slug: karp-zhang-2023-conjecture-5-2
bibkey: karpzhang2023pochhammer
doi: 10.1007/s13540-023-00238-0
url: https://arxiv.org/abs/2305.09029
triage: theorem
motivation_gids:
  - D5/S3/Analytic/KarpZhangConjecture/Result.claim
  - D5/S3/Analytic/KarpZhangConjecture/Result.result

Coefficientwise log-concavity of the two Karp–Zhang series

Problem

Dmitrii Karp and Yi Zhang, Log-concavity and log-convexity of series containing multiple Pochhammer symbols, arXiv:2305.09029v3, page 12, Conjecture 5.2:

Suppose that $\{y_k{\}}_{k=0}^{\infty }$ is a log-concave sequence. Then the functions $\mu \to \tilde{y}(\mu ;x)$ and $\mu \to \hat{y}(\mu ;x)$ are coefficient-wise log-concave on $\mu >0$.

The two series are

$$\tilde{y}(\mu ;x)=\sum _{k=0}^{\infty }{y}_k\frac{(2\mu {)}_k}{(\mu {)}_{k}k!}{x}^k,\hat{y}(\mu ;x)=\sum _{k=0}^{\infty }{y}_k\frac{\Gamma (\mu +k)}{\Gamma (2\mu +k)k!}{x}^k.$$

Here $(a{)}_k$ is the rising factorial. Definition 1.1, page 2:

Let $\{f_k{\}}_{k=0}^{\infty }$ be a non-negative nontrivial real sequence. We call $\{f_k{\}}_{k=0}^{\infty }$ log-concave (or Pólya frequency two – $P{F}_2$, or doubly positive) if $f_k^2\ge f_{k-1}{f}_{k+1}$ for each $k\in \mathbb{N}$, and $\{f_k{\}}_{k=0}^{\infty }$ is supported on an interval of integers (i.e., $f_N=0$ implies either $f_{N+i}=0$ for all $i\in {\mathbb{N}}_0$ or $f_{N-i}=0$ for $i=0,\dots ,N$).

The Lean predicate retains nonnegativity, nontriviality, adjacent inequalities for positive natural indices, and the printed zero-support disjunction. Natural subtraction is used only with the stated bounds.

For either series $f$, coefficientwise log-concavity means that every coefficient of

$$f(\mu +\alpha ;x)f(\mu +\beta ;x)-f(\mu ;x)f(\mu +\alpha +\beta ;x)$$

is nonnegative for every $\mu >0$ and every real $\alpha ,\beta \ge 0$. This is an assertion about finite Cauchy sums in each natural degree, so it requires no convergence or growth assumption on $y$.

Formal correspondence. All Lean names below have prefix D5.S3.Analytic.

Motivation

The conjecture asks for preservation of coefficientwise log-concavity by two different parameter-dependent transforms of an arbitrary log-concave sequence. Nonnegativity of a series at positive arguments would not suffice: the assertion must hold separately in every coefficient and for arbitrary nonnegative real shifts.

Gap

Status: Proved (Lean) for both halves of Conjecture 5.2. The result has neither a degree cutoff nor an integrality restriction on either shift. Conjectures 3.8 and 4.2 of the same paper remain open in this development.

Route

For the reciprocal series, put $c_k(t)=(2t{)}_k/((t{)}_{k}k!)$ and

$$F_{m,L,U}(s,t)=\sum _{k=L}^U{c}_k(s)c_{m-k}(t).$$

On a line $s+t=S$, normalize the summands to a probability vector $\pi$. The second derivative of the logarithm of their sum is the score variance minus the expected negative logarithmic curvature. Reversible birth-and-death rates on the finite index interval, with rates zero at its endpoints, give a curvature-weighted variance inequality. The coefficient curvature estimate absorbs its edge terms, proving concavity of $\log F_{m,L,U}(s,S-s)$. Only this direction of concavity is needed. For a symmetric interval, symmetry under $s\leftrightarrow t$ then gives

$$F_{m,q,m-q}(\mu ,\mu +\alpha +\beta )\le F_{m,q,m-q}(\mu +\alpha ,\mu +\beta ).$$

Log-concavity and interval support of $y$ imply that $W_q=y_q{y}_{m-q}$ increases towards the central index. Finite Abel summation expresses the reciprocal Turanian coefficient as $W_0$ times the full tail plus the increments $W_{q+1}-W_q$ times the successive symmetric tails. Every factor is nonnegative. This is KarpZhangReciprocal.Result.reciprocal_half.

