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The Kronecker theta function and a decomposition theorem for theta functions I
The Ramanujan Journal, 2021The Kronecker theta function \(K_y(z|\tau)\) is a quotient of the Jacobi theta functions, given by \[ K_y(z|\tau)=\frac{\theta'_1(0|\tau)\theta_1(z+y|\tau)}{\theta_1(z|\tau)\theta_1(y|\tau)}, \tag{1} \] where \(\operatorname{Im}\tau>0\) and \begin{align*} \theta_1(z|\tau) &=2\sum_{n=0}^\infty(-1)^nq^{(2n+1)^2/8}\sin(2n+1)z,\\ \theta'_1(0|\tau) &=2q^{1 ...
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Conic theta functions and their relations to theta functions
2013It is natural to ask when the spherical volume defined by the intersection of a sphere at the apex of an integer polyhedral cone is rational. We use number theoretic methods to study a new class of polyhedral functions called conic theta functions, which are closely related to classical theta functions.
Amanda Folsom +2 more
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Theta Functions and Transcendence
The Ramanujan Journal, 1997Des résultats de transcendance et d'indépendance algébrique concernant les valeurs de fonctions modulaires ont été récemment obtenus (travaux de K. Barré et al., Yu. Nesterenko, P. Philippon). Dans cet article, l'A. propose une relecture de ces résultats et d'autres, plus anciens, à l'aide des fonctions thêta de Jacobi \[ \begin{aligned} \theta_2(q) & =
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Mathematical Journal of Okayama University, 1996
Continuing along the lines of the earlier article [ibid. 36, 35-44 (1994; Zbl 0841.11021)], the author derives a number of formula -- many of them well-known -- relating theta functions, elliptic functions and classical modular forms.
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Continuing along the lines of the earlier article [ibid. 36, 35-44 (1994; Zbl 0841.11021)], the author derives a number of formula -- many of them well-known -- relating theta functions, elliptic functions and classical modular forms.
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Mathematical Journal of Okayama University, 1994
Starting with the classical theta function \(\theta(x,q)= \sum_{n\in \mathbb{Z}} x^n q^{n^2}\), \(q= e^{\pi i\tau}\), the author discusses, in turn, a number of well-known modular forms and modular functions. These include: Dedekind's \(\eta (\tau)\) (weight \({1\over 2}\) on \(\Gamma (1)\)), \(\theta_2 (\tau)\) (weight \({1\over 2}\) on \(\Gamma_0 (2)\
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Starting with the classical theta function \(\theta(x,q)= \sum_{n\in \mathbb{Z}} x^n q^{n^2}\), \(q= e^{\pi i\tau}\), the author discusses, in turn, a number of well-known modular forms and modular functions. These include: Dedekind's \(\eta (\tau)\) (weight \({1\over 2}\) on \(\Gamma (1)\)), \(\theta_2 (\tau)\) (weight \({1\over 2}\) on \(\Gamma_0 (2)\
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Commentarii mathematici Universitatis Sancti Pauli = Rikkyo Daigaku sugaku zasshi, 1994
Automorphic forms are by definition functions on a (real or adelic) Lie group that are invariant under a discrete subgroup and are eigenfunctions of a suitable Hecke algebra. It follows that these are smooth functions. In the paper under consideration the author states the definition of automorphic forms in a way general enough as to include ...
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Automorphic forms are by definition functions on a (real or adelic) Lie group that are invariant under a discrete subgroup and are eigenfunctions of a suitable Hecke algebra. It follows that these are smooth functions. In the paper under consideration the author states the definition of automorphic forms in a way general enough as to include ...
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A relation between Riemann theta functions and Jacobi theta functions.
2001The reviewer feels that it is best to quote from the paper literally to give an impression: ``Corollary 2: The relation between \(R_3\) and \(thata_3\) is \(R_3/theta3 =1\). Proof is to celar. Corollary 3: Proof is clear. '' From a mathematical viewpoint, the reviewer has problems e.g.
IŞIK, AHMET, KONYALIOĞLU, Alper Cihan
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Three-parameter mock theta functions
Journal of Mathematical Analysis and Applications, 2022Qing-Hu Hou, S -P Cui, Nancy Gu
exaly
1985
The expansions, in infinite series, of the functions ℘(u), ζ(u), and σ(u), which we have so far considered, are not best suited to numerical computation. It is of advantage therefore to introduce another function, denoted by θ(υ, τ), which has a rapidly convergent expansion in infinite series, and which is directly connected with the σ-function of ...
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The expansions, in infinite series, of the functions ℘(u), ζ(u), and σ(u), which we have so far considered, are not best suited to numerical computation. It is of advantage therefore to introduce another function, denoted by θ(υ, τ), which has a rapidly convergent expansion in infinite series, and which is directly connected with the σ-function of ...
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