Related papers: Computing power series expansions of modular forms
This work presents an algorithm for numerically computing Maass forms and their eigenvalues for Fuchsian groups of infinite covolume. By Patterson-Sullivan theory, this has the added benefit of computing Hausdorff dimensions of the limit…
We present a new combinatorial approach to the computation of the (real) Fourier expansions of $\cos^n(t)$ and $\sin^n(t)$, where $n\geq 1$ is an integer. As an application, we compute the Fourier expansions of $f(t)=\frac{1}{a-\cos t}$ and…
A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided…
In this paper we study the Kummer extensions of the power series field $K=k((X_1,...,X_n)$, where $k$ is an algebraically closed field of arbitrary characteristic.
A new series expansion for the function (arctan(x)/x)^n is developed and some properties of the expansion coefficients are obtained.
In this article we study modules over wild canonical algebras which correspond to extension bundles [9] over weighted projective lines. We prove that all modules attached to extension bundles can be established by matrices with coefficients…
Let $U_q$ be the quantum group corresponding to a complex simple Lie algebra $\mathfrak g$ with root system $R$. Assume the quantum parameter $q\in \C$ is a root of unity. In this paper we study the extensions between simple modules in the…
A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number…
We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier…
In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
The problem of computing modulo prime-power sums is investigated in distributed source coding as well as computation over Multiple-Access Channel (MAC). We build upon group codes and present a new class of codes called Quasi Group Codes…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
Many authors have investigated the congruence relations amongst the coefficients of power series expansions of modular forms $f$ in modular functions $t$. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and…
We use the techniques of group cohomology to give explicit computations of the local fundamental class. As an application, we discuss how to compute the Tate canonical class for the extension $\mathbb Q(\zeta_{p^\nu})/\mathbb Q$, where…
We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in…
It is shown that detailed and accurate information about the mass spectrum of the massive Schwinger model can be obtained using the technique of strong-coupling series expansions. Extended strong-coupling series for the energy eigenvalues…
In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums of the form $\sum_{k=1}^{\lfloor x\rfloor}k^{m}$ and $\sum_{k=1}^{n}k^{m}$, where…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
This paper describes an algorithm for determining radii of convergence of power expansions for algebraic functions and the testing done to check it. Since the current methods for computing these series are iterative, standard methods for…