Coefficients of the two-block hypergeometric Turanian

S. I. Kalmykov and D. B. Karp, Log-concavity and Turán-type inequalities for the generalized hypergeometric function, arXiv:1611.06724v1, Section 3, page 6:

Conjecture 2. Suppose p1 ≤ q1 , p2 ≤ q2 and conditions (13) hold for a1 , b1 and a2 , b2 . Then the function µ → f (µ; x) defined in (3) is log-concave on [1, ∞) and the corresponding generalized Turánian ∆f (α, β; x) has nonnegative coefficients at all powers of x.

The standing assumption on page 2 is:

We will always assume that a2 is not empty (i.e. p2 ≥ 1), while all other subvectors are allowed to be empty.

For positive parameter vectors in two blocks, equation (3) defines

$$ f(\mu;x)=\frac{\Gamma(\mathbf a_2+\mu)}{\Gamma(\mathbf b_2+\mu)} ,{}_pF_q(\mathbf a_1,\mathbf a_2+\mu; \mathbf b_1,\mathbf b_2+\mu;x). $$

Here a Gamma value at a vector means the product over its entries. Writing $(a)_n$ for the rising factorial and $c(t)$ for the shifted Gamma product ratio, the Taylor coefficient is

$$ F_n(\mu)= \frac{\prod_i(a_{1,i})n}{\prod_j(b{1,j})n} \frac{c(\mu+n)}{n!},\qquad c(t)=\frac{\prod_i\Gamma(a{2,i}+t)}{\prod_j\Gamma(b_{2,j}+t)}. $$

Equation (2) defines

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

whose degree-$m$ coefficient is

$$ \delta_m=\sum_{k=0}^m \bigl[F_k(\mu+\alpha)F_{m-k}(\mu+\beta) -F_k(\mu)F_{m-k}(\mu+\alpha+\beta)\bigr]. $$

For each block separately, condition (13) is

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

with $e_0=1$. The coefficient clause quantifies over $p_1\le q_1$, $1\le p_2\le q_2$, all four positive vectors satisfying their respective chains, every $\mu\ge1$, all $\alpha,\beta\ge0$, and every natural degree $m$.

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

| Source object | Lean name | Correspondence | | --- | --- | --- | | Unshifted rising factorial ratio | TwoBlockResult.pochRatio | Products of (ascPochhammer ℝ n).eval at the unshifted parameters. | | Equation (3) Taylor coefficient | TwoBlockResult.seriesCoeff | Rising factorial ratio times Defs.coeff for the shifted block at $\mu+n$, divided by $n!$. | | Equation (2) Cauchy coefficient | TwoBlockResult.twoBlockTuranCoeff | Exact finite sum with the factorials contained in seriesCoeff. | | Shifted Gamma ratio and condition (13) | Defs.coeff, Defs.esymm, Defs.cond13 | Reused definitions, with the ratio chain imposed separately on both blocks. | | Conjecture 2 coefficient clause | TwoBlockResult.claim | All dimensions, positivity assumptions, both chains, nonempty shifted numerator, shifts and coefficient degrees. | | Negative coefficient family | TwoBlockResult.half_parameter_family_neg | Empty unshifted blocks, shifted numerator $(1)$, repeated half denominators, $\mu=1$, $\alpha=\beta=1/2$, and every $q\ge7$. | | Refutation | TwoBlockResult.result | Exactly $\neg,\texttt{TwoBlockResult.claim}$. |

A negative coefficient under all of the stated hypotheses refutes the coefficient clause. Conjecture 2 requires the shifted numerator to be nonempty, so a counterexample with no shifted numerator does not meet its hypotheses. The witness here retains the shifted numerator $(1)$.

The explicit calculation in Route refutes the coefficient clause. The D5 modules of this delivery compile in the repository CI and encode the family inequality and its specialization at $q=7$. The separate assertion of log-concavity at a fixed nonnegative $x$ is not settled here.

Take $\mathbf a_1=\mathbf b_1=()$, $\mathbf a_2=(1)$, $\mathbf b_2=(1/2,\ldots,1/2)\in\mathbb R^q$, $\mu=1$, $\alpha=\beta=1/2$, and $m=1$. The empty rising factorial products equal one, so the coefficient reduces to Defs.turanCoeff for the shifted block. Now

$$ c(t)=\frac{\Gamma(1+t)}{\Gamma(1/2+t)^q},\qquad \delta_1=2c(3/2)c(5/2)-c(2)^2-c(1)c(3). $$

The integer and half-integer Gamma identities give

$$ \begin{aligned} c(3/2)&=\Gamma(5/2),& c(5/2)&=\Gamma(7/2)2^{-q},\ c(2)&=2\Gamma(5/2)^{-q},& c(1)&=\Gamma(3/2)^{-q},& c(3)&=6\Gamma(7/2)^{-q}, \end{aligned} $$

where $\Gamma(3/2)=\sqrt\pi/2$, $\Gamma(5/2)=3\sqrt\pi/4$ and $\Gamma(7/2)=15\sqrt\pi/8$. Consequently

$$ \delta_1=\frac{45\pi}{16},2^{-q} -4\left(\frac{16}{9\pi}\right)^q -6\left(\frac{16}{15\pi}\right)^q. $$

The bound $\pi<3.15$ gives $32/(9\pi)>640/567>1$. Exact rational arithmetic gives $(640/567)^7>45\cdot3.15/64$, and monotonicity in the natural exponent gives

$$ \left(\frac{32}{9\pi}\right)^q> \frac{45\pi}{64}\qquad(q\ge7). $$

Multiplying by $4\cdot2^{-q}>0$ shows that the middle term in the closed form exceeds its positive leading term. The final subtracted term is strictly positive, so $\delta_1<0$ for every $q\ge7$.

