Transcendence of Jacobi's theta series

Arithmetic Gevrey Series and Transcendence. a Survey

Theta hypergeometric series

We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hy-pergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.

Harmonic theta series and equidistribution

is a holomorphic modular form of weight 4 for the subgroup [1] Γθ, from the fact that there are no weight 4 cuspforms for Γθ, and from explicit computation of the Fourier coefficients of the two types of weight 4 Eisenstein series for Γθ. Toward a less frivolous result, recall that the irreducibility of spaces Hd of homogeneous, degree d harmonic polynomials f on R as O(n)-spaces gives Hecke’s ...

Functional Equation for Theta Series

In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in [3, p. 189].

Theta Series, Eisenstein Series and Poincaré Series over Function Fields

We construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.