A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915918573993984 |
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| author | Shibukawa, G. Tsuchimi, S. |
| author_facet | Shibukawa, G. Tsuchimi, S. |
| contents | In this paper, we introduce the little $μ$-function, which is obtained as a degenerate limit of the generalized $μ$-function. We derive the little $μ$-function as the image of the $q$-Borel summation of a divergent solution to the Ramanujan equation which is the most degenerate second order linear $q$-difference equations of Laplace type excluding those of constant coefficients. Moreover, we present several formulas, such as symmetries and connection formulas for the little $μ$-function, similar to those for the generalized $μ$-function. Furthermore, we establish contiguous relations related to the $q,t$-Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03693 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation Shibukawa, G. Tsuchimi, S. Classical Analysis and ODEs 33D15, 11P84, 39A13, 30D05, 33D70 In this paper, we introduce the little $μ$-function, which is obtained as a degenerate limit of the generalized $μ$-function. We derive the little $μ$-function as the image of the $q$-Borel summation of a divergent solution to the Ramanujan equation which is the most degenerate second order linear $q$-difference equations of Laplace type excluding those of constant coefficients. Moreover, we present several formulas, such as symmetries and connection formulas for the little $μ$-function, similar to those for the generalized $μ$-function. Furthermore, we establish contiguous relations related to the $q,t$-Fibonacci sequences and Wronskian relations involving the Rogers-Ramanujan continued fraction. |
| title | A degeneration of the generalized Zwegers' $μ$-function according to the Ramanujan difference equation |
| topic | Classical Analysis and ODEs 33D15, 11P84, 39A13, 30D05, 33D70 |
| url | https://arxiv.org/abs/2603.03693 |