The first block's chain is vacuous. In the shifted block, $p_2=1$ and $e_1(\mathbf a_2)=e_0(\mathbf a_2)=1$. Since the symmetric function of $q$ repeated entries $x$ in degree $k$ is $\binom qk x^k$, the only chain inequality is

$$ 2^{-q}\le q,2^{1-q}, $$

which holds for $q\ge1$. At $q=7$ all dimension, positivity and shift requirements are satisfied, contradicting coefficient nonnegativity.

A source requirement that either unshifted block be nonempty would invalidate this correspondence; the printed standing assumption expressly permits both to be empty. The calculation contains both degree-one Cauchy summands, the shifted numerator Gamma factors and all factorials. Condition (13) for the shifted block is checked explicitly. The claim retains all coefficient degrees and both chains. The D5 modules of this delivery compile in the repository CI.

The source is arXiv:1611.06724v1, pages 2 and 6, equations (2), (3), (13) and Conjecture 2; journal publication Analysis Mathematica 43 (2017), 567–580, DOI 10.1007/s10476-017-0503-z. The Lean source is D5/S3/Analytic/HypergeometricShiftTuran/TwoBlockResult.lean. TwoBlockResult.result is written with type exactly ¬ TwoBlockResult.claim. The D5 modules of this delivery compile in the repository CI.

The orchestrator-reported literature scope comprises all 15 OpenAlex citing works, arXiv and general web search, and repository search for the source identifier. No settlement of Conjecture 2 was found in that scope. Sokal's arXiv:2204.01045 concerns Conjecture 3. These source and literature findings are orchestrator-reported, and do not certify priority or exhaustive coverage.

The numerical entry uses the driver in the-omega-institute/trureturing-experiments at revision ccf77c1917e9d1a2f3d101bcd815841d9ca7dcb1: experiments/kalmykov-karp-pfq-turanian/two_block_scan.py, sha256 1a009ce42782cc93ea7bb3d21c4c154c1c4f6336dd1bed57790b6f6822e9c07d, with common.py, sha256 99d8d5329e2fd15025ebf1f46f67bbfa245787a2628075c93c10b48a98c5d2d2. Command: python3 two_block_scan.py, run from that directory with Python 3.12 and mpmath 1.4.1, exit 0. At 40 decimal digits, for $q=1,\ldots,11$ the closed form equals the direct Cauchy sum to a maximum absolute discrepancy of $4.6\cdot10^{-41}$. The shifted numerator is $[1]$, the denominator is $[1/2]^q$, $\mu=1$, $\alpha=\beta=1/2$ and the degree is one. The first coefficient is approximately $0.01693$ at $q=5$, $-0.002483$ at $q=6$ and $-0.008420$ at $q=7$.

What the settlement shows

The failure mechanism is the exponential dependence on denominator multiplicity. In the degree-one coefficient, the negative balanced product $c(2)^2$ eventually dominates the positive product $2c(3/2)c(5/2)$. A nonempty shifted numerator contributes the fixed factors $45\pi/16$, $4$ and $6$ in the closed form, and the exponential dominance persists. Thus neither $\mu\ge1$ nor the two ratio chains prevent this failure when both shifts are $1/2$.

• PROVED IN PROSE (Route): the closed form above and its strict negativity for every $q\ge7$, with all source hypotheses satisfied at $q=7$, refute the coefficient clause. The D5 argument is TwoBlockResult.half_parameter_family_neg followed by TwoBlockResult.result; the D5 modules of this delivery compile in the repository CI.
• COMPUTED: python3 two_block_scan.py, exit 0, in experiments/kalmykov-karp-pfq-turanian of the-omega-institute/trureturing-experiments at revision ccf77c1917e9d1a2f3d101bcd815841d9ca7dcb1, driver sha256 1a009ce42782cc93ea7bb3d21c4c154c1c4f6336dd1bed57790b6f6822e9c07d (common.py sha256 99d8d5329e2fd15025ebf1f46f67bbfa245787a2628075c93c10b48a98c5d2d2), Python 3.12 and mpmath 1.4.1: for $q=1,\ldots,11$ the closed form equals the direct sum with maximum discrepancy $4.6\cdot10^{-41}$ at 40 digits. The first coefficient is approximately $0.01693$ at $q=5$, $-0.002483$ at $q=6$ and $-0.008420$ at $q=7$. Thus the proved sufficient threshold $q\ge7$ is not claimed to be sharp.
• PROVED in the source, not formalized here: Theorem 3's integer-shift conclusions remain unaffected by this witness. With empty unshifted blocks the series is equation (10), and the shifts used here are noninteger. The counterexample does not invalidate an already proved statement restricted to integer shifts.
• OPEN: the sign of the full Turanian and the log-concavity clause of Conjecture 2 for this family at a fixed $x\ge0$; a classification of the parameter ranges where coefficient nonnegativity holds; and whether a lower bound on $\mu$ depending on denominator multiplicity restores it. Conjectures 1 and 2 have different stated domains, so their coefficient clauses require their respective witnesses. The source's proved results retain their hypotheses; an extension relying on Conjecture 2's full coefficient assertion cannot use that assertion after this refutation.

Axiom closure and Reg compilation: not recorded here. Emitted Blueprints and the four-slot registration's binding evidence: not recorded here. The source and literature checks are supplied by the orchestrator; exhaustive literature coverage, citation-index completeness and priority are unverified. Numerical agreement and signs are finite multiprecision observations rather than proofs of a uniform statement. No claim about the sign of the full Turanian or log-concavity at a fixed $x$ follows solely from the negative coefficient. 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.