Sun's q-log-convexity conjecture for binomial sums and the polynomials S_n^{(m)}

The repaired fourth-power binomial conjecture fails for every sufficiently large index

Source and essential prior attribution. In Conjecture 1.1(i) of Zhi-Wei Sun, A family of polynomials and related congruences and series, Frontiers in Combinatorics and Number Theory 2 (2026), 1–22, arXiv:2505.02767v4, one proposed assertion is that the polynomial sequence

$$R_n^{(m)}(q) = \sum_{j=0}^{n}\binom{n+j}{j}^{m}q^j \qquad(n\geq 0) \tag{1}$$

is $q$-log-convex for every integer $m\geq 2$.

The unrestricted assertion was already disproved by Qiping Zhou on July 24, 2026. In an answer to Sun's original MathOverflow question, Zhou exhibited the existing counterexample

$$[q^2]\left( R_6^{(5)}(q)R_8^{(5)}(q)-R_7^{(5)}(q)^2 \right) =-499140. \tag{2}$$

We make no claim to the first disproof of that unrestricted assertion.

However, in response on July 25, 2026, Sun proposed the explicitly narrower remaining possibility: “I believe that the conjecture still holds for $m=2,3,4$.” The new result below disproves the previously surviving case $m=4$, with infinitely many counterexamples. More generally, it proves failure for every fixed exponent $m\geq 4$. The cases $m=2,3$ and the other polynomial families in Conjecture 1.1 are not resolved here.

The notation $R_n^{(m)}$ is used to distinguish the rising-binomial polynomial (1) from the unrelated double-sum polynomial denoted $S_n^{(m)}$ elsewhere in Sun's paper.

1. An exact coefficient obstruction

A sequence of polynomials is $q$-log-convex if every coefficient of

$$\Delta_n^{(m)}(q) = R_{n-1}^{(m)}(q)R_{n+1}^{(m)}(q) -R_n^{(m)}(q)^2 \tag{3}$$

is nonnegative for every $n\geq 1$.

For $r\geq 2$, the first three coefficients of (1) are

$$[q^0]R_r^{(m)}(q)=1, \qquad [q]R_r^{(m)}(q)=(r+1)^m, \qquad [q^2]R_r^{(m)}(q) =\binom{r+2}{2}^{m}. \tag{4}$$

Consequently, for every $n\geq 3$,

$$\begin{aligned} D_m(n) &:=[q^2]\Delta_n^{(m)}(q) \\ &= \left(\frac{n(n+1)}{2}\right)^m +\left(\frac{(n+2)(n+3)}{2}\right)^m -2\left(\frac{(n+1)(n+2)}{2}\right)^m \\ &\hspace{1.2em} +\bigl(n(n+2)\bigr)^m -(n+1)^{2m}. \end{aligned} \tag{5}$$

This is a polynomial in $n$ with rational coefficients, obtained directly from the three terms contributing to a coefficient of degree two in each product.

2. The author's repaired case $m=4$

For $m=4$, the polynomial (5) factors exactly:

$$D_4(n) = -\frac{n+2}{2} \left( n^5-17n^4-108n^3-227n^2-211n-78 \right). \tag{6}$$

Write

$$F(n)=n^5-17n^4-108n^3-227n^2-211n-78.$$

For every real $t\geq 0$, expansion around $n=23$ gives

$$\begin{aligned} F(23+t) ={}&t^5+98t^4+3618t^3 \\ &+60033t^2+389800t+239996 >0. \end{aligned} \tag{7}$$

Combining (6) and (7),

$$\boxed{ [q^2]\left( R_{n-1}^{(4)}(q)R_{n+1}^{(4)}(q)-R_n^{(4)}(q)^2 \right)<0 \qquad\text{for every }n\geq 23. } \tag{8}$$

In particular, the first such index has the completely explicit certificate

$$\begin{aligned} [q^2]\left( R_{22}^{(4)}(q)R_{24}^{(4)}(q)-R_{23}^{(4)}(q)^2 \right) &= 276^4+325^4-2\cdot300^4+575^4-24^8 \\ &=-2999950<0. \end{aligned} \tag{9}$$

At the preceding index,

$$D_4(22)=1119504>0. \tag{10}$$

Direct expansion of all $528$ coefficients of $\Delta_n^{(4)}(q)$ for $1\leq n\leq22$ shows that they are nonnegative. Thus $n=23$ is the first failure of the full fourth-power $q$-log-convexity assertion, not merely the first failure detected by (8).

3. Every exponent $m\geq 4$ fails infinitely often

Let

$$H_m(x) = \left(\frac{(x+1)(x+2)}{2}\right)^m.$$

Then (5) can be rewritten as

$$D_m(n) = H_m(n-1)-2H_m(n)+H_m(n+1) +\bigl((n+1)^2-1\bigr)^m-(n+1)^{2m}. \tag{11}$$

Since $H_m(x)$ has leading term $2^{-m}x^{2m}$, its central second difference has leading term

$$H_m(n-1)-2H_m(n)+H_m(n+1) = \frac{(2m)(2m-1)}{2^m}n^{2m-2} +O\left(n^{2m-3}\right). \tag{12}$$

The binomial theorem gives

$$\bigl((n+1)^2-1\bigr)^m-(n+1)^{2m} = -mn^{2m-2}+O\left(n^{2m-3}\right). \tag{13}$$

Therefore

$$D_m(n) = m\left( \frac{2m-1}{2^{m-1}}-1 \right)n^{2m-2} +O\left(n^{2m-3}\right). \tag{14}$$

For every integer $m\geq4$,

$$2^{m-1}>2m-1. \tag{15}$$

Indeed, (15) holds at $m=4$ because $8>7$, and doubling the left-hand side preserves the strict inequality at every subsequent integer. Consequently, the leading coefficient in (14) is negative. Thus

$$\boxed{ \text{For every fixed integer }m\geq4, \quad [q^2]\Delta_n^{(m)}(q)<0 \quad\text{for all sufficiently large }n. } \tag{16}$$

For comparison, the same coefficient for the two still-open exponents is

$$\begin{aligned} D_2(n)&=(n+2)(n+3)>0, \\ D_3(n)&= \frac{3(n+1)(n+2)(n^2+11n+16)}{4}>0 \qquad(n\geq3). \end{aligned} \tag{17}$$

These identities concern only the coefficient of $q^2$ and do not establish full $q$-log-convexity for $m=2$ or $m=3$.

Hence Zhou's earlier $m=5$ counterexample already settled the original unrestricted formulation, whereas (8)–(16) newly rule out Sun's subsequent $m=4$ refinement and, in fact, every exponent $m\geq4$.