In this paper, we express the generating functions of the partion numbers of certain kinds by theta constants with rational characteristics. Our result is the generalizations of the result of Farkas~[1] for modulus 4 and 7.

Sharpened version of the Julia-Carathéodory Theorem for functions with fixed initial coefficients is proved.

Let $(K,\varphi)$ be a perfect valued field of rank 1, let $\overline{\varphi}$ be an extension of the absolute (multiplicative) value $\varphi$ to a fixed algebraic closure $\overline{K}$ and let $\| .\|_{\varphi}$ be the corresponding spectral norm on $K$. Let $(\widetilde{\overline{K}},\| .\|_{\varphi}^{\,\tilde{}})$ be a fixed completion of $(\overline{K},\| .\|_{\varphi})$. In this paper we generalize a result of A. Ostrowski~[8] relative to the absorbent property of a subfield, from the case of a complete non-Archimedian valued field of characteristic 0 to our ring $(\widetilde{\overline{K}},\| .\|_{\varphi}^{\,\tilde{}})$ (see Theorem 1, Theorem 4). We also apply these results to discuss in a more general context the following conjecture due to A. Zaharescu (2009): $\langle$For any $x,y\in\mathbf{C}_{p}$-the complex $p$-adic field, there exists $t\in\mathbf{Q}_{p}$-the $p$-adic number field, such that $\widetilde{\mathbf{Q}_{p}(x,y)}=\widetilde{\mathbf{Q}_{p}(x+ty)}$, where $\widetilde{L}$ means the $p$-adic topological closure of a subfield $L$ of $\mathbf{C}_{p}$ in $\mathbf{C}_{p}\rangle$.

If a locally finite rational representation $V$ of a connected reductive algebraic group $G$ has uniformly bounded multiplicities, the multiplicities may have good properties such as stability. Let $X$ be a quasi-affine spherical $G$-variety, and $M$ be a $(\mathbf{C}[X],G)$-module. In this paper, we show that the decomposition of $M$ as a $G$-representation can be controlled by the decomposition of the fiber $M/\mathfrak{m}(x_{0})M$ with respect to some reductive subgroup $L \subset G$ for sufficiently large parameters. As an application, we apply this result to branching laws for simple real Lie groups of Hermitian type. We show that the sufficient condition on multiplicity-freeness given by the theory of visible actions is also a necessary condition for holomorphic discrete series representations and symmetric pairs of holomorphic type. We also show that two branching laws of a holomorphic discrete series representation with respect to two symmetric pairs of holomorphic type coincide for sufficiently large parameters if two subgroups are in the same $\epsilon$-family.