Incomplete beta coefficients

Abstract

Definitions for the coefficient form of the incomplete beta conjecture.

Definition 1.1 (Gamma coefficient).

Lean statement: D5/S3/Analytic/IncompleteBeta/Defs.coeff

Citation. D. B. Karp (2016). 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.

Commentary.

The coefficient indexed by j is Gamma(a+b+j) divided by the product Gamma(b)Gamma(a+j+1).

Definition 1.2 (Coefficient difference).

Lean statement: D5/S3/Analytic/IncompleteBeta/Defs.phi

Citation. D. B. Karp (2016). 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.

Commentary.

The nth coefficient difference is the finite convolution difference obtained from two parameter shifts.

Definition 1.3 (Positivity claim).

Lean statement: D5/S3/Analytic/IncompleteBeta/Defs.claim

Citation. D. B. Karp (2016). 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.

Commentary.

For positive real parameters, every finite coefficient difference is positive.