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

First defined and studied by N. NĂ¶rlund 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)

### MSC:

 11B68 Bernoulli and Euler numbers and polynomials