In this correspondence, it is shown that Legendre sequences of period p can be explicitly represented using the trace function defined on the finite field with 2n elements, whenever p = 2n 1 is prime for some R 2 3.

Let D = 1 or D be a fundamental discriminant [1]. The Kronecker-Jacobi-Legendre symbol (D/n) is a completely multiplicative function on the positive integers: µ D n ¶ = ⎧ ⎪ ⎨

We prove identities involving sums of Legendre and Jacobi polynomials. The identities are related to Green’s functions for powers of the invariant Laplacian and to the Minakshisundaram-Pleijel zeta function.

Present work of this paper deals with the unification classical polynomials in which we have defined a generalized polynomial set analogous to that associated Legendre P (x) m n by taking 5the use Operator. Also derived explicit form, Operational Formulae generating functions for function.

Corresponding Author: Simon Liao The University of Winnipeg, Winnipeg, Manitoba, Canada, R3B 2E9, Canada Email: [email protected] Abstract: In this research, a numerical integration method is proposed to improve the computational accuracy of Legendre moments. To clarify the improved computation scheme, image reconstructions from higher order of Legendre moments, up to 240, are conducted. Wi...

Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to constraint systems, the procedure of finding a Hamiltonian for a singular Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairaut transf...

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to systems with constraints, the procedure of finding a Hamiltonian for a degenerate Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairau...

In the recent paper [3] by Asai et al., the growth order of holomorphic functions on a nuclear space has been considered. For this purpose, certain classes of growth functions u are introduced and many properties of Legendre transform of such functions are investigated. In [4], applying Legendre transform of u under the conditions (U0), (U2) and (U3) (see §2), the Gel’fand triple [E ]u ⊂ (L) ⊂ ...