We prove that sums of the form S = p−1 ∑ x=0 ( g(x) p ) f(x) with f(X), g(X) ∈ Z[X] can be explicitly computed whenever f and g are subject to some certain conditions which are defined in the paper.

In this paper, a new stochastic operational matrix for the Legendre wavelets is presented and a general procedure for forming this matrix is given. A computational method based on this stochastic operational matrix is proposed for solving stochastic Itô-Voltera integral equations. Convergence and error analysis of the Legendre wavelets basis are investigated. To reveal the accuracy and efficien...

We study the following second-order super-half-linear impulsive differential equations with delay r t φγ x′ t ′ p t φγ x t − σ q t f x t − σ e t , t / τk, x t akx t , x′ t bkx′ t , t τk , where t ≥ t0 ∈ R, φ∗ u |u|∗−1u, σ is a nonnegative constant, {τk} denotes the impulsive moments sequence with τ1 < τ2 < · · · < τk < · · · , limk→∞τk ∞, and τk 1 − τk > σ. By some classical inequalities, Ricca...

dynamically adaptive numerical methods have been developed to find solutions for differential equations. thesubject of wavelet has attracted the interest of many researchers, especially, in finding efficient solutions fordifferential equations. wavelets have the ability to show functions at different levels of resolution. in this paper, a numerical method is proposed for solving the second pain...

We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman’s Legendre-Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1 − x) ∑∞ n=0 { n+k n } x analogous to that of the Eulerian numbers, where { n k } are Everitt, Littlejohn, and Wellman’s Legendre-Stirling numbers of the second kind. Finally we use a r...

Introduction/purpose: Certain integrals involving the generalized Mittag-Leffler function with different types of polynomials are established. Methods: The properties used in conjunction kinds such as Jacobi, Legendre, and Hermite order to evaluate their integrals. Results: Some integral formulae Legendre function, Bessel Maitland hypergeometric functions derived. Conclusions: results obtained ...

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multiorder fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is ve...

We derive a new deterministic algorithm for the computation of a sparse Legendre expansion f of degree N with M N nonzero terms from only 2M function resp. derivative values f (1), j = 0, . . . , 2M − 1 of this expansion. For this purpose we apply a special annihilating filter method that allows us to separate the computation of the indices of the active Legendre basis polynomials and the evalu...

We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown, despite the various results on the irregularity ...