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Let be a polynomial with integral coefficients. Shapiro showed that if the values of at infinitely many blocks of consecutive integers are of the form , where is a polynomial with integral coefficients, then for some polynomial . In this paper, we show that if the values of at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form where is an integer greater than 1, then for some polynomial .
Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying with a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, ApostolEuler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.
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