The classical Mehler-Heine formula shows that the Bessel function \(J_0\) can be obtained as a limit in an appropriate sense of the Legendre polynomials \(P_N\) of order \(N\), thus relating the spherical functions for the Gelfand pair \((M(n),SO(n))\) with those of the Gelfand pair \((SO(n+1),SO(n))\) for the case \(n=2\) (see the paper under review and the references therein for more details). This work has since been carried forward in [A. H. Dooley and J. W. Rice, Math. Proc. Camb. Philos. Soc. 94, 509–517 (1983; Zbl 0532.22014)] where using a family of group contractions between \(SO(n+1)\) and \(M(n)\) the authors establish such a connection between the spherical functions of \((M(n),SO(n))\) and \((SO(n+1),SO(n))\) for all \(n\). In [A. H. Dooley and J. W. Rice, Trans. Am. Math. Soc. 289, 185–202 (1985; Zbl 0546.22017)] a similar result has been proved for the spherical functions for the Gelfand pairs \((M(n),SO(n))\) and \((SO_0(n,1),SO(n)))\) using a family of contractions between \(M(n)\) and \(SO_0(n,1)\).
In another work, F. Ricci and A. Samanta [Adv. Math. 338, 953–990 (2018; Zbl 1398.43002) have studied the vector-valued spherical functions associated with strong Gelfand pairs. The strong Gelfand pairs are defined as follows:
Let \(G\) be a Lie group, \(K\) be a compact subgroup of \(G\) and let \(\tau\) be an irreducible unitary representation of \(K\) on the vector space \(V_\tau\). The triple \((G,K,\tau)\) is called commutative if the algebra \(L^1_\tau(G,\mathrm{End}(V_\tau))\) of vector-valued functions \(F \in L^1(G,\mathrm{End}(V_\tau))\) satisfying the following transformation property \[F(k_1xk_2)= \tau(k_2^{-1}) F(x)\tau(k_1^{-1}) \] for \(k_1,k_2 \in K\) and \(x\in G\), is commutative under convolution.
(In case \(\tau\) is the trivial representation, the triple reduces to the familiar case of Gelfand pair \((G,K)\).)
The pair \((G,K)\) is called strong Gelfand pair if \((G,K,\tau)\) is commutative for every \(\tau \in \hat{K}\).
In the paper under review, the authors use the group contractions described in [A. H. Dooley and J. W. Rice, Math. Proc. Camb. Philos. Soc. 94, 509–517 (1983; Zbl 0532.22014)] and [A. H. Dooley and J. W. Rice, Trans. Am. Math. Soc. 289, 185–202 (1985; Zbl 0546.22017)] to show that the vector-valued spherical functions associated with strong Gelfand pairs \((M(n),SO(n))\) can be obtained as appropriate limits of the vector-valued spherical functions associated with each of the strong Gelfand pairs \((SO(n+1),SO(n))\) and \((SO_0(n,1),SO(n))\).