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We study the asymptotic behaviour of Bessel functions associated of rootsystems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as therank $n$ tends to infinity. In both cases, we characterize the possible limitfunctions and the Vershik-Kerov type sequences of spectral parameters for whichsuch limits exist. In the type $A$ case, this generalizes known results aboutthe approximation of the (positive-definite) Olshanski spherical functions ofthe space of infinite-dimensional Hermitian matrices over $\mathbb F = \mathbbR, \mathbb C, \mathbb H$ (with the action of the associated infinite unitarygroup) by spherical functions of finite-dimensional spaces of Hermitianmatrices. In the type B case, our results include asymptotic results for thespherical functions associated with the Cartan motion groups of non-compactGrassmannians as the rank goes to infinity, and a classification of theOlshanski spherical functions of the associated inductive limits.

Limits of Bessel functions for root systems as the rank tends to infinity