arXiv subjects

Hironobu Kimura

Publications and source records attributed to Hironobu Kimura.

6 recordsLinked to original sources

arXiv2025

Weyl group action on Radon hypergeometric function and its symmetry

For positive integers $r,n,N:=rn$, we consider the Radon hypergeometric function (Radon HGF) associated with a partition $λ$ of $n$ defined on the Grassmannian $Gr(m,N)$ for $r<m<N$, which is obtained as the Radon transform of a character of the group $H_λ\subset G:=GL(N)$. We study its symmetry described by the Weyl group analogue $N_{G}(H_λ)/H_λ$. We consider the Hermitian matrix integral analogue of the Gauss HGF and its confluent family, which are understood as the Radon HGF on $Gr(2r,4r)$ for partitions $λ$ of $4$, we apply the result of symmetry to these particular cases and derive a transformation formula for the Gauss analogue which is known as a part of "24 solutions of Kummer" for the classical Gauss HGF. We derive a similar transformation formula for the analogue Kummer's confluent HGF.

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arXiv2025

Capelli identity and contiguity relations of Radon hypergeometric function on the Grassmannian

The Radon hypergeometric function (Radon HGF) F(z;\a) of type łon the Grassmannian Gr(m,N), N=rn for some integers r,n>0, is defined as a Radon transform of the character of the universal covering of the Lie group H_ł\subset GL(N) specified by a partition łof n, where \a \in C^{n} is the parameter to in the character. For this Radon HGF, we give the contiguity relations of the form L^{(i,j)}F(z;\a)=\b(\a_{0}^{(j)})F(z;\a+\e^{(i)}-\e^{(j)}) with a differential operator L^{(i,j)} of order r and the polynomial \b(s) called b-function. As its application, we derive the contiguity relation for the beta and gamma functions defined by Hermitian matrix integral. In establishing the contiguity relations for Radon HGF, the classical Capelli identity and Cayley's formula play an essential role.

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arXiv2025

On Radon hypergeometric functions on the Grassmannian manifold

We give a definition of Radon hypergeometric function (Radon HGF) of confluent and nonconfluent type, which is a function on the Grassmannian Gr(m,nr) obtained as a Radon transform of a character of the universal covering group of H_λ\subset GL(nr) specified by a partition λof n, where H_{(1,\dots,1)}\simeq(GL(r))^{n}. When r=1, the Radon HGF reduces to the Gelfand HGF on the Grassmannian. We give a system of differential equations satisfied by the Radon HGF and show that the Hermitian matrix integral analogues of Gauss HGF and its confluent family: Kummer, Bessel, Hermite-Weber and Airy function, are obtained in a unified manner as the Radon HGF on Gr(2r,4r) corresponding to the partitions (1,1,1,1), (2,1,1), (2,2), (3,1) and (4), respectively.

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arXiv2025

Gelfand hypergeometric functions as solutions to the 2-dimensional Toda-Hirota equations II

We construct solutions of the $2$-dimensional Toda-Hirota equation (2dTHE) expressed by the Gelfand hypergeometric function (Gelfand HGF) on the Grassmannian $\mathrm{GM}(2,N)$ of confluent or non-confluent type, which is labeled by a partition of $N$. A system of hyperbolic equations in $N$ complex variables is obtained from the differential equations which form a main body of the Gelfand hypergeometric system and characterize the image of Radon transform. We use the Laplace sequence of the system of hyperbolic operators to find an elementary seed solution of the 2dTHE and then use the Bäcklund transformation to obtain higher solutions expressed in terms of Gelfand HGF. In constructing the Laplace sequence, the contiguity relations (operators) for the Gelfand HGF play an important role.

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arXiv2025

On the exterior power structure of the cohomology groups for the general hypergeometric integral

In this article we study the exterior power structure of the algebraic de Rham cohomology group associated with the Gelfand hypergeometric function and its confluent family. The hypergeometric function F(z) is a function on the Zariski open subset $Z_{r+1}\subset\mat{r+1,N}$, called the generic stratum, defined by an r-dimesional integral on $\Ps^{r}$. For $z\in Z_{r+1}$, the algebraic de Rham cohomology group is associated to the integral. When z belongs to the particular subset of $Z_{r+1}$, called the Veronese image, we show that this cohomology group can be expressed as the exterior power product of the de Rham cohomology group associated with the hypergeometric function defined by 1-dimensional integral.

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arXiv2025

Gelfand hypergeometric function as a solution to the 2-dimensional Toda-Hirota equation

We construct solutions of the 2-dimensional Toda-Hirota equation (2dTHE) expressed by the solutions of the system of so-called Euler-Poisson-Darboux equations (EPD) in N complex variables. The system of EPD arises naturally from the differential equations which form a main body of the system characterizing the Gelfand hypergeometric function (Gelfand HGF) on the Grassmannian GM$(2,N)$. Using this link and the contiguity relations for the Gelfand HGF, which are constructed from root vectors for the root $ε_i-ε_j$ for $\mathfrak{gl}(N)$, we show that the Gelfand HGF gives solutions of the 2dTHE.

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