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Formulae of $\imath$-divided powers in ${\bf U}_q(\mathfrak{sl}_2)$, II

The coideal subalgebra of the quantum $\mathfrak{sl}_2$ is a polynomial algebra in a generator $t$ which depends on a parameter $κ$. The existence of the $\imath$-canonical basis (also known as the $\imath$-divided powers) for the coideal subalgebra of the quantum $\mathfrak{sl}_2$ were established by Bao and Wang. We establish closed formulae for the $\imath$-divided powers as polynomials in $t$ and also in terms of Chevalley generators of the quantum $\mathfrak{sl}_2$ when the parameter $κ$ is an arbitrary $q$-integer. The formulae were known earlier when $κ=0,1$.