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\(q\)-Gevrey asymptotic expansion and Jacobi theta function. (Développement asymptotique \(q\)-Gevrey et fonction thêta de Jacobi.) (French) Zbl 1025.39014
Summary: Some notions of \(q\)-Gevrey asymptotic expansion have been studied by C. Zhang [Ann. Inst. Fourier 49, 227–261 (1999; Zbl 0974.39009); C. R. Acad. Sci., Paris, Sér. I, Math. 331, 31–34 (2000; Zbl 1101.33307)]. Recently we became interested in a new notion of asymptotic expansion, it is related to a Jacobi theta function and allows one to establish the natural link between the asymptotics of \(q\)-difference equations and the theory of elliptic functions. The purpose of this Note is to give some new results related to this notion of asymptotic expansion.

MSC:
39A13 Difference equations, scaling (\(q\)-differences)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
30D20 Entire functions of one complex variable (general theory)
33E05 Elliptic functions and integrals
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