slug: kalmykov-karp-2018-qgamma-turanian-coefficients
bibkey: kalmykovkarp2018qgamma
doi: 10.1016/j.jmaa.2017.11.041
triage: theorem
motivation_gids:
  - D5/S3/Analytic/QGammaTuran/Result.result
url: https://arxiv.org/abs/1702.03788

Refutation of reciprocal q-gamma coefficient positivity

Problem

S. I. Kalmykov and D. B. Karp, Inequalities for series in q-shifted factorials and q-gamma functions, J. Math. Anal. Appl. 460 (2018), 332–351, DOI 10.1016/j.jmaa.2017.11.041, arXiv:1702.03788v1, Conjecture 1, printed p. 12:

All conclusions of Theorem 3 are valid for all α > 0, while conditions on other parameters remain intact. In particular, µ → g(µ; x) is log-concave for each fixed 0 < x < Rg.

Theorem 3, printed p. 11:

For a real sequence {gₙ}ₙ≥₀ and fixed 0 < q < 1 define

$g(\mu ;x)=\sum _{n=0}^{\infty }\frac{g_n{x}^n}{{\Gamma }_q(\mu +n)}.$ (16)

Suppose {gₙ}∞ₙ₌₀ is doubly positive. Then (16) has a positive radius of convergence Rg and the generalized Turánian

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

is positive for 0 < x < Rg if α ∈ N, β > 0 and µ ≥ max(−β, −1). If, in addition, α ≤ β + 1, then Δg(α, β; x) has positive power series coefficients at all powers of x.

The sequence is nontrivial, nonnegative, log-concave and has no internal zeros, the definition of doubly positive on printed p. 3. Equation (6), printed p. 4, defines

$${\Gamma }_q(z)=(1-q{)}^{1-z}\frac{(q;q{)}_{\infty }}{(q^z;q{)}_{\infty }}.$$

Its reciprocal is extended by zero at the poles $z\in -{\mathbb{N}}_0$. Only the coefficient-positivity conclusion is refuted here.

Formal reading

The new Lean names have prefix D5.S3.Analytic.QGammaTuran.Result.

Motivation

The settling declaration is D5/S3/Analytic/QGammaTuran/Result.result. The source note is D5/L/Asymptotics/kalmykovkarp2018qgamma. The unit sequence meets the sequence hypotheses. The boundary triple $(-t,t,t)$ is permitted by the conjectured coefficient clause and exposes the same cancellation pattern as in the classical reciprocal Gamma problem.

Gap

Status: Refuted for the coefficient clause. The witness is $g_n=1$, $q=(99/100{)}^{10}$, $\mu =-7/10$, $\alpha =\beta =7/10$, $m=2$. Its shifts satisfy $\mu =\max (-\beta ,-1)$ and $\alpha \le \beta +1$. Theorem 3 itself concerns integral $\alpha$; the witness has $\alpha =7/10$, so it does not contradict that theorem. Function positivity and parameter log-concavity remain unaddressed.

Route

Write $R_q(x)=1/{\Gamma }_q(x)$ and $[x{]}_q=(1-q^x)/(1-q)$. The product definition yields $R_q(x)=[x{]}_q{R}_q(x+1)$, $R_q(0)=0$ and $R_q(1)=1$. At $t=7/10$, the degree-two Cauchy sum reduces to

$${\delta }_2=1-PC,P=R_q(3/10)R_q(7/10),$$

$$C=\frac{[-7/10{]}_q}{[7/10{]}_q[17/10{]}_q}+\frac{1}{[7/10{]}_q}+\frac{1}{[3/10{]}_q}.$$

For $r=99/100$ and $q=r^{10}$, every bracket appearing here is rational because $q^{n/10}=r^n$. The complementary exponents in $P$ give

$$P=\frac{(r^3;q{)}_{\infty }(r^7;q{)}_{\infty }}{(q;q{)}_{\infty }^2(1-q)}.$$

For $0\le a<1$, a partial product bounds $(a;q{)}_{\infty }$ from above. Its lower bound is the partial product multiplied by $1-a{q}^N/(1-q)$, using

$$\prod _{j\ge N}(1-a{q}^j)\ge 1-\frac{a{q}^N}{1-q}.$$

These bounds convert $PC>1$ to a finite rational comparison and yield ${\delta }_2<0$.

