Congruences relating rational values of Bernoulli and Euler polynomials (Q2721673)

For each natural number \(n\), the \(n\)-th Bernoulli polynomial \(B _n (t)\) and the \(n\)-th Euler polynomial \(E _n (t)\) are defined to be the \(n\)-th coefficients in the Taylor expansions of the generating functions NEWLINE\[NEWLINExe ^{tx}/(e ^x - 1) = \sum _{m=0} ^{\infty } B _m (t).x ^m/m!, \quad|x |< 2\pi,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE2e ^{tx}/(e ^x + 1) = \sum _{m=0} ^{\infty } E _m (t).x ^m/m!, \quad |x |< \pi,NEWLINE\]NEWLINE respectively. The \(n\)-th Bernoulli number \(B _n\) and the \(n\)-th Euler number \(E _n\) are defined by the formulae \(B _n = B _n (0)\) and \(E _n = 2 ^nE _n(1/2)\), \(\forall n \in {\mathbb N}\). It is easily verified that these polynomials are related as follows: NEWLINENEWLINENEWLINE(i) \(B _n (t + 1) - B _n (t) = nt ^{n-1}\); (ii) \(B _n (1 - t) = (-1) ^nB _n (t)\);NEWLINENEWLINENEWLINE(iii) \(E _n (t + 1) + E _n (t) = 2t ^n\), and (iv) \(E _n (1 - t) = (-1) ^nE _n (t)\). NEWLINENEWLINENEWLINEThis enables one to express the coefficients of \(B _n (t)\) and \(E _n (t)\) in terms of the Bernoulli numbers \(B _0,\dots, B _n\) (and thereby to show that they are rational numbers). NEWLINENEWLINENEWLINEThe paper under review deals with the study of the divisibility properties of \(B _n (t)\) and \(E _n (t)\) at rational values of \(t\). The author's starting point is the divisibility properties of \(B _n (1/6)\), \(B _n (1/4)\), \(B _n (1/3)\), \(E _n (1/6)\) and \(E(1/3)\) established by \textit{N. Nörlund} [see Vorlesungen über Differenzenrechnung. Berlin, Springer (1924; JFM 50.0315.02)]. The main results of the paper prove the validity of the following congruences: NEWLINENEWLINENEWLINE(i) \(2B _n(2r/s) - nE _{n-1} (2r/s) \equiv 2 ^{n+1}B _n\) (mod \(2 ^{n+1})\), provided that \(n \in {\mathbb N}\), and \((r, s)\) is a pair of integers such that \(\text{gcd}(2r, s) = 1\); NEWLINENEWLINENEWLINE(ii) For \(n\) an even positive integer, NEWLINE\[NEWLINE\begin{aligned} n5 ^nE _{n-1} (1/5) &\equiv -(2 ^n(5 ^n - 5) + 5 ^n(2 ^{n+1} + 2))B _n\pmod {2 ^{n+1} + 2};\\ n5 ^nE _{n-1} (2/5) &\equiv -(5 _n - 5 + 5 ^n(2 ^{n+1} + 2))B _n\pmod {2 ^{n+1} + 2};\\ (2 ^n + 1)10 ^nB _n(1/10) &\equiv -(5 ^n(2 ^{2n} + 2 ^n - 2) + 2 ^{n-1}(5 ^n - 5))B _n\pmod {5n(2 ^{n-1} + 1)};\\ (2 ^n + 1)10 ^nB _n(3/10) &\equiv (5 ^n(2 ^{2n} + 2 ^n - 2) + (2 ^{2n-1} - 1)(5 ^n - 5))B _n\pmod {5n(2 ^{n-1} + 1)}. \end{aligned}NEWLINE\]NEWLINE The author presents several interesting consequences to these results. For example, he shows that if \(p\) is a prime number congruent to \(11\) or \(19\) modulo \(40\), then \(p\) divides the numerators of \(B _{(p+1)/2} (1/10)\) and \(B _{(p+1)/2} (3/10)\).