A class of big (p,q)-Appell polynomials and their associated difference equations

Abstract

In the present paper, we introduce and investigate the big (p, q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p, q)-Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p, q)-Bernoulli polynomials and the big (p, q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q → 1−, the big (p, q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p, q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p, q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p, q)-result is fairly straightforward.

On a quaternionic sequence with Vietoris’ numbers

“…As the authors have mentioned, this sequence of rational numbers "is a sequence that can be considered on the crossroad of positivity of trigonometric sums, stable behavior of some classes of holomorphic functions and a set of Appell polynomials in several hypercomplex variables" [6, p. 77]. Regarding the class of Appell polynomials and their generalizations, one can find in the literature some research of these polynomials from several points of view, see for example [4,5,7] and [25][26][27]. Recalling some combinatorial properties from Leopold Vietoris (1891Vietoris ( -2002 [33], some interesting generalizations can be seen in [24].…”