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arXiv · 2506.04638

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

Hironobu Kimura
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Abstract

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 $\epsilon_i-\epsilon_j$ for $\mathfrak{gl}(N)$, we show that the Gelfand HGF gives solutions of the 2dTHE.

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Hironobu Kimura. 2025-06-05. Gelfand hypergeometric function as a solution to the 2-dimensional Toda-Hirota equation. https://arxiv.org/abs/2506.04638

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