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Statement

Notation as on the Statement (I) page; the nonnegative zeros of $p_n$ are $j=0,1,\ldots,n$ and those of $q_n$ are $j=0,1,\ldots,n+1$.

Slopes at the zeros (Section 4, pp. 297--298). With $0!=1$,

$$|p'_n(j)|=(n+j)!\,(n-j)!,\qquad j=0,1,\ldots,n, \tag{16}$$ $$|q'_n(j)|=(n+j)!\,(n+1-j)!,\qquad j=0,1,\ldots,n+1. \tag{18}$$

Hence (p. 297, (17))

$$|p'_n(j+1)|-|p'_n(j)|=(2j+1)(n+j)!\,(n-j-1)!>0,\qquad j=0,1,\ldots,n-1,$$

and (p. 298, (19))

$$|q'_n(j+1)|-|q'_n(j)|=(2j)(n+j)!\,(n-j)!>0,\qquad j=1,\ldots,n.$$

So $|p'_n(j)|$ increases for $j=0,\ldots,n$, and $|q'_n(j)|$ increases for $j=1,\ldots,n+1$, while $|q'_n(0)|=|q'_n(1)|=n!\,(n+1)!$, for each fixed $n$.

Statement (III) (p. 298). The sequences $\{|p'_n(j)|\}$, $j=0,\ldots,n$, and $\{|q'_n(j)|\}$, $j=1,\ldots,n+1$, are each absolutely monotonic.

Here a sequence $\{a_j\}$ is absolutely monotonic when every defined difference is non-negative: $\Delta^m a_j\ge0$ for all $m$ and $j$, where $\Delta^0a_j=a_j$ and $\Delta^{m+1}a_j=\Delta^m a_{j+1}-\Delta^m a_j$ (p. 298). For $p_n$ this is the inequality

$$\Delta^m\{(n+j)!\,(n-j)!\}\ge0,\qquad j=0,\ldots,n;\quad m=0,\ldots,n-j, \tag{20}$$

and the paper shows it with strict inequality. Remark (i) (p. 299) says that similar results hold for these sequences taken at fixed $j$ as $n$ varies.

Read depth. Claims checked: (16)--(20) and (III) were read on the page images of pp. 297--299. The proofs were followed but not checked step by step.

Proof pointer

Pp. 297--299. (16) comes from the product rule applied to (1): at an interior zero $j$ only one term survives, and the two remaining products evaluate to $(2j)!$ and $\prod_{k=j+1}^{n}(k^2-j^2)$, whose product is $(n+j)!(n-j)!$; the cases $j=0,1,n$ are checked directly. (18) follows from $q'_n=(x-n-1)p'_n+p_n$, with $q'_n(n+1)=p_n(n+1)=(2n+1)!$. For (20) the paper proves by induction that $\Delta^m\{(n+j)!(n-j)!\}$ equals $(n+j)!(n-j-m)!$ times a polynomial in $j,m,n$ with non-negative integer coefficients, not all zero; the case of $q_n$ is said to follow in the same fashion. Remark (ii) (p. 299) reports D. J. Newman's observation that a beta function integral for these differences makes (20) obvious.

Dependencies

The definitions (1) and (2).

Bears on

None recorded.