The Monomiality Principle Applied to Extensions of Apostol-Type Hermite Polynomials

Abstract

In this research paper, we present a  class of polynomials referred to as Apostol-type Hermite-Bernoulli/Euler polynomials   $\mathcal{U}_\nu(x,y;\rho;\mu)$, which can be given by the following generating function
\begin{equation*} 
\displaystyle \frac{2-\mu+\frac{\mu}{2}\xi}{\rho e^{\xi}+(1-\mu)}e^{x \xi+y \xi^2} =\displaystyle\sum\limits_{\nu=0}^{\infty}
\mathcal{U}_\nu(x,y;\rho;\mu)\frac{\xi^\nu}{\nu!},
\end{equation*}
for some particular values of $\rho$ and $\mu$. Further, the summation formulae and determinant forms of these polynomials are derived. This novel family encompasses both the classical Appell-type polynomials and their noteworthy extensions. Our investigations heavily rely on generating function techniques, supported by illustrative examples to demonstrate the validity of our results. Furthermore, we introduce derivative and multiplicative operators, facilitating the definition of the  Apostol-type Hermite-Bernoulli/Euler polynomials as a quasi-monomial set.