Derivatives of a finite class of orthogonal polynomials defined on the positive real line related to F-distribution

Among the six classes of classical orthogonal polynomials, three of them are infinite, namely Jacobi, Hermite and Laguerre and the remaining three are finite and characterized by Masjed Jamei (2002) [5]. In this work, we consider derivatives of one such finite class of orthogonal polynomials that are orthogonal with respect to the weight function which is related to the probability density function of the F distribution. For this derivative class, besides orthogonality we find various other related properties such as the normal form and the self adjoint form. The corresponding Gaussian quadrature formulae are also given. Examples are provided to support the advantages of considering this derivative class of the finite class of orthogonal polynomials.

[1]  W. A. Al-Salam.  Characterization Theorems for Orthogonal Polynomials , 1990 .

[2]  J. K. Ord,et al.  Families of frequency distributions , 1973 .

[3]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[4]  Wolfgang Hahn,et al.  Über die Jacobischen Polynome und zwei verwandte Polynomklassen , 1935 .

[5]  F. Marcellán,et al.  Orthogonal polynomials on the unit circle and their derivatives , 1991 .

[6]  V. B. Uvarov,et al.  Special Functions of Mathematical Physics: A Unified Introduction with Applications , 1988 .

[7]  A. Stroud,et al.  Approximate Calculation of Integrals , 1962 .

[8]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[9]  W. Gautschi.  Orthogonal polynomials-Constructive theory and applications * , 1985 .

[10]  M. Masjed‐Jamei.  Three Finite Classes of Hypergeometric Orthogonal Polynomials and Their Application in Functions Approximation , 2002 .

[11]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[12]  Esmail Babolian,et al.  On numerical improvement of Gauss-Lobatto quadrature rules , 2005, Appl. Math. Comput..

[13]  Mohammad Masjed-Jamei,et al.  On numerical improvement of closed Newton-Cotes quadrature rules , 2005, Appl. Math. Comput..

[14]  Dirk Laurie,et al.  Calculation of Gauss-Kronrod quadrature rules , 1997, Math. Comput..

[15]  H. L. Krall,et al.  On derivatives of orthogonal polynomials. II , 1936 .

[16]  W. Gautschi.  Construction of Gauss-Christoffel quadrature formulas , 1968 .

[17]  Mohammad Masjed-Jamei,et al.  On numerical improvement of Gauss-Radau quadrature rules , 2005, Appl. Math. Comput..