The Hermite polynomials form the basis of a Hilbert space and may be used to get an expansion of the probability density function. Usually, this series is called Gram-Charlier. For practical purposes, only the first few terms of this expansion are taken into consideration. The resulting truncated series may be viewed as the normal probability density function multiplied by a polynomial that acc...

The probability density function (PDF) of a random variable associated with the solution partial differential equation (PDE) parameters is approximated using truncated series expansion. PDE solved two stochastic finite element methods, Monte Carlo sampling and Galerkin method global polynomials. functional PDE, such as average over physical domain. are obtained considering number terms in Gram-...

Abstract By means of a truncation condition on the parameters, elliptic Ruijsenaars difference operators are restricted onto finite lattice points encoded by bounded partitions. A corresponding orthogonal basis joint eigenfunctions is constructed in terms polynomials spectrum. In trigonometric limit, this recovers diagonalization truncated Macdonald finite-dimensional polynomials.

This is a preliminary version of a Chapter on Algebraic Algorithms in the upcoming Computing Handbook Set Computer Science (Volume I), CRCPress/Taylor and Francis Group. Algebraic algorithms deal with numbers, vectors, matrices, polynomials, formal power series, exponential and differential polynomials, rational functions, algebraic sets, curves and surfaces. In this vast area, manipulation wit...

The introduced method in this study consists of reducing a system of infinite boundary integro-differential equations (IBI-DE) into a system of al- gebraic equations, by expanding the unknown functions, as a series in terms of Laguerre polynomials with unknown coefficients. Properties of these polynomials and operational matrix of integration are rst presented. Finally, two examples illustra...

Let $$n\ge 3$$ be an integer and p a prime with $$p\equiv 1\pmod {n}$$ . In this paper, we show that $$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} \frac{n-1}{n}&{}\frac{n-1}{n}&{}\ldots &{}\frac{n-1}{n}\\ &{}1&{}\ldots &{}1\end{array}\bigg | \, 1\bigg ]_{p-1}\equiv -\Gamma _p\bigg (\frac{1}{n}\bigg )^n\pmod {p^3}, \end{aligned}$$ where the truncated hypergeometric series x_1&{}x_2&{}\...

A method is derived to obtain an expansion formula for the magnetic field $\boldsymbol{B}(\boldsymbol{r})$ generated by a closed planar wire carrying steady electric current. The parametric equation of loop $\mathcal{R}(\phi)=R+Hf(\phi)$, with $R$ radius circle, $H \in [0,R)$ radial deformation amplitude, and $f(\phi)\in[-1,1]$ periodic function. based on replacement $1/|\boldsymbol{r} - \bolds...

Littlewood polynomials are polynomials with each of their coefficients in the set {−1, 1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1. We show that the arithmetic means of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1 are asymptotically e−γ/2 √ n and Γ(1+ p/2)1...

The theory of derivatives and integrals of fractional in fractional calculus have found enormousapplications in mathematics, physics and engineering so for that reason we need an efficient and accurate computational method for the solution of fractional differential equations. This paper presents a numerical method for solving a class of linear and nonlinear multi-order fractional differential ...

The truncated Fourier series exhibits oscillation that does not disappear as the number of terms in the truncation is increased. This paper introduces 2-D fractional Fourier series (FrFS) according to the 1-D fractional Fourier series, and finds such a Gibbs oscillation also occurs in the partial sums of FrFS for bivariate functions at a jump discontinuity. In this study, the 2-D inverse polyno...