Hermite based poly-bernoulli polynomials with a g-parameter

We introduce the Hermite based poly-Bernoulli polynomials with a q parameter and give some of their basic properties including not only addition property, but also derivative properties and integral representations. We also define the Hermite based A-Stirling polynomials of the second kind, and then provide some relations. Moreover, we derive several correlations and identities including the Hermite-Kampé de Fériet (or Gould-Hopper) family of polynomials, the Hermite based poly-Bernoulli polynomials with a q parameter and the Hermite based A-Stirling polynomials of the second kind