Exponentially-improved asymptotics for $q$-difference equations: ${}_2ϕ_0$ and $q{\rm P}_{\rm I}$
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| Format: | Preprint |
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2024
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| _version_ | 1866914701269532672 |
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| author | Joshi, Nalini Daalhuis, Adri Olde |
| author_facet | Joshi, Nalini Daalhuis, Adri Olde |
| contents | Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size $q^{-\frac12 n(n-1)}$, in which $q\in(0,1)$ is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function ${}_2ϕ_0$ and for solutions of the $q$-difference first Painlevé equation $q{\rm P}_{\rm I}$. These are optimal truncated expansions, and re-expansions in terms of new $q$-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02196 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exponentially-improved asymptotics for $q$-difference equations: ${}_2ϕ_0$ and $q{\rm P}_{\rm I}$ Joshi, Nalini Daalhuis, Adri Olde Classical Analysis and ODEs 33D15, 34M30, 34M40, 39A13 Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size $q^{-\frac12 n(n-1)}$, in which $q\in(0,1)$ is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function ${}_2ϕ_0$ and for solutions of the $q$-difference first Painlevé equation $q{\rm P}_{\rm I}$. These are optimal truncated expansions, and re-expansions in terms of new $q$-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena. |
| title | Exponentially-improved asymptotics for $q$-difference equations: ${}_2ϕ_0$ and $q{\rm P}_{\rm I}$ |
| topic | Classical Analysis and ODEs 33D15, 34M30, 34M40, 39A13 |
| url | https://arxiv.org/abs/2403.02196 |