arXiv subjects
Publications and source records attributed to Kazuhide Matsuda.
In this series of papers, we introduce higher level versions of the theta group $Γ_θ.$ In this paper, we treat the theta group of level $5$, $Γ_{θ,5},$ and construct modular forms on $Γ_{θ,5}$. Moreover we compute their multiplier systems. For this purpose, we derive transformation formula of theta function with characteristics.
In this paper, we introduce higher level versions of the theta group $Γ_θ.$ In particular, we treat level 3 and 4 versions of the theta group, $Γ_{θ,3}$ and $Γ_{θ,4}$ and prove that $\displaystyle F(τ)=η\left(\frac{τ-1}{3} \right) η\left(\frac{τ+1}{3} \right)$ and $\displaystyle G(τ)=η\left(\frac{τ-1}{4} \right) η\left(\frac{τ+1}{4} \right)$ are modular forms on $Γ_{θ,3}$ and $Γ_{θ,4}$ respectively. Moreover we compute their multiplier systems, $ν_{F}$ and $ν_{G}$.
The multiplier systems of $η^{2k}(τ)$ and $θ^{2k}(τ)$ $(k\in\mathbb{Z})$ are characters. In this paper, we determine their kernels, Ker$\,ν_{η^{2k}}$ and Ker$\,ν_{ θ^{2k} } $.
This paper describes the derivation of the level 5 versions of Ramanujan's system of ordinary differential equations satisfied by the Eisenstein series, $E_2(q),E_4(q)$, and $E_6(q).$
The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by modular forms of level six and to construct extensions of the differential field of the cubic theta functions, generalizing the classical Ramanujan and Halphen fields. We treat both modular forms that appear in the literature and others that do not. We find Riccati equations satisfied by level six modular forms, explore applications to number theory, and find new relations among the Eisenstein series.
The aim of this paper is to derive new results about Jacobi's inversion formulas for modular forms of levels 5 and 6. For this purpose, we use Farkas and Kra's theory of theta functions with rational characteristics.
In this paper, we derive systems of ordinary differential equations (ODEs) satisfied by modular forms of level three, which are level three versions of Ramanujan's system of ODEs satisfied by the classical Eisenstein series.
In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.
In this paper, we realize high-level versions of Jacobi's derivative formula to all the rational characteristics corresponding to level $k \,\,(k=3,4,5,6).$ For this purpose, we propose the method to obtain derivative formulas by means of the residue theorem. We believe that this method can be also applied to all the rational characteristics corresponding to level $k\ge 7.$
We completely classify the rational solutions of the Sasano system of type $D5^{(1)}.$ The rational solutions are classified to four classes by the Bäcklund transformation group.
In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.
In this paper, we derive quintic versions of the cubic identities of Farkas and Kra. We believe that our results can be easily generalized to $k$ th power versions,$(k=7,9,11,\ldots).$ Moreover, we investigate the algebraic structure of theta constants of level five.
In this paper, we determine all the positive integers $a, b$ and $c$ such that every nonnegative integer can be represented as $$ f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with} \,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we prove that $f^{a,b}_c$ can represent all the nonnegative integers if it represents $n=1,2,3,5,6,10.$
In this paper, we obtain analogues of Jacobi's derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural number $n,\,\,(n=1,2,\ldots)$ as the sum of two squares, or the sum of a square and twice a square.
We classify the rational solutions of the Sasano systems of types $B_4^{(1)},$ $D_4^{(1)}$ and $D_5^{(2)},$ which are all given by the coupled $P_{\rm III}$ systems and have the affine Weyl group symmetries of types $B_4^{(1)},$ $D_4^{(1)}$ and $D_5^{(2)}.$ The rational solutions are classified as one type by the Bäcklund transformation groups.
In this paper, we completely classify the raional solutions of the Noumi and Yamada system of type A_4^{(1)}, which is a generalization of the forth Painlevé equation. The rational solutions are classified to three types by the Bäcklund transformation group.
We completely classify all of rational solutions of the Noumi and Yamada system of type $A_5^{(1)}$, which is a generalization of the fifth Painlevé equation. The rational solutions are classified to five classes by the Bäcklund transformation group.
In this paper, we completely classify the rational solutions of the Sasano system of type $A_5^{(2)}$, which is given by the coupled Painlevé III system. This system of differential equations has the affine Weyl group symmetry of type $A_5^{(2)}$.