Coefficientwise log-concavity of the triple rising-factorial series

Dmitrii Karp and Yi Zhang, Log-concavity and log-convexity of series containing multiple Pochhammer symbols, arXiv:2305.09029v3, page 8, Conjecture 3.8:

Assume that ${f_n}_{n=0}^{\infty}$ is log-concave and independent of $\mu$. Then the function

$$ \mu\to f(\mu;x)=\sum_{n=1}^{\infty} f_n\frac{(\mu)_{3n}}{(3n-1)!}x^n $$

is coefficient-wise log-concave.

Definition 1.1, page 2, uses nonnegative, nontrivial real sequences, $f_k^2\ge f_{k-1}f_{k+1}$ for positive natural $k$, and interval support. In particular, internal zeros are excluded, but zero endpoints and finite supports are allowed. Define

$$ a_0(t)=0,\qquad a_n(t)=\frac{(t)_{3n}}{(3n-1)!}\quad(n\ge1), $$

and, for every natural $m$ and nonnegative real $\mu,\alpha,\beta$,

$$ \delta_m=\sum_{k=0}^{m} f_kf_{m-k} \left[a_k(\mu+\alpha)a_{m-k}(\mu+\beta) -a_k(\mu)a_{m-k}(\mu+\alpha+\beta)\right]. $$

The assertion is $\delta_m\ge0$ for every such sequence, parameter triple and degree. The endpoint summands vanish. These are finite formal Cauchy coefficients, without convergence or growth assumptions.

The coefficient-wise convention on printed page 1 reads:

if the coefficients $\delta_k$ at all powers of $x$ in (2) are non-negative (non-positive) for all $\alpha,\beta\ge0$ we will say that $\mu\to f(\mu;x)$ is coefficient-wise log-concave (log-convex ).

The Lean namespace prefix is D5.S3.Analytic. The coefficient is KarpZhangNinh.Result.Kernel.a, the finite Turanian is KarpZhangNinh.Result.delta, the sequence predicate is the reused KarpZhangReciprocal.Result.IsLogConcave, and the universal proposition is KarpZhangNinh.Result.claim.

The conjecture is the remaining multiplier-three case of the source's rising-factorial family. Positivity of evaluated series, integer shifts, or a bounded set of degrees does not imply its coefficient assertion. The sequence must be fixed independently of the parameters.

The required conclusion is the exact universal Result.claim. Result.result proves it through a nonnegative full convolution sum, one-sign-change central tails and the existing finite Abel decomposition. The homogeneous and beta-lifting inequalities are uniform in the degree.

Put $N=3m$ and

$$ P_N(x,y)=\sum_{\substack{0\le k\le N\3\mid k}} k(N-k)\binom Nk x^ky^{N-k}. $$

The binomial residue counts obey Pascal recurrence and a six-periodic closed form. For the finite differences $A_n$ and $T_n(N)$ this gives

$$ 3A_n=2^{n-1}+\alpha_n,\qquad 3T_n(N)=(N-n)2^{n-1}+(N-1)\alpha_n-(n-1)\beta_n, $$

where the formulas for $\alpha_n,\beta_n$ are selected by $n$ modulo six. Elementary uniform exponential bounds make both quantities nonnegative for $1\le n<N$ and $3\mid N$. For $d>0$, the shear $y=x+d$ gives

$$ \frac{(\partial_x-\partial_y)P_N(x,y)}{y-x}\bigg|{y=x+d} =\sum{n=1}^{N-1}n\binom Nn T_n(N)x^{n-1}d^{N-n-1}. $$

A positive term makes $t\mapsto P_N(t,1-t)$ strictly increasing on $(0,1/2]$ when $m\ge2$.

For a symmetric homogeneous polynomial $P(x,y)=\sum_{k=0}^N c_kx^ky^{N-k}$, beta moments identify

$$ \frac{\sum_{k=0}^N c_k(u)k(v){N-k}}{(u+v)_N} =\frac{1}{B(u,v)}\int_0^1 P(t,1-t)t^{u-1}(1-t)^{v-1},dt \quad(u,v>0). $$

The reused symmetric beta comparison, extended from strictly increasing to nondecreasing integrands by a positive quadratic perturbation and a limit, transfers balancing monotonicity to the rising-factorial lift. No coefficient-sign assumption is needed for this preservation theorem. Factorial normalization identifies the lift of $P_{3m}$ with $(3m)!\sum_{k=0}^m a_k(u)a_{m-k}(v)$, giving the nonnegative full tail.

