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
the 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 ${q \rightarrow 1{-}},$
the big $(p,q)$-Appell polynomials reduce to
the usual Appell polynomials. Therefore, the
recurrence relation and difference equation
obtained for the big $(p,q)$-Appell
polynomials coincide with the recurrence
relation and differential equation
satisfied by the usual Appell polynomials.