To the complex vector space \(V=M(n,p:\mathbb{C})\), the space of \(n \times p\) complex matrices, with the inner product given by \((z|w)= \text{tr}(zw^{*})\), one associates the Heisenberg group \(H=V\times \mathbb R\) with the product
\[ (z,t)(z', t')=(z+z', t+t'+ \text{Im}(z|z')). \]
The group \(K=U(n)\times U(p)\) acts on \(H\) by \((u,v)\cdot (z,t)=(uzv^*,t)\) if \((u,v) \in K\). Then \((G,K)\) with \(G=K\ltimes H\) is a Gelfand pair. A continuous complex-valued function \(\varphi\) on \(H\) is said to be spherical if it satisfies the following functional equation
\[ \int_{K}\varphi(z+k\cdot z', t+t'+ \text{Im}(z|k\cdot z'))\alpha(dk) =\varphi(z,t)\varphi(z',t') \]
for \((z,t),(z', t') \in H\) (\(\alpha\) denotes the normalized Haar measure on \(K\)).
The author determines the spherical functions related to the Gelfand pair \((G,K)\). This analysis involves the asymptotics of shifted Schur functions. The author also studies the asymptotics of these spherical functions for large \(n\) and \(p\). The asymptotics of spherical functions for large dimensions are related to spherical functions for Olshanski spherical pairs. The author also considers inductive limits of Gelfand pairs associated to the Heisenberg group.