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

Abstract: The $\imath$-divided powers (depending on a parity) form the $\imath$canonical basis for the split rank 1 $\imath$quantum group and they are a basic ingredient for $\imath$quantum groups of higher rank. We obtain closed formulae for the structure constants for multiplication of the $\imath$-divided powers. Closed formulae for the comultiplication of the $\imath$-divided powers are also obtained. These structure constants are integral and positive.