Diophantine equations with Bernoulli polynomials

Abstract

The Bernoulli polynomials B n (x) are defined by the generating series The Bernoulli polynomials B n (x) are related to the sums of nth powers of natural numbers as follows. For any n ≥ 1, the sum 1 n + 2 n + · · · + k n is a polynomial function S n (k) of k andn + 1 .In this paper, for nonzero rational numbers a, b and rational polynomials C(y), we study the Diophantine equation aB m (x) = bB n (y) + C(y)

Polynomial values of (alternating) power sums

“…Recently, Rakaczki and Kreso [22] proved an analogous result for equations (14) and (15). For further related results we refer to the papers of Kulkarny and Sury [17,18], and of Bennett [4].…”

Complete decomposition of Dickson-type polynomials and related Diophantine equations

“…Since by (24) the coefficient [x 2k−1 ] of f 2k (x) equals zero, one step in Proposition 12 gives α 1 = 0,q = x 2 and thus (12). It remains to show that h k is indecomposable whenever k = 4.…”

“…In recent years, much interest has been focused on using the criterion of Theorem 1 to Diophantine equations of the form p m (x) = p n (y) and p m (x) = g(y), where {p k } k 0 denotes some specific polynomial family and g(x) is an arbitrary polynomial over Q. The interested reader may consult [2,11,17,18,26] for equations with binomial coefficient polynomials, [12][13][14] for equations with Bernoulli polynomials, [2,19] for power-sum polynomials, [14] for truncated Taylor polynomials of the exponential function and [1,9,10,25,27,28] for polynomials in three-term recurrences. As a principle, the difficulty consists in proving a uniform indecomposability theorem for {p k }.…”

Diophantine equations with Appell sequences

“…Recently, Rakaczki and Kreso [11] proved an analogous result for the case when P n (x) = (E n (0)±E n (x))/2 (which is not an Appell polynomial anymore). For further related results we refer to [7,8].…”