Changgui Zhang, University of La Rochelle, France

Jacobi theta function and $q$-exponential functions


Recently, the classical Borel-Laplace summation method has extended to the cases of $q$-Gevrey formal power series. This permits to construct, for each given formal power series satisfying a linear meromorphic $q$-difference equation, an actual solution in the ``origin" of the logarithm function Riemann surface. See~:  Developpements asymptotiques $q$-Gevrey et series $Gq$-sommables, Ann. Institut de Fourier 49 (1999) 227-261, and, by Web~: 99-4. Sur la fonction $q$-Gamma de Jackson. 99-10. (with F. Marotte) Multisommabilite des solutions formelles d'une equation aux $q$-differences lineaire analytique. We shall present a new summation method for $q$-series, using $\sum_{n\ge 0}q^{-n(n-1)/2}x^n$ as $q$-exponential function. Some classical results will also be given (as simple consequence of Stokes phenomenen).