Congruences and recurrences for Bernoulli numbers of higher order. (English) Zbl 0820.11009

First defined and studied by N. Nörlund in the 1920s, the Bernoulli polynomials of order \(k\), for any integer \(k\), may be defined by \[ {{x^ k e^{xz}} \over {(e^ x- 1)^ k}}= \sum_{n=0}^ \infty B_ n^{(k)} (z) {{x^ n} \over {n!}}. \] In particular, \(B_ n^{(k)} (0)= B_ n^{(k)}\) is the Bernoulli number of order \(k\), and \(B_ n^{(1)}= B_ n\) is the ordinary Bernoulli number. Later, these were the subject of many papers by L. Carlitz and others. For example, Carlitz proved \(B_ p^{(p)}=- {1\over 2} p^ 2 (p- 1)!\pmod {p^ 5}\), for primes \(p>3\), and this was later extended by F. R. Olson.
In an earlier paper, the author examined the number \(B_ n^{(n)}\) and proved that, for \(p\) prime, \(p>3\), \(r\) odd, and \(p+1\geq r\geq 5\), \[ B_ p^{(p)} \equiv- \sum_{j=1}^{r-4} {1\over {j+1}} s(p,j) p^{j+1} \pmod {p^ r}, \] where \(s(p,j)\) is the Stirling number of the first kind. The purpose of the present paper is to examine the divisibility properties of \(B_ n^{(k)}\) for arbitrary \(n\) and \(k\). A summary of the main results follows:
(1) Bernoulli polynomials have the property \(B_{n+k}^{(n)} \bigl(z+ {\textstyle {1\over 2}}n \bigr)= (-1)^{n+k} B^{(n)}_{n+ k} \bigl(- z+ {\textstyle {1\over 2}}n \bigr)\).
(2) For \(p>5\), \[ \begin{aligned} B_{p+2}^{ (p+1)} &\equiv- {\textstyle {1\over 12}} (p+2)! p^ 2 \pmod {p^ 6},\\ B^{(p)}_{ p+2} &\equiv {\textstyle {1\over 24}} p^ 2 (p+2)! (p+ 12b_{p+1} )\pmod {p^ 7},\\ B^{(p)}_{ p+4} &\equiv {\textstyle {1\over 12}} p^ 2 (p+4)! (3p+ 2)b_{p +3} \pmod {p^ 4}, \end{aligned} \] where \(b_ n\) is the Bernoulli number of the second kind.
(3) If \(n\) is odd, composite, and \(n>9\), then \(B_{n+2}^{ (n+1)} \equiv 0\pmod {n^ 4}\).
(4) For \(k\geq 0\), define \(A_ k (p; n)= {{(-1)^ n p^{[ n/( p-1)]}} \over {n!}} B_ n^{(n -k)}\). Then \(A_ k(p; n)\) is integral \((\text{mod }p)\). \[ \begin{aligned} A_ k \bigl( p; r(p-1) \bigr) &\equiv (-1)^ r \left( \begin{smallmatrix} r+k\\ k\end{smallmatrix} \right) \pmod p,\\ A_ k \bigl( p; r(p-1)+1 \bigr) &\equiv {\textstyle {1\over 2}} (-1)^{r-1} (r+k-1) \left( \begin{smallmatrix} r+k\\ k\end{smallmatrix} \right) \pmod p \qquad (p>2). \end{aligned} \tag{5} \] (6) For \(k\geq 0\), \[ {{B_ n^{( n-k +1)}} \over {n!}}= \sum_{r=0}^ n {{(-1)^{n-r}} \over {n+1-r}} {{B_ r^{(r-k)}} \over {r^ k}}, \qquad {{(-1 )^{n+k} B^{(n)}_{ n+k}} \over {(n+k)!}}= \sum_{r=0}^ n {n \choose r} {{B^{(r )}_{r+k}} \over {(r+k)!}}. \]
Reviewer: M.Wyneken (Flint)


11B68 Bernoulli and Euler numbers and polynomials