Coefficients of the shifted Gamma product Turanian

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

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

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

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

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 $\mu\ge1$, $\alpha,\beta\ge0$ and $\alpha\le\beta+1$.

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

| Source object | Lean name | Correspondence | | --- | --- | --- | | Equation (10) coefficient ratio | Defs.coeff | Product of Gamma values for numerator parameters divided by the product for denominator parameters. | | Equation (9) coefficient | Defs.turanCoeff | Exact finite Cauchy sum for the degree-$m$ coefficient of the shifted Turanian. | | Condition (13) | Defs.esymm, Defs.cond13 | Elementary symmetric functions and their consecutive ratio chain. | | Conjecture 1 coefficient clause | Defs.claim | Universal nonnegativity for the printed parameter and shift range. | | Negative coefficient family | Result.half_parameter_family_neg | For $p=0$, repeated $b_j=1/2$, $\mu=1$, $\alpha=\beta=1/2$, every $q\ge6$ gives $\delta_1<0$. | | Refutation | Result.result | Exactly $\neg,\texttt{Defs.claim}$. |

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.

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.

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 $q\ge6$. 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.

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.

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.

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 $q\ge6$ 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 $\alpha\ge1$; the threshold $\mu\ge1$ 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 $p=0$, $b=(1/2,\ldots,1/2)\in\mathbb R^q$, $\mu=1$, $\alpha=\beta=1/2$, $$\delta_1=2\cdot2^{-q}-\left(\frac{16}{9\pi}\right)^q- \left(\frac{16}{15\pi}\right)^q,$$ and $\delta_1<0$ for every $q\ge6$; 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 $q=1,\ldots,8$, and $\delta_1<0$ already at $q=5$ ($\delta_1\approx-4.03\cdot10^{-5}$). 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 $\alpha,\beta\le0.75$ and never for $\alpha\ge1$. 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 $p\le q\le4$, positive parameters, $\mu\ge1$ and $\alpha,\beta\in[0,3]$, 2095 satisfy condition (13), and none of them has a negative $\delta_m$ for $m\le15$.
• COMPUTED (50-digit mpmath, series truncated at 400 terms, $x\in{0,1/4,\ldots,99.75}$, 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 $b=(0.01,\ldots,0.01)$, $\min_x\Delta_f/f(\mu;x)^2$ is $\approx-2.2\cdot10^{-4}$ (at $x=25.25$) for $q=7$ and $\approx-6.0\cdot10^{-3}$ (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 ($\Delta_f\ge0$ for $x\ge0$) 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.

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.