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Setting

The paper normalizes a polynomial with simple, equally spaced, real zeros by a change of scale to one of two forms (p. 293). Odd degree $2n+1$:

$$p_n(x)=x\prod_{k=1}^{n}(x^2-k^2),\qquad n=1,2,\ldots, \tag{1}$$

with zeros at the integers $-n,\ldots,n$ and centre of symmetry $0$. Even degree $2n+2$, $n=0,1,\ldots$:

$$q_0(x)=x(x-1),\qquad q_n(x)=(x-n-1)\,p_n(x),\qquad n=1,2,\ldots, \tag{2}$$

with zeros at the integers $-n,\ldots,n+1$ and centre of symmetry $\tfrac12$. By symmetry the paper states its results for positive $x$. An arch is the part of the graph between two consecutive zeros.

Statement

Statement (I) (p. 294). Let $x$ be a non-integer with $0<x<n$. Then

$$|p_n(x+1)|>|p_n(x)|\qquad\text{and}\qquad|q_n(x+1)|>|q_n(x)|. \tag{3}$$

Consequently the areas, and the maxima, under the successive arches of $|p_n(x)|$ and of $|q_n(x)|$, for $x>0$, each form an increasing sequence.

The paper notes (p. 293) that this arch-to-arch monotonicity, for a fixed polynomial, does not depend on the normalizations (1) and (2). A footnote (p. 294) reports D. J. Newman's remark that parts of (I) are implicit in Milne-Thomson's The Calculus of Finite Differences (1960), especially from p. 131 on.

Read depth. Claims checked: the statement and the setting were read clause by clause on the page images of pp. 293--294. The short proof was followed.

Proof pointer

P. 294. Shifting the argument by one gives the exact ratio $p_n(x+1)=\frac{x+n+1}{x-n}\,p_n(x)$, and similarly $q_n(x+1)=\frac{x+n+1}{x-n-1}\,q_n(x)$. For $0<x<n$ the numerator exceeds the absolute value of the denominator, and neither side vanishes at a non-integer.

Dependencies

None beyond the definitions (1) and (2).

Bears on

Problem 1114: context only. The paper calls (I) an analogue of Bálint's verification of Erdős's conjecture, since both describe a monotonicity in passing from one arch of a fixed polynomial to the next (p. 294). Statement (I) concerns the values of the polynomial, not the gaps between zeros of its derivative, and says nothing about those gaps.