Exponential Transforms and Appell Polynomials

Exponential Transforms and Appell Polynomials.

Appell Transformation and Canonical Transforms

The interpretation of the optical Appell transformation, as previously elaborated in relation to the free-space paraxial propagation under both a rectangular and a circular cylindrical symmetry, is reviewed. Then, the caloric Appell transformation, well known in the theory of heat equation, is shown to be amenable for a similar interpretation involving the Laplace transform rather than the Four...

Appell Polynomials and Their Relatives

This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials when polynomials in noncommuting variables are considered. They also fit well into the framework of free probability. For the free Appell polyn...

On Multiple Appell Polynomials

In this paper, we first define the multiple Appell polynomials and find several equivalent conditions for this class of polynomials. Then we give a characterization theorem that if multiple Appell polynomials are also multiple orthogonal, then they are the multiple Hermite polynomials.

Combinatorial Polynomials as Moments, Hankel Transforms, and Exponential Riordan Arrays

In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.