Falsifier

One permitted sequence and parameter choice with a nonpositive coefficient contradicts the universal strict-positivity clause. The displayed unit-sequence witness has a negative second coefficient. An error in the Cauchy reduction, the product-tail bound or its rational comparison would invalidate this argument without establishing the coefficient clause.

Evidence

D5/S3/Analytic/QGammaTuran/Result.result has the type result : ¬ claim, with no additional hypotheses. The sequence hypothesis is supplied by the existing D5.S3.Analytic.ReciprocalGammaTuran.Defs.admissible_one.

The experiments are available at the immutable q-gamma experiment directory. The following readings concern the scripts at that revision.

• COMPUTED. Command: python3 /Users/macstudio/omega-op/s3688/kk-qgamma-cert.py; exit code 0; script kk-qgamma-cert.py, SHA-256 a0b18ca90e4bfd31b2c5d3871f53212c3b53c9e66cbb3be2d04042ce85065f85. The script uses 800 product factors with outward rounding to multiples of $2^{-200}$ and the geometric tail bound. Its exact rational checks are $C>0$ and $P_{\mathrm{lower}}C>1064/1000$, hence $PC>1064/1000$. Its printed value for $1-P_{\mathrm{lower}}C$, approximately $-0.06428299060000457$, is a floating approximation; the asserted rational inequalities are exact.
• COMPUTED. Command: python3 /Users/macstudio/omega-op/s3688/kk-qgamma-scan.py; exit code 0; script kk-qgamma-scan.py, SHA-256 e1dcb5bf944905427dbdf16ea8b07267d074c4628cd75cbe14f421619c3b15c0. This finite scan uses 30 decimal digits and truncated products, with $q\in \{0.5,0.9,0.97\}$, $t=k/20$ for $1\le k\le 19$, $\alpha =\beta =t$, $\mu \in \{-t,-t+0.05,0,0.5\}$, retaining $\mu \ge \max (-t,-1)$, and degrees zero through four. It finds negative coefficients for $q=0.9$ and $q=0.97$ and none for $q=0.5$. The respective minimum values are approximately $-0.0636984$, $-0.07192$ and $0.000120102$; the negative minima occur at $(t,\mu ,m)=(0.7,-0.7,2)$ and the positive minimum at $(0.05,-0.05,4)$. These are finite numerical observations.

The supplied literature scope comprises the cited-by works recorded for OpenAlex W2603745271: arXiv:1808.05105 cites the source but supplies no settlement of Conjecture 1; the 2019 journal version adds no settlement in that scope; DOI 10.1142/s1793042120400187 treats a different q-binomial question. The classical reciprocal Gamma refutation, arXiv:2607.02511 and the Zenodo preprint on failure of parameter log-concavity concern $q=1$. No settlement was found in the supplied Formal Conjectures and conjectures.io scope. This bounded search does not establish exhaustive coverage or priority.

Triage

What the settlement shows

• PROVED. The universal coefficient clause is false: D5.S3.Analytic.QGammaTuran.Result.result. The witness uses the boundary cancellation at $(-t,t,t)$, $t=7/10$, and $q=(99/100{)}^{10}$. Thus the boundary failure seen in the classical $q=1$ problem persists at this $q$ close to one.
• COMPUTED. At this witness the exact rational certificate gives $PC>1064/1000$, as checked by kk-qgamma-cert.py with the command, exit code and SHA-256 in Evidence. The approximate printed bound is separate from the exact comparison.
• COMPUTED. The finite scan in Evidence finds no negative coefficient at $q=0.5$ through degree four on its stated grid.
• OPEN. The dependence of coefficient positivity on $q$, including any uniform interval near $q=1$ or a positive result at $q=0.5$, is not settled by this witness or finite scan.
• OPEN. The function-positivity and parameter log-concavity clauses of Conjecture 1 are not addressed. A negative coefficient alone does not establish failure of function positivity.
• OPEN. The remaining conjectural extensions, and any results requiring them, receive no conclusion from this coefficient counterexample. The integral-shift assertion of Theorem 3 is outside the witness’s parameter choice.

ASSUMED-UNVERIFIED

Exhaustive literature coverage, citation-index completeness and priority are unverified. The bounded literature scope is supplied bibliographic evidence; no current network search is asserted. The scan’s truncated product values are numerical observations, not rigorous signs on a continuous parameter range. Its absence of negative values at $q=0.5$ does not prove coefficient positivity there.