On the extensions of some classical distributions

Abstract

Some properties of polynomials associated with strong distribution functions are given, including conditions for the polynomials to satisfy a three term recurrence relation. Strong distributions that are extensions to the four classical distributions are given as examples.

The strong c-Symmetric distribution

“…It is easily verified that the examples of strong symmetric distributions given in [4] are strong c-symmetric distributions (with c -ab in the first two examples and c = a in the other).…”

“…These polynomials are very much related to the Laurent polynomials considered for example in [2]. Here, first we present some results, a good part of which are given in [4], and the rest easily follow from these.…”

“…Sri Ranga [2] For any strong distribution, di//(t), the relations (i) Q n {z) is a monic polynomial of degree n, for n > 0, In the three examples of strong symmetric distributions given in [4], namely, the strong Tchebycheff, the strong Legendre and the strong Hermite distributions, we note that the coefficients a n and P n of the associated recurrence relations satisfy, in addition to the expected condition fi n = 0, n > 1, the condition that a 2n are constant for all n > 1. We show here that this is due to these distributions being members of a "special" class of strong symmetric distributions.…”

On the convergence of certain Gauss-type quadrature formulas for unbounded intervals

“…In order to compute the moments c k , let us introduce the auxiliary weight function studied by Ranga [20]: (4.5) and setc…”

Symmetric orthogonal polynomials and the associated orthogonal 𝐿-polynomials

“…In [9,10,11,12] some results can be found regarding the polynomials defined by (1.3) for weight functions satisfying a property different from (1.2). For some studies of polynomials given by a variation of (1.3), see [5,6,7].…”