A determinant identity for the sum of contour integral matrices
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915961408323584 |
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| author | Liu, Zhipeng Tripathi, Tejaswi |
| author_facet | Liu, Zhipeng Tripathi, Tejaswi |
| contents | We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2π\mathrm{i}}\oint_0 z^{i-j-1}p_i(z)f_j(z)\mathrm{d} z$ and $B(i,j)= \frac{1}{2π\mathrm{i}}\int_Γ q_i(z)g_j(z) \mathrm{d} z$. Under suitable assumptions on the functions $p,q,f,g$, we show that $\det(A+B)$ can be expressed as a Fredholm determinant $\det(\mathrm{I} +K)$, where $K$ is an integral kernel acting on the contour $Γ$. This result generalizes a recent identity obtained in \cite{Baik-Liao-Liu26}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A determinant identity for the sum of contour integral matrices Liu, Zhipeng Tripathi, Tejaswi Classical Analysis and ODEs Probability We derive an identity for the determinant of the sum of two $n\times n$ matrices, $A$ and $B$, whose entries are defined via contour integrals. Specifically, we consider $A(i,j)=\frac{1}{2π\mathrm{i}}\oint_0 z^{i-j-1}p_i(z)f_j(z)\mathrm{d} z$ and $B(i,j)= \frac{1}{2π\mathrm{i}}\int_Γ q_i(z)g_j(z) \mathrm{d} z$. Under suitable assumptions on the functions $p,q,f,g$, we show that $\det(A+B)$ can be expressed as a Fredholm determinant $\det(\mathrm{I} +K)$, where $K$ is an integral kernel acting on the contour $Γ$. This result generalizes a recent identity obtained in \cite{Baik-Liao-Liu26}. |
| title | A determinant identity for the sum of contour integral matrices |
| topic | Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2604.24747 |