A family of sequences of binomial type

TL;DR

This paper explores polynomial sequences of binomial type linked to specific delta operators, revealing connections with Fuss numbers, Bessel-Carlitz polynomials, and inverse Gaussian distributions, and introduces related probability distributions.

Contribution

It derives new polynomial sequences of binomial type from delta operators and establishes their relations with Fuss numbers and certain probability distributions.

Findings

Identified polynomial sequences of binomial type for specific delta operators.

Connected Bessel-Carlitz polynomials with moments of inverse Gaussian distributions.

Discovered probability distributions for which Bessel polynomials are moment sequences.

Abstract

For delta operator $aD-bD^{p+1}$ we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case $D-\frac{1}{2}D^2$ we show that the corresponding Bessel-Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions $\nu_{t}$, $t>0$, for which $\left\{y_{n}(t)\right\}$, the Bessel polynomials at $t$, is the moment sequence.