slug: kalmykov-karp-pfq-conjecture-one-coefficients
bibkey: kalmykovkarp2017pfq
doi: 10.1007/s10476-017-0503-z
url: https://arxiv.org/abs/1611.06724
triage: theorem
motivation_gids:
  - D5/S3/Analytic/HypergeometricShiftTuran/Defs.claim
  - D5/S3/Analytic/HypergeometricShiftTuran/Result.result

Coefficients of the shifted Gamma product Turanian

Problem

S. I. Kalmykov and D. B. Karp, Log-concavity and Turán-type inequalities for the generalized hypergeometric function, arXiv:1611.06724v1, Section 3, Conjecture 1:

All conclusions of Theorem 3 hold for all α, β ≥ 0 if µ ≥ 1.

For positive parameter vectors of lengths $p$ and $q$, put

$$c(t)=\frac{{\prod }_{i=1}^p\Gamma (a_i+t)}{{\prod }_{j=1}^q\Gamma (b_j+t)},f(\mu ;x)=\sum _{n=0}^{\infty }c(\mu +n)\frac{x^n}{n!}.$$

Equation (9) defines

$${\Delta }_f(\alpha ,\beta ;x)=f(\mu +\alpha ;x)f(\mu +\beta ;x)-f(\mu ;x)f(\mu +\alpha +\beta ;x),$$

and the coefficient of degree $m$ is

$${\delta }_m=\sum _{k=0}^m\frac{c(\mu +\alpha +k)c(\mu +\beta +m-k)-c(\mu +k)c(\mu +\alpha +\beta +m-k)}{k!(m-k)!}.$$

For $p\le q$, condition (13) consists of

$$\frac{e_{q-j}(\mathbf{b})}{e_{p-j}(\mathbf{a})}\le \frac{e_{q-j-1}(\mathbf{b})}{e_{p-j-1}(\mathbf{a})}(0\le j<p),$$

where $e_0=1$. Empty parameter vectors are allowed; in particular, the condition is vacuous at $p=0$. Defs.claim retains all parameter vectors and all coefficient degrees with $\mu \ge 1$, $\alpha ,\beta \ge 0$ and $\alpha \le \beta +1$.

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

Motivation

The coefficient conclusion is one of the conclusions asserted by the conjecture. A single negative coefficient under these hypotheses suffices to refute that assertion, independently of the value of the Turanian at nonnegative real arguments.

Gap

Status: Refuted. The coefficient clause fails for repeated denominator parameters at $p=0$, while Theorem 3’s integer-shift case remains outside this counterexample. The first conclusion concerning the sign of ${\Delta }_f$ is addressed below by finite numerical evidence only.

Route

Take $p=0$, $b_j=1/2$, $\mu =1$, $\alpha =\beta =1/2$ and $m=1$. Then $c(t)=\Gamma (t+1/2{)}^{-q}$, and the two summands give

$${\delta }_1=2(1/2{)}^q-{\left(\frac{16}{9\pi }\right)}^q-{\left(\frac{16}{15\pi }\right)}^q.$$

The half-integer Gamma identities supply the displayed values. The bound $\pi <3.15$ gives $32/(9\pi )>640/567>1$, and exact rational arithmetic gives $(640/567{)}^6>2$. Monotonicity of natural powers gives $(32/(9\pi ){)}^q>2$ for $q\ge 6$. Thus the middle term exceeds the positive leading term, and the last term is strictly positive before subtraction. Specialization at $q=6$ contradicts coefficient nonnegativity.

Falsifier

A source mismatch concerning empty parameter vectors, the symmetric ratio condition, or the conjecture’s shift restrictions would invalidate this correspondence. The finite sum must retain both degree-one terms and both factorial denominators. The identity for the coefficient and the family inequality must compile without additional hypotheses, proof placeholders, or new axioms.

Evidence

The mathematical source is arXiv:1611.06724v1, Section 3, with the journal publication Analysis Mathematica 43 (2017), 567–580, DOI 10.1007/s10476-017-0503-z. The Lean sources are D5/S3/Analytic/HypergeometricShiftTuran/Defs.lean and D5/S3/Analytic/HypergeometricShiftTuran/Result.lean; Result.result has type exactly ¬ Defs.claim.

The counterexample family and the numerical readings were obtained with the assistance of the Claude Opus 5.5 model, and the Lean proof was drafted with the assistance of the gpt-6.1-sol model. Searches covered the works citing the paper in OpenAlex, arXiv and general web search. Sokal’s arXiv:2204.01045 refutes Conjecture 3 of the same paper, not Conjecture 1. Liu’s Zenodo record 22729164 concerns the 2012 reciprocal Gamma conjecture of the same authors. No settlement of Conjecture 1 was found in that scope; this is not a priority certification.

