bibkey: karp2016incompletebeta
authors: D. B. Karp
year: 2016
title: "Normalized incomplete beta function: log-concavity in parameters and other properties"
doi: 10.1007/s10958-016-2958-z
url: https://arxiv.org/abs/1509.05120v1
claim: "Conjecture 1 asserts strict positivity of every coefficient in the normalized incomplete beta Turanian for positive real parameters and shifts."
strata_touched:
  - D5/S3/Analytic/IncompleteBeta/Defs
  - D5/S3/Analytic/IncompleteBeta/Moment
  - D5/S3/Analytic/IncompleteBeta/Comparison
  - D5/S3/Analytic/IncompleteBeta/Result
license: citation-only
triage: anchor

Coefficient-wise Wright log-concavity of normalized incomplete beta functions

Verified locator

DOI: 10.1007/s10958-016-2958-z

Source: https://arxiv.org/abs/1509.05120v1

Pinned source: §3, Conjecture 1; definitions (1), (3), (4). The integer-shift case is proved via Lemma 2 and Theorem 3. Pinned source: §3, Conjecture 2, equation (6), and Corollary 1; equation (1) supplies the coefficients. Journal publication: Journal of Mathematical Sciences 217(1) (2016), 91–107.

Mathematical correspondence

Equation (1) writes the normalized incomplete beta function as

$$x^{-a}(1-x{)}^{-b}{I}_x(a,b)=\sum _{j\ge 0}{c}_j(a,b)x^j,c_j(a,b)=\frac{\Gamma (a+b+j)}{\Gamma (b)\Gamma (a+j+1)}.$$

Equations (3) and (4) define the Turanian and its coefficients:

$${\phi }_{a,b,\alpha ,\beta }(x)=I_x(a,b+\alpha )I_x(a,b+\beta )-I_x(a,b)I_x(a,b+\alpha +\beta )=x^{2a}(1-x{)}^{2b+\alpha +\beta }\sum _{n\ge 0}{\phi }_n{x}^n,$$

$${\phi }_n=\sum _{k=0}^n[c_k(a,b+\alpha )c_{n-k}(a,b+\beta )-c_k(a,b)c_{n-k}(a,b+\alpha +\beta )].$$

coeff, phi and claim in D5/S3/Analytic/IncompleteBeta/Defs encode these coefficients and their strict positivity for all positive real $a,b,\alpha ,\beta$ and all natural $n$. The coefficients do not depend on $x$; the source’s restriction $x\in (0,1)$ imposes no further coefficient condition.

Moment.convolution_eq_betaMean represents the coefficient convolution as a beta moment. Moment.W_hasDerivAt supplies the telescoping derivative of its symmetric kernel. Comparison.betaMean_strictAntiOn compares symmetric beta means as the shape imbalance increases. Result.convolution_strictAnti_imbalance applies this comparison at fixed shape sum, and Result.result proves the full coefficient claim.

The source proves the restricted case $\alpha \in \mathbb{N}$, $\beta \ge \alpha -1$ and states the general positive-shift case as Conjecture 1. The bounded literature scope and its limits are given in Problems/karp-2016-incomplete-beta-coefficient-log-concavity.md.