Lemma 3.1, printed page 4, states:

Let $k,m\in\mathbb N_0$ with $k\le m/2$, $\mu\ge0$ and $\alpha,\beta>0$. Set

$$ T_{k,m}:=(\mu+\alpha)k(\mu+\beta){m-k}+(\mu+\alpha)_{m-k}(\mu+\beta)k -(\mu)k(\mu+\alpha+\beta){m-k}-(\mu){m-k}(\mu+\alpha+\beta)_k. $$

If $T_{k,m}\le0$ for some $1\le k\le m/2$, then $T_{k-1,m}<0$.

Formula (8) is KarpZhangNinh.Result.Kernel.pairPoch, with $a,b,c,N$ for $\mu,\alpha,\beta,m$. The theorem KarpZhangNinh.Result.Kernel.paired_sign_step uses the positive-$\mu$ specialization of the source's $\mu\ge0$ implication.

For positive parameters the paired Pochhammer kernel has at most one negative-to-positive sign change. Backward length recurrence proves this before factorial division; applying it three times gives the needed multiplier-three assertion. The existing finite one-sign-change weighting principle gives every central tail. The existing Abel decomposition then expresses $\delta_m$ as a sum of these tails times nonnegative increments of $f_kf_{m-k}$. Polynomial continuity supplies all zero-parameter cases.

A failure includes reversing the Turanian, dropping either factorial, allowing internal zeros without proof, weakening arbitrary real shifts to integers, imposing a degree cutoff, or counting an even middle index twice. The beta comparison must use the same constant parameter sum throughout. The settlement must prove the exact claim without additional axioms, placeholders, or an assumed full-tail inequality.

The source quotation and locator are in Library/Asymptotics/karpzhang2023pochhammer.md; the preregistration is https://github.com/the-omega-institute/trureturing/issues/21124. The candidate, source and literature work was produced by the Claude Opus 5.5 orchestrator. The paper proof was written with the assistance of two serial codex-cli research seats (gpt-6.1-sol, high) and checked by an independent codex-cli review seat; the Lean proofs were drafted with the assistance of codex-cli seats. The research and checking seats belong to a single model family, so their agreement is not a claim of model diversity. The paper proof gives the all-degree homogeneous positivity, the beta-lifting theorem and the paired-sign/Abel reduction.

The Lean sources are D5/S3/Analytic/KarpZhangNinh/Beta.lean (including the Homogeneous namespace) and D5/S3/Analytic/KarpZhangNinh/Result.lean. Result.result : Result.claim is kernel-checked by the scoped settling-module build. This is distinct from freezing and Scribe resolution admission. The generic beta comparison and gamma–Pochhammer moment machinery are reused from their existing D5 owners. The sequence predicate, balanced products, finite pairing and Abel identity are reused from the existing Karp–Zhang developments.

What the settlement shows

• PROVED: The decisive mechanism is homogeneous trisection positivity, followed by symmetric beta lifting, paired-sign propagation and finite Abel weighting. The exact universal target is KarpZhangNinh.Result.result : KarpZhangNinh.Result.claim.
• PROVED: The reusable tool is the general theorem KarpZhangNinh.Beta.lift_monotoneOn: a symmetric homogeneous polynomial whose unit-sum slice is nondecreasing towards its centre has a rising-factorial lift with the same balancing property. The coefficients may have either sign; closed endpoints and total parameter zero are included.
• OPEN: Independent verification of the source-reported numerical boundary is not supplied. The source report on page 4 states: “In this section we will prove this conjecture for r = 2 and disprove numerically for r = 4 (strongly suggesting that is it also wrong for r > 4). The case r = 3 remains open and is formulated in the form of Conjecture 3.8.” Remark 3.7, page 8, reports that the analogous unweighted family is neither coefficient-wise log-convex nor coefficient-wise log-concave for $r\ge4$ by numerical computations. These are attributed source computations, not a formal all-parameter refutation or a new local experiment. No reproducible experiment entry is claimed for the authors' computation.
• OPEN: The homogeneous positivity argument uses trisection. No formal multiplier-four refutation or theorem for every multiplier greater than four is included; the numerical failure boundary retains its source attribution.
• OPEN: All-degree coefficient positivity of the full sheared rising-factorial derivative quotient and joint log-concavity of arbitrary interval convolutions are outside the settled claim.
• PROVED: KarpZhangNinh.Result.result gives the exact multiplier-three conclusion and resolves the source's remaining case; the source's multiplier-two result and reported multiplier-four failure keep their respective scopes.

Exhaustive literature coverage, citation-index completeness and priority are unverified. The preregistration supplies the prior literature checks and attribution of the research proof. No new online search is included. Kernel compilation does not establish freezing, publication or required CI. The new public theorems have no completed four-slot escape registrations; the missing owner registrations are associated with preregistration #21124. The scoped Scribe emission includes the Proved resolution node. This does not establish freezing; this implementation does not write frozen state.