Asymptotic behavior of multiplicative spherical integrals and S-transform

In this paper, we study the asymptotics of a spherical integral that is a multiplicative counterpart to the well-known Harish-Chandra–Itzykson–Zuber integral. This integral can also be expressed in terms of the Heckman–Opdam hypergeometric function. When the argument of this spherical integral is of finite support and of order [Formula: see text], these asymptotics involve a modified version of the [Formula: see text]-transform of the limit measure of the matrix argument and its largest eigenvalue. To prove our main result, we are leveraging a technique of successive conditioning used in [12]. In particular we prove in a “mathematically rigorous” manner a result from [22] in the case [Formula: see text] and we generalize it for multiple arguments.