Related papers: Chebyshev Series Representation of Feigenbaum's Pe…

We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation…

For a function that is analytic on and around an interval, Chebyshev polynomial interpolation provides spectral convergence. However, if the function has a singularity close to the interval, the rate of convergence is near one. In these…

We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact…

Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \begin{align*} g_{\lambda}^{*}(f)(x) =\bigg(\iint_{\mathbb R^{n+1}_{+}}…

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite…

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq…

The aim of the present work is to introduce a method based on Chebyshev polynomials for the numerical solution of a system of Cauchy type singular integral equations of the first kind on a finite segment. Moreover, an estimation error is…

To each weakly holomorphic modular function $f\not \equiv 0$ for $\mathrm{SL}(2,\mathbb{Z})$, which is non-negative on the geodesic arc $\{e^{it} : \pi/3\leq t\leq 2\pi/3\}$, we attach a $\mathrm{GL}(2,\mathbb{Z})$-invariant map…

Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called…

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for…

In this paper, we study the optimal general convergence rates for quadratures derived from Chebyshev points. By building on the aliasing errors on integration of Chebyshev polynomials, together with the asymptotic formulae on the…

It is needed to solve generalized eigenvalue problems (GEP) in many applications, such as the numerical simulation of vibration analysis, quantum mechanics, electronic structure, etc. The subspace iteration is a kind of widely used…

A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional)…

Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha…

Two representations of a reductive group G are spectrally equivalent if the same irreducible representations appear in both of them. The semigroup of finite dimensional representations of G with tensor product and up to spectral equivalence…

In this paper, we investigate the revivals of the one-dimensional periodic Schr\"odinger equation with a piecewise $C^2$ potential function. As has been observed through numerical simulations of the equation with various initial data and…

The sequence of period 6 starting with 1, 1, 0, -1, -1, 0 appears in many different disguises in mathematics. Various q-versions of this sequence are found, and their relations with Euler's pentagonal numbers theorem and Chebyshev…

A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent coefficients, which may depend on $x$. We assume that the spatial coordinate $x$…

Analytic expressions for the Clebsch-Gordan (CG) coefficients of the SO(5) group that involve the 14-dimensional representation can be found in an old paper of M. K. F. Wong. A careful analysis yields that roughly 30% of the coefficients…

Pointwise H\"older exponents describe the degree of regularity of a function near a point. For a function $f:\mathbb{R}\to\mathbb{R}$, a number $\alpha>0$ and a point $t_0\in\mathbb{R}$, write $f\in C^\alpha(t_0)$ if there is a constant $C$…