Exponentially-improved asymptotics for q-difference equations: 2ϕ0 and qPI

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 nth coefficient is often of the size q−12n(n−1), in which q∈(0,1) is fixed. Hence, the divergence is much s...

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Bibliographic Details
Published in:Indagationes mathematicae Vol. 36; no. 6; pp. 1555 - 1571
Main Authors: Joshi, Nalini, Olde Daalhuis, Adri
Format: Journal Article
Language:English
Published: Elsevier B.V 01.11.2025
Subjects:
Asymptotic expansions
Basic hypergeometric functions
Hyperterminant
q-difference equations
Stokes phenomenon
Asymptotic expansions
39A13
Basic hypergeometric functions
q-difference equations
Hyperterminant
33D15
34M30
34M40
Stokes phenomenon
ISSN:0019-3577
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Summary: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 nth coefficient is often of the size q−12n(n−1), in which q∈(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 2ϕ0 and for solutions of the q-difference first Painlevé equation qPI. These are optimal truncated expansions, and re-expansions in terms of new q-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.
ISSN:0019-3577
DOI:10.1016/j.indag.2025.02.002