Let $G\left(\lambda \right)$ be the Hecke group of parameter $\lambda >0$, defined by $G\left(\lambda \right)=\langle S_{\lambda },T\rangle$, where $S_{\lambda }z=z+\lambda$, $Tx=-1/z$. (Note: Hecke proved that $G\left(\lambda \right)$ is discrete if and only if $\lambda \ge 2$ or $\lambda =2cos\pi /q$, with $q\in \mathbb{Z}$, $q\ge 3$.) This article initiates the study of “nonanalytic automorphic integrals (AI's)” on $G\left(\lambda \right)$. These functions, defined in the upper half-plane $\mathscr{H}$, generalize “analytic AI's” on $G\left(\lambda \right)$. The latter, in turn, are generalizations of analytic automorphic forms (AF's) on $G\left(\lambda \right)$. (Of course, nontrivial AF's exist on $G\left(\lambda \right)$ only if $\lambda >0$ is chosen so that $G\left(\lambda \right)$ is discrete.)

Analytic AI's on $G\left(\lambda \right)$ are distinguished from analytic AF's on $G\left(\lambda \right)$ in that the former have additive “period functions” in the transformation law under the inversion $T$. These can have several forms; in the present context the period functions are “log-polynomials sums”, finite sums of terms $z^{\alpha }{(logz)}^t$, with $\alpha \in ℂ$ and $t\in \mathbb{Z}$, $t\ge 0$. Analytic AI's on $G\left(\lambda \right)$ are power series in the variable $e^{2\pi iz/\lambda }$, $z\in \mathscr{H}$.

Here, for the first time, the author puts forward a concept of nonanalytic AI on $G\left(\lambda \right)$. His formulation is motivated by two requirements:

(i) the general shape of a nonanalytic AI should be preserved under application of seven weight-changing operators (some of them newly introduced here);

(ii) the nonanalytic Eisenstein series of Maass are to be included.

The author's nonanalytic AI's on $G\left(\lambda \right)$ are series of terms of the form $a{y}^w{e}^{2\pi i/\lambda (n,z-n_2\overline{z})}$, where $y=\text{Im}z$ and the $a$'s and $w$'s are in $ℂ$. The summation is over all nonnegative integers $n_1,n_2$ and only finitely many $w$ appear. Such a series is periodic, of period $\lambda$, and a polynomial growth restruction on the $a$'s guarantees that it is real-analytic in $\mathscr{H}$. In place of the usual single weight there is a pair of complex “coweights”. Finally, the additive period function in the transformation formula under $T$ is an “axial log-polynomial sum”, that is, a function defined in $\mathscr{H}$ which agrees with a log-polynomial sum on the positive imaginary axis.

Through the use of classical analytic AF's, analytic and nonanalytic Eisenstein series and the weight-changing operators, Pasles constructs a multitude of nonanalytic AI's, in his sense, but with zero period functions. (Starting with analytic AI's, instead of AF's, one could apply Pasles' approach to construct a variety of nonanalytic AI's.) Pasles ends the paper by proving that a nonanalytic AI of complex coweights, which is holomorphic in $\mathscr{H}$, is either constant or an analytic AI of a (single) complex weight. This nice result suggests that the author's conception of nonanalytic AI is a natural one. The suggestion is strengthened by the fact that in a sequel the author proves a Riemann-Hecke-Bochner correspondence theorem for his nonalytic AI's. (It remains an open question whether nontrivial analytic or nonanalytic AI's exist on nondiscrete $G\left(\lambda \right)$).