We discuss the large N limit of the supersymmetric CP N models as an illustration of Cecotti and Vafa's tt * formalism. In this limit the 'tt * equation' becomes the long wavelength limit of the 2D Toda lattice, an equation first studied in the context of self-dual gravity. We show how simple finite temperature and large N techniques determine the relevant solution, and verify analytically that...

Let E d denote the set of coe cient vectors (a1, . . . , ad) 2 Rd of contractive polynomials xd + a1x 1 + · · ·+ ad 2 R[x] that have exactly s pairs of complex conjugate roots and let v d = d(E (s) d ) be its (d-dimensional) Lebesgue measure. We settle the instance s = 1 of a conjecture by Akiyama and Pethő, stating that the ratio v d /v (0) d is an integer for all d 2s. Moreover we establish t...

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 ...

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining second-order backward difference in and Galerkin spectral space with Legendre polynomials. Through detailed analysis, we demonstrate that scheme is unconditionally stable. The proved to have min{2−α,2−β}-order convergence accuracy smooth solutions, where α,β are two exp...

We show how Turán’s inequality Pn(x) −Pn−1(x)Pn+1(x) ≥ 0 for Legendre polynomials and related inequalities can be proven by means of a computer procedure. The use of this procedure simplifies the daily work with inequalities. For instance, we have found the stronger inequality |x|Pn(x) 2 − Pn−1(x)Pn+1(x) ≥ 0, −1 ≤ x ≤ 1, effortlessly with the aid of our method.

This article is an attempt to understand the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations of Heisenberg algebras. The resulting framework that we describe is a generalization of the classical Boson-Fermion correspondence, from ...

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuou...

We show that with small modifications in diverse expressions for the shifted Legendre polynomials, we can obtain the corresponding relations for Daubechies polynomials.

In this paper, a numerical efficient method is proposed for the solution of time fractional mobile/immobile equation. The fractional derivative of equation is described in the Caputo sense. The proposed method is based on a finite difference scheme in time and Legendre spectral method in space. In this approach the time fractional derivative of mentioned equation is approximated by a scheme of ord...

Abstract. We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead t...