bibkey: kalmykovkarp2013logconcavity
authors: S. I. Kalmykov and D. B. Karp
year: 2013
title: Log-concavity for series in reciprocal gamma functions and applications
doi: 10.1080/10652469.2013.764874
url: https://arxiv.org/abs/1206.4814
claim: "The Moreover clause of Conjecture 1 asserts positive coefficients for the shifted reciprocal Gamma Turanian under Theorem 2's sequence hypotheses."
strata_touched:
  - D5/S3/Analytic/ReciprocalGammaTuran/Defs
  - D5/S3/Analytic/ReciprocalGammaTuran/Series
  - D5/S3/Analytic/ReciprocalGammaTuran/Result
license: citation-only
triage: anchor

Reciprocal Gamma Turanian coefficients

Verified locator

DOI: 10.1080/10652469.2013.764874

Source: https://arxiv.org/abs/1206.4814

Pinned version: https://arxiv.org/abs/1206.4814v1, Conjecture 1, equation (20), Theorem 2. Journal publication: Integral Transforms and Special Functions 24(11) (2013), 859–872.

Mathematical correspondence

Equation (20) defines

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

Theorem 2 assumes a nontrivial nonnegative log-concave sequence without internal zeros. Conjecture 1 asserts log-concavity in the positive parameter and, in its Moreover clause, positivity of all coefficients of

$$g(\mu +\alpha ,x)g(\mu +\beta ,x)-g(\mu ,x)g(\mu +\alpha +\beta ,x)$$

for $\alpha ,\beta >0$, $\mu \ge -1$, $\mu +\alpha ,\mu +\beta \ge 0$. The reciprocal Gamma value at a pole is zero.

Defs.rgamma, Defs.AdmissibleSeq, Defs.turanCoeff and Defs.claim encode these objects and the coefficientwise clause. Series.coeff_mul identifies the finite sum with the formal Cauchy coefficient. Boundary.d_two_neg gives a negative second coefficient for the unit sequence at every boundary shift $\mu =-t$, $\alpha =\beta =t$, $0<t<1$. Interior.interior_failure places a negative second coefficient strictly inside the stated parameter range. Result.result refutes only the Moreover clause; it does not address the positive-parameter log-concavity assertion.

The bounded literature scope and its limitations are stated in Problems/kalmykov-karp-2013-reciprocal-gamma-turanian-coefficients.md.