Quasi-elliptic functions

A. Ya. Khrystiyanyn   ; Dz. V. Lukivska

Ufa mathematical journal, Tome 9 (2017) no. 4, pp. 127-134

Résumé

We study certain generalizations of elliptic functions, namely quasi-elliptic functions. Let $p = e^{i\alpha},$ $q = e^{i\beta},$ $\alpha,\, \beta \in \mathbb{R}.$ A meromorphic in $\mathbb{C}$ function $g$ is called quasi-elliptic if there exist $\omega_1, \omega_2 \in \mathbb{C}^{*},$ $\mathrm{Im} \frac{\omega_2}{\omega_1} > 0,$ such that $g(u+\omega_1)=pg(u)$, $g(u+\omega_2)=qg(u)$ for each $u\in\mathbb{C}$. In the case $\alpha = \beta = 0 \mod 2\pi$ this is a classical theory of elliptic functions. A class of quasi-elliptic functions is denoted by $\mathcal{QE}.$ We show that the class $\mathcal{QE}$ is nontrivial. For this class of functions we construct analogues $\wp_{\alpha \beta}$, $\zeta_{\alpha \beta}$ of $\wp$ and $\zeta$ Weierstrass functions. Moreover, these analogues are in fact the generalizations of the classical $\wp$ and $\zeta$ functions in such a way that the latter can be found among the former by letting $\alpha=0$ and $\beta=0$. We also study an analogue of the Weierstrass $\sigma$ function and establish connections between this function and $\wp_{\alpha \beta}$ as well as $\zeta_{\alpha \beta}$. Let $q, p \in\mathbb{C}^*,$ $|q|1.$ A meromorphic in $\mathbb{C^{*}}$ function $f$ is said to be $p$-loxodromic of multiplicator $q$ if for each $z \in \mathbb{C}^{*}$ $f(qz) = pf(z).$ We obtain telations between quasi-elliptic and $p$-loxodromic functions.

Détail

Keywords: quasi-elliptic function, the Weierstrass $\wp$-function, the Weierstrass $\zeta$-function, the Weierstrass $\sigma$-function, $p$-loxodromic function.

Bibliographie

[1] Y. Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic Press, San Diego, 2002 | MR | Zbl

[2] V. S. Khoroshchak, A. A. Kondratyuk, “Some steps to nonlinear analysis”, Book of Abstracts of “XVIII-th Conference on analytic functions and related topics” (Chełm, Poland, 2016), 38–39

[3] A. Hurwitz, R. Courant, Function theory, Nauka, M., 1968 (in Russian)

[4] V. S. Khoroshchak, A. A. Kondratyuk, “Generalization of the Weierstrass $\wp$-function”, Book of Abstracts of Ukrainian Scientific Conference “Modern problems of probability theory and mathematical analysis” (Vorokhta, Ukraine, 2016), 93 (in Ukrainian)

[5] V. S. Khoroshchak, A. Ya. Khrystiyanyn, D. V. Lukivska, “A class of Julia exceptional functions”, Carpathian Math. Publ., 8:1 (2016), 172–180 | DOI | MR | Zbl

[6] A. A. Kondratyuk, V. S. Khoroshchak, D. V. Lukivska, “p-Elliptic functions”, Visnyk Lviv Univ. Ser. Mech. Math., 81 (2016), 121–129

[7] O. Rausenberger, Lehrbuch der Theorie der Periodischen Functionen Einer variabeln, Druck und Ferlag von B. G. Teubner, Leipzig, 1884

[8] G. Valiron, Cours d'Analyse mathématique: Théorie des fonctions, Masson, Paris, 1948 | MR