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Statement

Notation as on the relations (7)--(11) page; in addition $x''_{nj}$ and $\xi''_{nj}$ denote the zeros of $p''_n$ and $q''_n$ (p. 299). All differences $\Delta^m$ are taken with respect to $j$.

Conjecture (21) (p. 299). Numerical calculations by Anastase Mastoras suggest that

$$(-1)^m\Delta^m x'_{nj}\ge0\quad(m=2,3,\ldots,n-1),\qquad (-1)^m\Delta^m x''_{nj}\ge0\quad(m=2,3,\ldots,n-2),$$

and similarly for $\xi'_{nj}$ and $\xi''_{nj}$. The paper notes that for $x'_{nj}$ and $\xi'_{nj}$ with $m=2$ this is the substance of the Erdős--Bálint result (p. 299).

The calculations covered polynomials of degrees 5, 10, 15, 20, 50, 101, 102, 307 and 1000. Some entries did not conform to (21), but only where a high degree, a high rank of the zero and at least a moderately high order of differencing combine; the paper says these may be round-off errors and that it would still be reasonable to conjecture (21) (pp. 299--300).

A weaker-hypothesis conjecture (p. 300). If (21) holds for $x''_{nj}$ and $\xi''_{nj}$, the paper suggests replacing equal spacing by the hypothesis $(-1)^m\Delta^m x_{nj}\ge0$ for $m=2,3,\ldots,n$ on the zeros themselves (with, as it notes, $\Delta x_{nj}>0$), and expecting still $(-1)^m\Delta^m x'_{nj}\ge0$ for $m=2,3,\ldots,n$, and likewise with $\xi$ in place of $x$. It observes that this, if proved, together with the first part of (21), would give the second part of (21) and the analogue for all higher derivatives. It adds that analogous results, with differencing in $n$ rather than $j$, may hold for the sequences in (13) and (14).

Read depth. Claims checked: (21) and the conjectures around it were read on the page images of pp. 299--300. The paper proves none of them; the case $m=2$ for $x'_{nj}$ and $\xi'_{nj}$ it attributes to Bálint.

Dependencies

The definitions on the relations (7)--(11) page.

Bears on

Problem 1114: a proposed generalization. By the paper's own account (p. 299), the case $m=2$ for $x'_{nj}$ and $\xi'_{nj}$ is the Erdős--Bálint result, which is the problem's monotonicity of gaps; the cases $m\ge3$, the second-derivative part and the weaker-hypothesis form are conjectures the paper poses and does not prove.