The spectral decomposition of $$|\theta |^2$$

Let $$\theta $$ θ be an elementary theta function, such as the classical Jacobi theta function. We establish a spectral decomposition and surprisingly strong asymptotic formulas for $$\langle |\theta |^2, \varphi \rangle $$ ⟨ | θ | 2 , φ ⟩ as $$\varphi $$ φ traverses a sequence of Hecke-translates of a nice enough fixed function. The subtlety is that typically $$|\theta |^2 \notin L^2$$ | θ | 2 ∉ L 2 . Applications to the subconvexity, quantum variance and 4-norm problems We determine all pairs $$(A_{f,g},A_{g,h})$$ ( A f , g , A g , h ) of generalized weighted quasi-arithmetic means being square iterative roots of $$(A_{F,G},A_{G,H})$$ ( A F , G , A G , H ) , that is, the equation $$( A_{f,g},A_{g,h}) \circ ( A_{f,g},A_{g,h}) =(A_{F,G},A_{G,H}),$$ ( A f , g , A g , h ) ∘ ( A f , g , A g , h ) = ( A F , G , A G , H ) , is solved under three times differentiability of the functions f, g, h, F, G, H. As an application, some special cases are presented. are indicated.

Log-concavity results for a biparametric and an elliptic extension of the q-binomial coefficients

We establish discrete and continuous log-concavity results for a biparametric extension of the [Formula: see text]-numbers and of the [Formula: see text]-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turán’s inequality.

Generating functions of planar polygons from homological mirror symmetry of elliptic curves

We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.

Algebraic results for certain values of the Jacobi theta-constant $\theta_3(\tau)$

In its most elaborate form, the Jacobi theta function is defined for two complex variables $z$ and τ by $\theta (z|\tau ) =\sum _{\nu =-\infty }^{\infty } e^{\pi i\nu ^2\tau + 2\pi i\nu z}$, which converges for all complex number $z$, and τ in the upper half-plane. The special case \[ \theta _3(\tau ):=\theta (0|\tau )= 1+2\sum _{\nu =1}^{\infty } e^{\pi i\nu ^2 \tau } \] is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function $\theta (z|\tau )$. In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant $\theta _3(\tau )$. For example, the three values $\theta _3(\tau )$, $\theta _3(n\tau )$, and $D\theta _3(\tau )$ are algebraically independent over $\mathbb{Q} $ for any τ such that $q=e^{\pi i\tau }$ is an algebraic number, where $n\geq 2$ is an integer and $D:=(\pi i)^{-1}{d}/{d\tau }$ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values $\theta _3(\tau )$ and $\theta _3(2^m\tau )$ for $m\geq 1$. As an application of our main theorem, the algebraic dependence over $\mathbb{Q} $ of the three values $\theta _3(\ell \tau )$, $\theta _3(m\tau )$, and $\theta _3(n\tau )$ for integers $\ell ,m,n\geq 1$ is also presented.

Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

BASES FOR Sk(Γ1(4)) AND FORMULAS FOR EVEN POWERS OF THE JACOBI THETA FUNCTION

We find a basis for the space Sk(Γ1(4)) of cusp forms of weight k for the congruence subgroup Γ1(4) in terms of Eisenstein series. As an application, we obtain formulas for r2k(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares.