Summary: Orthogonality of free Sheffer systems
Michael Anshelevich
August 5, 2004
µ = infinitely divisible probability measure, {µt : t [0, )}
the corresponding convolution semigroup,
µt µs = µt+s.
Assume all their moments are finite, µ centered, variance
1.
k(z) = log
R
exz dµ(x) = log

n=0
1
n!
mn(µ)zn
is the cumulant-generating function, k(z) = 1
2z2 + . . ..
Let U = z+. . . be an analytic function / formal power se-
ries. The Sheffer polynomials (Sheffer 1937) are defined

Source: Anshelevich, Michael - Department of Mathematics, Texas A&M University

Collections: Mathematics