The numerical programs are in the-omega-institute/trureturing-experiments at revision c07877042ab4e3330e285b869fbe5d9e632a1c4b, directory experiments/kalmykov-karp-pfq-turanian (common.py sha256 99d8d5329e2fd15025ebf1f46f67bbfa245787a2628075c93c10b48a98c5d2d2, coeff_scan.py sha256 25ff5ca287cbae1bddc7deb49d652f5fb2f35f51557831470564572de435cd4b, random_control.py sha256 42d60926364e157fc9793fffb066dab72ef0973f7bc2d13a96f00b38b3a6e596, turanian_scan.py sha256 c03c640c50f9cd684e99bc690cd46e542894219ef58990e1a5e5af24a9f9610c), run from that directory with Python 3.12 and mpmath 1.4.1.

Triage

What the settlement shows

The conjecture fails through strong log-concavity of the Gamma ratio $c(t)=\Gamma (t+1/2{)}^{-q}$ for large $q$: the degree-one coefficient pairs the balanced product $c(2{)}^2=\Gamma (5/2{)}^{-2q}$ against the unbalanced products $c(3/2)c(5/2)$, and since $(32/(9\pi ){)}^q>2$ for $q\ge 6$ the balanced product dominates. In this family both shifts equal $1/2$. Theorem 3 of the source covers $\alpha \in \mathbb{N}$ and gives no guarantee for noninteger shifts, including noninteger $\alpha \ge 1$; the threshold $\mu \ge 1$ does not compensate once $q$ is large. The nearest statements that survive are the integer-shift case of Theorem 3 and, numerically, the regime of few denominator parameters.

• PROVED (Lean, Result.half_parameter_family_neg, Result.result): For $p=0$, $b=(1/2,\dots ,1/2)\in {\mathbb{R}}^q$, $\mu =1$, $\alpha =\beta =1/2$, $${\delta }_1=2\cdot 2^{-q}-{\left(\frac{16}{9\pi }\right)}^q-{\left(\frac{16}{15\pi }\right)}^q,$$ and ${\delta }_1<0$ for every $q\ge 6$; hence the coefficient clause of Conjecture 1 fails.
• PROVED in the source, not formalized here: Theorem 3’s integer-shift case ($\alpha \in \mathbb{N}$) is unaffected.
• COMPUTED (40-digit mpmath, outside the Lean development; command python3 coeff_scan.py, exit 0): the closed form above agrees with the direct sum for $q=1,\dots ,8$, and ${\delta }_1<0$ already at $q=5$ (${\delta }_1\approx -4.03\cdot 10^{-5}$). With $b=1/2$ and $\alpha =\beta =1/2$ the least $q$ giving ${\delta }_1<0$ is $5,7,8,10$ for $\mu =1,1.5,2,3$. For $q=6$, $b=1/2$, $\mu =1$, on the grid $\alpha \in \{0.1,0.25,0.5,0.75,1,1.25,1.5,2\}$, $\beta \in \{0.1,0.25,0.5,0.75,1,1.5,2\}$, ${\delta }_1<0$ occurs only at points with $\alpha ,\beta \le 0.75$ and never for $\alpha \ge 1$. For $p=1$, $a\in \{1,2,3\}$, $b=(1/2{)}^6$, $\mu =1$, $\alpha =\beta =1/2$: ${\delta }_1/{\delta }_0\approx -0.0024,-0.0073,-0.0122$.
• COMPUTED (40-digit mpmath, outside the Lean development; command python3 random_control.py 3 3000, exit 0): of 3000 random draws with $p\le q\le 4$, positive parameters, $\mu \ge 1$ and $\alpha ,\beta \in [0,3]$, 2095 satisfy condition (13), and none of them has a negative ${\delta }_m$ for $m\le 15$.
• COMPUTED (50-digit mpmath, series truncated at 400 terms, $x\in \{0,1/4,\dots ,99.75\}$, outside the Lean development; commands python3 turanian_scan.py 0.01 7 10, python3 turanian_scan.py 0.5 6 11, python3 turanian_scan.py 1 7 11, each exit 0): with $p=0$, $\mu =1$, $\alpha =\beta =1/2$ and $b=(0.01,\dots ,0.01)$, ${\min }_x{\Delta }_f/f(\mu ;x{)}^2$ is $\approx -2.2\cdot 10^{-4}$ (at $x=25.25$) for $q=7$ and $\approx -6.0\cdot 10^{-3}$ (at $x=36.5$) for $q=10$; for $b=1/2$ ($q=6,11$) and $b=1$ ($q=7,11$) the minimum over the grid is positive. This is finite numerical evidence that the first conclusion of Conjecture 1 (${\Delta }_f\ge 0$ for $x\ge 0$) also fails; it is not a theorem.
• OPEN: whether the coefficient clause holds for $p=q$, where the log-concavity of the Gamma ratio is weaker; whether a lower bound on $\mu$ depending on $q$ restores it; a proof that ${\Delta }_f<0$ somewhere on $(0,\infty )$ for the parameters above; and Conjecture 2 of the same paper.

ASSUMED-UNVERIFIED

Exhaustive literature coverage, citation-index completeness and priority are unverified. The numerical readings are finite multiprecision observations, not proofs; the sign of ${\Delta }_f$ itself is established only on the stated grid and truncation. Four-slot escape registrations of the shared public Gamma and power helpers (and, for the two-block module, of first_coefficient_eq) are not completed; the gap is recorded at https://github.com/the-omega-institute/trureturing/issues/20966.