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Identities and congruences involving the geometric polynomials. (English) Zbl 1438.05015

Summary: In this paper, we investigate the umbral representation of the geometric polynomials \(\mathfrak{w}_{x}^{n}:=w_{n} (x) \) to derive some properties involving these polynomials. Furthermore, for any prime number \(p\) and any polynomial \(f\) with integer coefficients, we show \( (f (\mathfrak{w}_{x}))^{p}\equiv f(\mathfrak{w}_{x}) \pmod p\) and we give other curious congruences.

MSC:

05A18 Partitions of sets
05A40 Umbral calculus
11A07 Congruences; primitive roots; residue systems
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