Classical orthogonal polynomials with weight function ((ax + b)2 + (cx + d)2)−p exp(q Arctg((ax + b)/(cx + d))), x ∈ (−∞, ∞) and a generalization of T and F distributions

Abstract

From the main equation One can extract six ‘infinite’ and ‘finite’ sequences of classical orthogonal polynomials. As we know, three of them, i.e. Jacobi, Laguerre and Hermite polynomials are ‘infinitely’ orthogonal for every n ∈ Z +. In this work, we introduce one of the three other cases, which is ‘finitely’ orthogonal, and denote its general applications in functions approximation and numerical integration. This polynomial class is orthogonal with respect to the weight function ((ax + b)2 + (cx + d)2)−p exp (q Arctg ((ax + b)/(cx + d))) on (−∞, ∞) and has a linear property that can be applied to estimate the value of polynomials at some specific points. The weight function of this class can also be considered as an important special distribution function so that by having its explicit criterion, we can generalize the T-sampling distribution and prove that it tends to the Normal distribution like T-student distribution. Moreover, the same method can be applied to generalize the F-sampling distribution and to prove that it tends to the Gamma distribution function when one of its parameters tends to infinity.