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.

A probabilistic interpretation of a modified Gegenbauer polynomial is furnished by its expression in terms of a combinatorial probability defined on a compound urn model. Also, a combinatorial interpretation of its coefficients is provided. In particular, probabilistic interpretations of a modified Chebyshev polynomial of the second kind and a modified Legendre polynomial together with combinat...

We provide a direct computation of the $F$-pure threshold degree four homogeneous polynomial in two variables and, more generally, certain polynomials with distinct roots. The depends on whether cross ratio roots satisfies specific M\"{o}bius transformation Legendre polynomial. then make connection between long lasting open question, involving relationship and log canonical threshold, over $\ma...

Applying the theorems of Mobius inverse and Dirichlet inverse, a general algorithm to obtain biorthogonal functions based on generalized Fourier series analysis is introduced. In the algorithm, the orthogonal function can be not only Fourier or Legendre series, but also can be any one of all orthogonal function systems. These kinds of biorthogonal function sets are used as scramble signals to c...

A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of ...

A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre-Gauss-Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of ...

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.

We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients of the powers of (x − 1)/2 in the n-th Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation that contain a point in a given open generalized orthant. We thus provide a geomet...