Related papers: Singular values of some modular functions
This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a…
The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in…
A study of products of factorials modulo p and complete residues.
Real or complex polynomial mappings between affines spaces admitting a Lipschitz-trivial value are completely characterized.
We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method…
We present some new results in theory of classical theta-functions of Jacobi and sigma-functions of Weierstrass: ordinary differential equations (dynamical systems) and series expansions. The paper is basically organized as a stream of new…
When extending the Ehrhart lattice point enumerator $L_P(t)$ to allow real dilation parameters $t$, we lose the invariance under integer translations that exists when $t$ is restricted to be an integer. This paper studies this phenomenon;…
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
Let $g$ be a principal modulus with rational Fourier coefficients for a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ between $\Gamma(N)$ or $\Gamma_0(N)^\dag$ for a positive integer $N$. Let $K$ be an imaginary quadratic field. We give…
We discuss the theory of generalized Weierstrass $\sigma$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal")…
We give explicit formulas for Whittaker functions for the class one principal series representations of the orthogonal groups $ SO_{2n+1}(\R) $ of odd degree. Our formulas are similar to the recursive formulas for Whittaker functions on…
We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that…
We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic zero. This extends prior…
Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions,…
Solutions of Schr\"oder-Poincar\'e's polynomial equations $f(az)=P(f(z))$ usually do not admit a simple closed-form representation in terms of known standard functions. We show that there is a one-to-one correspondence between zeros of $f$…
We prove multiplicative relations between certain Fourier coefficients of degree 2 Siegel eigenforms. These relations are analogous to those for elliptic eigenforms. We also provide two sets of formulas for the eigenvalues of degree 2…
The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. In this paper, the properties of this algebra are studied from the point of view of motivic periods.
We have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used…
We introduce and study the approximation properties of $g$-polynomials, defined as linear combinations of iterated Stieltjes integrals of a constant function. Focusing on the case where the derivator $g$ has finitely many discontinuities,…
Graphical functions are single-valued complex functions which arise from Feynman amplitudes. We study their properties and use their connection to multiple polylogarithms to calculate Feynman periods. For the zig-zag and two more families…