For the gamma series, introduce the formal exponential smoothings

$$P_p(z)=\sum _{r\ge 0}{y}_{p+r}\frac{z^r}{r!},H(a;z,w)=\sum _{j\ge 0}\frac{P_{2j}(z)}{j!\Gamma (a+j)}{w}^j.$$

Simultaneous induction in coefficient degree, using Pascal’s recurrence, shows that $P_{p+1}{P}_{q+1}-P_p{P}_{q+2}$ has nonnegative coefficients when $p\le q+1$. Consequently the coefficient weights in $P_{2k}{P}_{2(m-k)}$ increase towards the centre. Pair the reciprocal-gamma kernel at $k$ and $m-k$, counting the middle term once. A finite Vandermonde identity and log-convexity of Gamma show that the kernel’s total is nonnegative. The Gamma recurrence propagates nonpositivity of the four-term bracket backwards, establishing at most one change from negative to positive. The finite weighting principle therefore proves nonnegativity of every bivariate Turanian coefficient of $H$.

Finally Gamma duplication and a simultaneous recurrence in degree give

$$\frac{\Gamma (t+n)}{\Gamma (2t+n)n!}=\sqrt{\pi }{2}^{1-2t}\sum _{j=0}^{\lfloor n/2\rfloor }\frac{1}{(n-2j)!j!\Gamma (t+1/2+j)2^{n-2j}16^j}.$$

Thus the gamma series is $\sqrt{\pi }{2}^{1-2t}H(t+1/2;x/2,x^2/16)$. The scale products at $t+\alpha ,t+\beta$ and $t,t+\alpha +\beta$ agree and are positive. Each substituted coefficient is a finite nonnegative weighted sum of the bivariate coefficients. This proves KarpZhangGamma.Series.gamma_half. The settling result combines the two halves without changing any hypothesis.

Falsifier

The correspondence would fail if either coefficient omitted a factorial, reversed the Turanian sign, restricted a shift to integers, or replaced the source’s interval-support condition by adjacent inequalities alone. The paired kernel must count its middle index once. The positive diagonal substitution must retain every index satisfying $r+2j=n$. The exact universal claim and both imported halves must compile without proof placeholders or additional axioms.

Evidence

The source is arXiv:2305.09029v3, pages 2 and 12, with DOI 10.1007/s13540-023-00238-0; the quotation and sequence convention are recorded in Library/Asymptotics/karpzhang2023pochhammer.md. The reciprocal proof is in D5/S3/Analytic/KarpZhangReciprocal/{BirthDeath,IntervalConvolution,Result}.lean; the gamma proof is in D5/S3/Analytic/KarpZhangGamma/{Weighting,Smoothing,Result,Series}.lean. D5/S3/Analytic/KarpZhangConjecture/Result.lean states claim and proves it as result by the two imported half theorems.

The paper proofs were written with the assistance of codex-cli research seats and checked by independent codex-cli referee seats, the Lean proofs were drafted with the assistance of codex-cli seats, and the work was orchestrated with the assistance of Claude Opus 5.5.

The settlement is associated with preregistration https://github.com/the-omega-institute/trureturing/issues/20997. The scoped Lean build of D5.S3.Analytic.KarpZhangConjecture.Result verifies the settling theorem and its import closure. Its axiom closure is contained in $\{\text{propext},\text{Classical.choice},\text{Quot.sound}\}$. This local verification is distinct from publication, freezing and required CI.

Triage

What the settlement shows

The reciprocal mechanism is variance absorption for interval convolutions, followed by symmetric-tail balancing and Abel summation. The gamma mechanism is preservation of balanced minors under exponential smoothing, a single negative-to-positive sign change in the reciprocal-gamma kernel, and a positive diagonal transformation. Both mechanisms are uniform in coefficient degree and allow arbitrary nonnegative real shifts, including zero. The conclusions concern formal coefficients; analytic evaluation additionally requires the relevant convergence hypotheses. Conjecture 5.2 can therefore be used as a proved coefficient inequality under precisely its printed sequence conditions. This supplies neither a new hypothesis nor a proof of the paper’s other conjectures.

• PROVED (Lean, KarpZhangReciprocal.Result.reciprocal_half): Every reciprocal-series Turanian coefficient is nonnegative for every admissible sequence, positive base parameter and nonnegative real shifts.
• PROVED (Lean, KarpZhangGamma.Series.gamma_half): Every gamma-series Turanian coefficient is nonnegative under the same conditions.
• PROVED (Lean, KarpZhangConjecture.Result.result): The conjunction of both universal coefficient assertions, exactly claim.
• OPEN: Conjectures 3.8 and 4.2 of the same paper. Numerical support is reported by the research line; neither conjecture is proved here. The numerical scope has not been independently revalidated in this delivery.

ASSUMED-UNVERIFIED

Exhaustive literature coverage, citation-index completeness and priority are unverified. Numerical support for Conjectures 3.8 and 4.2 is supplied by the research line; no numerical experiment or uniform extension is certified by this delivery. The producer attribution above is supplied by the coordinating research line. Four-slot escape registrations for the public theorems of the seven supporting modules and for the settling result are not completed in this delivery; their mathematical proofs do not constitute those registrations.