A note on the linear homogeneous group over a finite field

In this paper are examined some classes of linear and non-linear analytical systems of partial differential equations. Compatibility conditions are found and if they are satisfied, the solutions are given as functional series in a neighborhood of a given point (x = 0).

The singular series for a single square

Carlitz, L. (1957)

Portugaliae mathematica

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On the normal systems of differential equations

Breves Filho, J.A. (1974)

Portugaliae mathematica

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Study of a generalized integral operator

Goyal, S.P. (1975)

Portugaliae mathematica

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The new properties of the theta functions

Stefan Czekalski (2013)

Annales mathématiques Blaise Pascal

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It is shown, that the function $\begin{matrix} H (x) & = \sum_{k = - \infty}^{\infty} e^{- k^{2} x} \\ satisfies the relation \\ H (x) & = \sum_{n = 0}^{\infty} \frac{{(2 π)}^{2 n}}{(2 n)!} H^{(n)} (x) . \end{matrix}$