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Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pairof Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as boundedsymmetric domains in complex vector spaces$\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universalcovering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergmanspace $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction tothe subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely,and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi'sformula in terms of the $K_1$-decomposition of the space$\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement$\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object ofthis article is to compute explicitly the inner product $\big\langlef(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for$f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$,$z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when$\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$,we compute this inner product explicitly by introducing a multivariategeneralization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as anapplication, we construct explicitly $\widetilde{G}_1$-intertwining operators(symmetry breaking operators)$\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ fromholomorphic discrete series representations of $\widetilde{G}$ to those of$\widetilde{G}_1$, which are unique up to constant multiple for sufficientlylarge $\lambda$.

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups