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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.

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
Full Text: DOI
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