relations are proven for $r\\leq 5$. 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Baseilhac, T.T. Vu","submitted_at":"2013-12-20T11:38:11Z","abstract_excerpt":"Let $\\{A_j|j=0,1,...,rank(g)\\}$ be the fundamental generators of the generalized $q-$Onsager algebra $\\cal O_{q}(\\widehat{g})$ introduced in \\cite{BB1}, where $\\widehat{g}$ is a simply-laced affine Lie algebra. New relations between certain monomials of the fundamental generators - indexed by the integer $r\\in\\mathbb{Z}^{+}$ - are conjectured. These relations can be seen as deformed analogues of Lusztig's $r-$th higher order $q-$Serre relations associated with ${\\cal U}_q({\\widehat g})$, which are recovered as special cases. The relations are proven for $r\\leq 5$. 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