Refutation of reciprocal Gamma coefficient positivity

S. I. Kalmykov and D. B. Karp, Log-concavity for series in reciprocal gamma functions and applications, arXiv:1206.4814v1, Conjecture 1:

Under hypotheses of Theorem 2 the function µ → g(µ, x) is log-concave on (0, ∞) for each fixed x ≥ 0. Moreover, the function x → g(µ + α, x)g(µ + β, x) − g(µ, x)g(µ + α + β, x) has positive power series coefficients for µ ≥ −1 and µ + α ≥ 0, µ + β ≥ 0, where α, β > 0.

Theorem 2's sequence hypotheses are nontriviality, nonnegativity, log-concavity $g_{n-1}g_{n+1}\le g_n^2$ for $n\ge1$, and no internal zeros. Equation (20) is

$$g(\mu,x)=\sum_{n=0}^{\infty}\frac{g_n x^n}{\Gamma(n+\mu)},$$

where reciprocal Gamma is zero at its poles. Only the “Moreover” clause is refuted; the first assertion is not addressed.

Formal reading

All Lean names below have prefix D5.S3.Analytic.ReciprocalGammaTuran.

| Source object | Lean name | Correspondence | | --- | --- | --- | | $1/\Gamma(s)$ | Defs.rgamma | Inverse real Gamma, zero at nonpositive integers. | | Theorem 2's sequence hypotheses | Defs.AdmissibleSeq | Nontrivial, nonnegative, log-concave, without internal zeros. | | Equation (20)'s formal series | Series.formalSeries | Degree $n$ coefficient $g_n/\Gamma(n+\mu)$, with no factorial weight. | | Degree $m$ Turanian coefficient | Defs.turanCoeff, Series.coeff_mul | Exact finite Cauchy sum over $0\le k\le m$. | | Moreover clause | Defs.claim | Every admissible sequence and all real shifts in the printed range, at every natural degree. | | Boundary failure | Boundary.d_two_neg | For every $0<t<1$, degree two is negative at $(-t,t,t)$. | | Interior failure | Interior.interior_failure | Positive shifted parameters and a negative second coefficient. | | Refutation | Result.result | Exactly $\neg,\texttt{Defs.claim}$. |

The settling declaration is D5/S3/Analytic/ReciprocalGammaTuran/Result.result. The source note is D5/L/Asymptotics/kalmykovkarp2013logconcavity. The constant unit sequence satisfies every sequence hypothesis. Its boundary shifts expose a negative coefficient, and continuity shows that this failure persists strictly inside the permitted range.

Status: Refuted. The second coefficient fails for the unit sequence at every boundary triple $(-t,t,t)$ with $0<t<1$, and also at strictly interior parameters. The first assertion on log-concavity for positive parameters is not addressed. Positivity of every coefficient throughout the principal range $\mu\ge0$ is not settled here.

Write $F_b(x)=\sum_{n\ge0}x^n/\Gamma(n+b)$ and $D_t(x)=F_0(x)^2-F_{-t}(x)F_t(x)=\sum_m d_m x^m$. The finite Cauchy identity gives

$$d_2=1-\frac{1+t^2}{t(1-t^2)\Gamma(t)\Gamma(1-t)}.$$

The formal proof uses Euler reflection and the strict bound

$$\sin(\pi t)>\frac{\pi t(1-t^2)}{1+t^2}\qquad(0<t<1).$$

For $u=t^2$, the finite Euler product is at least $(1-u)(1-u/4)(1+u/(n+2))/(1+u/2)$. The induction follows from

$$(1+u/a)(1-u/(a+1)^2)-(1+u/(a+1)) =\frac{u(1-u)}{a(a+1)^2}\ge0.$$

Its limit gives the displayed strict sine bound, hence $d_2<0$. The entire reciprocal Gamma function makes the second Cauchy coefficient jointly continuous. Increasing $\mu$ by a sufficiently small positive amount preserves negativity and makes both shifted parameters strictly positive. The unit sequence then contradicts universal positivity.

