Difference equations for a class of twice-iterated $\Delta _{h}$-Appell sequences of polynomials

Abstract

In the present paper, we introduce a family of the twice-iterated $\Delta _{h}$-Appell sequences of polynomials based upon the discrete Appell convolution of the $\Delta _{h}$-Appell sequence of polynomials $Q_{n}(x)$. For these twice-iterated $\Delta _{h}$-Appell polynomials, we prove an equivalence theorem and derive several determinantal properties in terms of the $\Delta _{h}$-Appell polynomial sequence $Q_{n}(x)$. We also find the recurrence relation, the shift operators and the difference equation satisfied by the twice-iterated $\Delta _{h}$-Appell polynomial sequences. By appropriately specializing our results, we obtain the corresponding properties for the sequences of the twice-iterated Bernoulli polynomials of the second kind, the twice-iterated Boole polynomials, the twice-iterated Boole–Bernoulli polynomials of the second kind, the twice-iterated Poisson–Charlier–Bernoulli polynomials of the second kind and the twice-iterated Poisson–Charlier–Boole polynomials.