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The aim of the present paper is three folds. Firstly, we complete the studyof the weighted hyperholomorphic Bergman space of the second kind on the ballof radius $R$ centred at the origin. The explicit expression of its Bergmankernel is given and can be written in terms of special hypergeometric functionsof two non-commuting (quaternionic) variables. Secondly, we introduce and studysome basic properties of an associated integral transform, the quaternionicanalogue of the so-called second Bargmann transform for the holomorphic Bergmanspace. Finally, we establish the asymptotic behavior as $R$ goes to infinity.We show in particular that the reproducing kernel of the weighted slicehyperholomorphic Bergman space gives rise to its analogue for the slicehyperholomorphic Bargamann-Fock space.

The slice hyperholomorphic Bergman space on $\mathbb{B}_R$: Integral representation and asymptotic behavior