相关论文: Elliptic Faulhaber polynomials and Lam\'e densitie…
We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure…
We study the generalized Lam\'e equation on an elliptic curve $E$ with multiple singularities. By restricting to the locus admitting solutions with quasi-periodic properties, we construct two curves: (i) The generalized Lam'e curve: with…
A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number--theoretical properties. A new class of…
Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lam\'e polynomials of arbitrary order. The models discussed are…
We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators…
We derive new matrix representation for higher order Daehee numbers and polynomials, the higher order lambda-Daehee numbers and polynomials and the twisted lambda-Daehee numbers and polynomials of order k. This helps us to obtain simple and…
For the oscillator-like systems, connected with the Laguerre, Legendre and Chebyshev polynomials coherent states of Glauber-Barut-Girardello type are defined. The suggested construction can be applied to each system of orthogonal…
Sum of powers 1^p+...+n^p, with n and p being natural numbers and n>=1, can be expressed as a polynomial function of n of degree p+1. Such representations are often called Faulhaber formulae. A simple recursive algorithm for computing…
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…
A new $q$-analogue of Appell polynomial sequences and their generalizations are introduced and their main characterizations are proved. As consequences new $q$-analogue of Bernoulli and Euler polynomials and numbers is introduced, their…
The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on…
A simple position probability density formulation is presented for the motion of a particle in a spherically symmetric potential. The approach provides an alternative to Newtonian methods for presentation in an elementary course, and…
In this paper, we give some recurrence formula and new and interesting identities for the poly-Bernoulli numbers and polynomials which are derived from umbral calculus.
We give criteria for finite dimensionality or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite dimensional scenario…
We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on…
The coefficients occurring in summation formulae of the Lubbock type are shown to be generalised Bernoulli polynomials which turn up in subdivision questions such as quantum field theory around a conical singularity and on spherical lunes.…
Quantisation on spaces with properties of curvature, multiple connectedness and non orientablility is obtained. The geodesic length spectrum for the Laplacian operator is extended to solve the Schroedinger operator. Homotopy fundamental…
Fix m >= 1 and let E be an elliptic curve over Q with complex multiplication. We formulate conjectures on the density of primes p (congruent to one modulo m) for which the pth Fourier coefficient of E is an mth power modulo p; often these…
Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying H\"ormander condition, the existence and uniqueness is proved in L_{p}-spaces of functions whose regularity is defined by a scalable,…
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…