Report
Exponentially-improved asymptotics for $q$-difference equations: ${}_2\phi_0$ and $q{\rm P}_{\rm I}$
| العنوان: | Exponentially-improved asymptotics for $q$-difference equations: ${}_2\phi_0$ and $q{\rm P}_{\rm I}$ |
|---|---|
| المؤلفون: | Joshi, Nalini, Daalhuis, Adri Olde |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Classical Analysis and ODEs, 33D15, 34M30, 34M40, 39A13 |
| الوصف: | 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 ${}_2\phi_0$ and for solutions of the $q$-difference first Painlev\'e 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. Comment: 15 pages, 1 figure |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2403.02196 |
| رقم الانضمام: | edsarx.2403.02196 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |