### 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.

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