In this contribution, we consider the sequence { Q n λ } n ≥ 0 of monic polynomials orthogonal with respect to the following inner product involving differences 〈 p , q 〉 λ = ∫ 0 ∞ p x q x d ψ ( a ) ( x ) + λ Δp ( c ) Δq ( c ) , $$\langle p,q\rangle_{\lambda }={\int}_{0}^{\infty }p\left( x\right) q\left( x\right) d\psi^{(a)}(x)+\lambda {\Delta} p(c){\Delta} q(c), $$
where λ ∈ ℝ + , Δ denotes the forward difference operator defined by Δf (x) = f (x + 1) − f (x), ψ(a) with a > 0 is the well-known Poisson distribution of probability theory d ψ ( a ) ( x ) = e − a a x x ! at x = 0 , 1 , 2 , … , $$d\psi^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at }x = 0,1,2,{\ldots} , $$
and c ∈ ℝ is such that ψ(a) has no points of increase in the interval (c,c + 1). We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second-order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of c such that c + 1 < 0, we obtain some results on the distribution of its zeros as decreasing functions of λ, when this parameter goes from zero to infinity.