Some properties of 2-orthogonal polynomials associated with inverse Gaussian distribution

Abstract

In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated polynomials. A particularly important property is that the associated polynomials satisfy a four-term recurrence relation. Then we state and calculate the famous creation and annihilation operators of the Heisenberg-Weyl Lie algebra that acting on IG polynomials P_n as the multiplicative operator X and the derivative one D on the monomials xⁿ. Finally, I used the Maple computer algebra system to program and Calculate the six first IG polynomials and their roots numerically.

Keywords:

• Natural exponential family
• variance function
• 2-orthogonal polynomials
• creation and annihilation operators
• roots of polynomials

AMS SUBJECT CLASSIFICATION:

• Primary 33E20
• Secondary 33C47
• 34L10
• 46T30