Some generalized Jacobi polynomials

Some Properties of Jacobi Polynomials

A main motivation for this paper is the search for the sufficient condition of the primality of an integer n in order that the congruence 1n−1 + 2n−1 + 3n−1 + · · · + (n− 1)n−1 ≡ −1 (mod n) holds. Some properties of Jacobi polynomials were investigated using certain Kummer results. Certain properties of Bernoulli polynomials as well as the Staudt–Clausen theorem for prime factors were also used...

A Family of Generalized Jacobi Polynomials

The family of orthogonal polynomials corresponding to a generalized Jacobi weight function was considered by Wheeler and Gautschi who derived recurrence relations, both for the related Chebyshev moments and for the associated orthogonal polynomials. We obtain an explicit representation of these polynomials, from which the recurrence relation can be derived.

Generalized Jacobi polynomials/functions and their applications

We introduce a family of generalized Jacobi polynomials/functions with indexes α,β ∈ R which are mutually orthogonal with respect to the corresponding Jacobi weights and which inherit selected important properties of the classical Jacobi polynomials. We establish their basic approximation properties in suitably weighted Sobolev spaces. As an example of their applications, we show that the gener...

An Integral Formula for Generalized Gegenbauer Polynomials and Jacobi Polynomials

Some Generalized Fibonacci Polynomials

We introduce polynomial generalizations of the r-Fibonacci, r-Gibonacci, and rLucas sequences which arise in connection with two statistics defined, respectively, on linear, phased, and circular r-mino arrangements.