The all-degree formula has the following separate prose proof. Put $C=1/(\Gamma(t)\Gamma(1-t))$, $f(v)=v^{1-t}-v^{1+t}$ and $A(s)=\int_0^1v^{s-1}/(1+v),dv$ for $s>0$. Splitting the beta integral on $(0,\infty)$ at one and substituting $v=1/u$ in its tail gives $A(t)+A(1-t)=\Gamma(t)\Gamma(1-t)$. Adding integrands gives $A(s)+A(s+1)=1/s$. Consequently

$$I_2:=A(2-t)-A(2+t) =\frac{1+t^2}{t(1-t^2)}-\Gamma(t)\Gamma(1-t), \qquad d_2=-C I_2.$$

The eventual ratio of successive terms of $F_b$ is $x/(n+b)$, so its radius is infinite and termwise differentiation is valid. Gamma recurrence, applied coefficientwise, gives $xF_b'=(x+1-b)F_b+(b-1)/\Gamma(b)$ and $xD_t'=2(x+1)D_t+R_t$. Thus, for $m\ge3$,

$$(m-2)d_m=2d_{m-1}+r_m,\quad r_m=\frac{t+1}{\Gamma(-t)\Gamma(m+t)} +\frac{1-t}{\Gamma(t)\Gamma(m-t)}.$$

The beta integral and integration by parts, using $f(0)=f(1)=0$, yield

$$r_m=\frac{C}{(m-2)!}\int_0^1 f'(v)(1-v)^{m-2},dv =\frac{C}{(m-3)!}\int_0^1 f(v)(1-v)^{m-3},dv.$$

For $I_m=\int_0^1 f(v)(1-v)^{m-2}/(1+v),dv$, the identity $(1-v)/(1+v)=2/(1+v)-1$ now proves inductively $d_m=-C I_m/(m-2)!$. Every integral is finite and strictly positive: $f(v)>0$ for $0<v<1$ and $0<t<1$. This argument is a prose result; the Lean proof uses only the second coefficient and continuity.

A single admissible sequence and admissible parameter triple with a negative coefficient refutes the universal clause. The unit sequence and the strictly interior triple supplied by Interior.interior_failure provide such a counterexample. An error in the Gamma evaluation or strict sine bound would invalidate this route without proving the clause.

D5/S3/Analytic/ReciprocalGammaTuran/Result.result has exactly the type result : ¬ claim, with no additional hypotheses. Its principal dependencies are Defs.admissible_one, Interior.interior_failure, Interior.continuous_turanCoeff_two, Boundary.d_two_neg, Series.turanCoeff_one_boundary and Series.coeff_mul. The boundary argument uses a finite Euler-product lower bound and Mathlib's reflection and sine-product theorems. The result's axiom closure is exactly {propext, Classical.choice, Quot.sound}. The coefficient identification is a formal power-series identity; no separate Lean theorem for convergence of $F_b$ or its differential recurrence is asserted by these modules.

The refutation route was derived with the assistance of the ChatGPT Pro model. No settlement of this clause was found in the searched scope: arXiv:1611.06724 §3 reports numerical counterexamples for a different numerator-and-denominator shift family with nonnegative parameters; the 2017 q-gamma paper, Karp–Zhang (2024), DOI 10.1007/s13540-023-00238-0, Formal Conjectures and conjectures.io did not supply a settlement of this clause in that scope. The existing Karp–Zhang quadratic-truncation and Karp–Sitnik Kummer results concern different statements. This bounded scope does not establish exhaustive citation-index coverage or priority.

• PROVED. For $g_n\equiv1$ and every $0<t<1$, the second coefficient at $\mu=-t$, $\alpha=\beta=t$ is negative, by Boundary.d_two_neg.
• PROVED. Failure occurs at parameters strictly inside the range $\mu+\alpha>0$, $\mu+\beta>0$, by Interior.interior_failure.
• PROVED in prose. At the boundary every coefficient of degree $m\ge2$ is negative: $d_m=-\frac{C}{(m-2)!}\int_0^1 \frac{v^{1-t}-v^{1+t}}{1+v}(1-v)^{m-2},dv$, where $C=1/(\Gamma(t)\Gamma(1-t))$ and $0<t<1$, as derived above.
• COMPUTED. Double-precision coefficients of degrees zero through three for $(\mu,\alpha,\beta)=(-0.5,0.6,0.6)$ are $0.228,0.097,-0.0018,-0.0144$ after rounding. For $\mu\in{0,0.5}$ and $\alpha=\beta=0.5$, all computed coefficients of degrees zero through twenty are positive. These finite computations are separate from the Lean proof.
• OPEN. For the unit sequence $g_n\equiv1$, coefficientwise positivity for all $\mu\ge0$ and $\alpha,\beta>0$ is left open here. Strict positivity cannot hold for every admissible sequence: $g_0=1$ and $g_n=0$ for $n>0$ is admissible and gives zero coefficients in every positive degree.

Exhaustive literature coverage, citation-index completeness and priority are unverified. The literature scope above is supplied bibliographic evidence; the external texts were not independently reread here. The all-degree integral formula is proved in prose, not Lean. Numerical coefficient signs are finite double-precision observations, not a proof for the principal range.