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Let D be the Lie ball in ℂ n , when n is odd. In this paper, we give an explicit expression of the generalized spherical function Φλ,m for D. More precisely, Φλ,m is given explicitly as sum of determinants of the classical Gauss hypergeometric functions. Note that our computation technic is done by integration by parts and induction. Namely, it is based on the key Lemma 2.2. The purpose of this article is to present an explicit expression of the generalized spherical function on odd-dimensional bounded symmetric domains of type IV (Lie balls). [ABSTRACT FROM AUTHOR]

For any n, [image omitted], such that n ≥ m, let { [image omitted]; Im - z * z positive definite} be the bounded symmetric domain of type I of rank m and let A'(S) be the space of all hyperfunctions over the Shilov boundary S of [image omitted]. The aim of this article is to give a necessary and sufficient condition on the generalized Poisson transform Pl,λ f of an element f in the space A'(S) for that f∈Lp(S), 1 < p < ∞. More precisely, we establish for any [image omitted] such that [image omitted] that: (i) Let F = Pl,λ f, f∈Lp(S). Then we have [image omitted] (ii) Let f be a hyperfunction on S such that its image F = Pl,λ f satisfies the growth condition [image omitted], then necessarily such f is in Lp(S). [ABSTRACT FROM AUTHOR]