The truncated exponential polynomials, the associated Hermite forms and applications

The truncated exponential polynomials, the associated Hermite forms and applications

We discuss the properties of the truncated exponential polynomials and develop the theory of new form of Hermite polynomials, which can be constructed using the truncated exponential as a generating function. We derive their explicit forms and comment on their usefulness in applications, with particular reference to the theory of flattened beams, used in optics.

Quadratic Hermite-Padé polynomials associated with the exponential function

The combinatorics of associated Hermite polynomials

We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization form...

Asymptotics for Quadratic Hermite-padé Polynomials Associated with the Exponential Function

The asymptotic behavior of quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I and pn, qn, rn ∈ P2n of type II associated with the exponential function are studied. In the introduction the background of the definition of Hermite-Padé polynomials is reviewed. The quadratic Hermite-Padé polynomials pn, qn, rn ∈ Pn of type I are defined by the relation pn(z) + qn(z)e z + rn(z)e 2z = O(z) ...

Asymptotic Distributions of Zeros of Quadratic Hermite-pad E Polynomials Associated with the Exponential Function