Related papers: Integrals of Legendre polynomials and approximatio…
In this paper, we provide a new and sharper bound for the Legendre coefficients of differentiable functions and then derive a new error bound of the truncated Legendre series in the uniform norm. The key idea of proof relies on integration…
We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and…
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.
Associated Legendre functions of fractional degree appear in the solution of boundary value problems in wedges or in toroidal geometries, and elsewhere in applied mathematics. In the classical case when the degree is half an odd integer,…
When treating problems of vector diffraction in electromagnetic theory, the evaluation of the integral involving Bessel and associated Legendre functions is necessary. Here we present the analytical result for this integral that will make…
In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…
We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic.
The article is devoted to comparative analysis of the efficiency of application of Legendre polynomials and trigonometric functions to the numerical integration of Ito stochastic differential equations in the framework of the method of…
We explore integrals of products of Legendre polynomials with a logarithmic weight function. More precisely, for Legendre polynomials $P_m$ and $P_n$ of orders $m$ and $n$, respectively, we provide simple derivations of the integrals $$\int…
It is shown that if a function defined on the segment [-1,1] has sufficiently good approximation by partial sums of the Legendre polynomial expansion, then, given the function's Fourier coefficients $c_n$ for some subset of $n\in[n_1,n_2]$,…
New constructions in the theory of fields for multiple integrals are designed. Generalizations of the Legendre - Weyl - Caratheodory transforms and corresponding invariant integrals are introduced and explored. Connection and curvature of…
In this paper, we consider the degenerate Frobenius-Euler polynomials and investigate some identities of these polynomials.
Algorithms for jointly obtaining projection estimates of the density and distribution function of a random variable using Legendre polynomials are proposed. For these algorithms, a problem of the conditional optimization is solved. Such…
In this paper, we investigate some identities of Laguerre polynomials involving Bernoulli and Euler polynomials which are derived from umbral calculus.
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear…
The Floquet eigenvalue problem and a generalized form of the Wangerin eigenvalue problem for Lam\'e's differential equation are discussed. Results include comparison theorems for eigenvalues and analytic continuation, zeros and limiting…
By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…
In this paper, we study nonlinear differential equations arising from Eulerian polynomials and their applications. From our study of nonlinear differential equations, we derive some new and explicit identities involving Eulerian and…