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Source Identities and Kernel Functions for the Deformed Koornwinder–van Diejen Models

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Abstract

We consider generalizations of the BC-type relativistic Calogero–Moser–Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh–Feigin–Sergeev–Veselov type deformation of these operators and their corresponding kernel functions, which generalize the known kernel functions for the Koornwinder–van Diejen models.

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Notes

  1. Throughout the paper, we write \(A_{N}(\mathbf {x} ; \mathbf {g}, \lambda ,\beta )\), etc., to indicate the arguments \(\mathbf {x}\) and the coupling parameters.

  2. We use the standard definition for the odd Jacobi theta function \(\vartheta _{1}(x) := \vartheta _{1}(x |\tau ) \) (\(\mathfrak {I}(\tau )>0\)) found in [WW40]; see also (23).

  3. The function s(x) in (50) should then be replaced with the odd Jacobi theta function \(\vartheta _{1}(x)\).

  4. These Gamma functions are the same as those Gamma functions used in [AHL14].

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Acknowledgements

We would like to thank E. Langmann for suggesting this project, and M. Noumi and S.N.M. Ruijsenaars for helpful discussions. This work has been supported by the Japan Society for the Promotion of Science and the author is a JSPS International Research Fellow (Grant Nos. P17768 and 17F17768).

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  1. Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan

    Farrokh Atai

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Communicated by H. T. Yau

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Atai, F. Source Identities and Kernel Functions for the Deformed Koornwinder–van Diejen Models. Commun. Math. Phys. 377, 2191–2216 (2020). https://doi.org/10.1007/s00220-020-03753-w

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