Some theorems on the Schur derivative (Q2650745)

In 1933 \textit{I. Schur} [Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, No. 3-4, 145--151 (1933; Zbl 0006.24801)] defined the derivatives of a sequence \(a_1, a_2, \ldots\) by \[ \Delta a_m=(a_{m+1}-a_m)/p^{m+1}\quad\text{ and }\quad \Delta^r a_m= \Delta(\Delta^{r-1} a_m). \] He examined \(a_m = a^{p^m}\) and \textit{M. Zorn} [Ann. Math. (2) 38, 451--464 (1937; Zbl 0016.34803)] the case \(a_m=\left(a^{(p-1)p^m}-1\right)/p^{m+1}\). The author considers also the case \(a_m = a^{k p^m}\). His most general result is: If \(p>2\), \(1\leq r\leq p\), \(m\geq 0\), \(k\geq 1\), \(q_m=\left(a^{(p-1)p^m}-1\right)/p^{m+1}\), then \[ \Delta^r a^{k p^m}\equiv a^{k p^m} q^r_m k^r \prod_{i=1}^r (p^i -1)/ r! (p-1)^r \pmod {p^m}, \] if \(r < p - 1\), the congruence is valid mod \(p^{m+1}\). Therefore \(\Delta^r a^{k p^m}\) is integral if \(r=1,\ldots, p-1\). The proofs are elementary but require many calculations. This theorem is extended to algebraic integers. The case \(k = p - 1\) is extended to prime ideals of arbitrary degree and for arbitrary \(k\) to prime ideals of first degree. The author also introduces \(\Delta_p a_m = (a_{mp} - a_m)^{1+r}\) where \(r\) designs the number of factors \(p\) of \(m\). It is proved if \((a, k) =1\), \(k = p_1^{\varepsilon_1}\cdots p_s^{\varepsilon_s}\) and \(r_i < p_i\), that \(\Delta_{p_1}^{r_1}\cdots \Delta_{p_s}^{r_s}a^k\) is integral for all \(k>1\). Finally, the main result is applied to polynomials especially to Euler and Bernoulli polynomials for integral (mod \(p\)) values of the argument.