Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform

In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $\beta =2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $\beta >0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this result one can calculate the asymptotic behavior of complete homogeneous symmetric polynomials.