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...

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...

The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of O(N3) for cut-off frequency N . In this paper, we propose a fast a...

We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with N − 5 unknowns. An efficient and accurate direct solution for algorithms based on the LegendreGalerkin approximations...