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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 nas the multiplicative operator  X  and the derivative one  D  on the monomials  x n. Finally, I used the Maple computer algebra system to program and Calculate the six first IG polynomials and their roots numerically.

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