A class of big \((p, q)\)-Appell polynomials and their associated difference equations. (English) Zbl 1499.11128
Summary: In the present paper, we introduce and investigate the big \((p,q)\)-Appell polynomials. We prove an equivalence 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 bigq-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 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.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
Keywords:
quantum or \(q\)-calculus; post quantum or \((p, q)\)-calculus; big \((p, q)\)-Bernoulli polynomials; big \((p, q)\)-Euler polynomials; big \(q\)-Bernoulli polynomials; big \(q\)-Euler polynomials; recurrence relations; difference equations; Cauchy productReferences:
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