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First parameter coefficient differences

Abstract

Coefficient inequalities for real shifts of the first incomplete beta parameter.

Definition 1.1 (Coefficient difference).

Lean statement: D5/S3/Analytic/IncompleteBetaShape/Defs.psi

Formalization. D5/S3/Analytic/IncompleteBetaShape/Defs.psi (✓ std3).

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 is the sum, for k from zero to n, of c_k(a+alpha,b)c_(n-k)(a+beta,b) minus c_k(a,b)c_(n-k)(a+alpha+beta,b). The Gamma coefficient c is reused from the incomplete beta coefficient definitions. By the series expansion and product identity, this is the coefficient of x^n after division of the product difference by x^(2a+alpha+beta)(1-x)^(2b). It is independent of x, so the source’s restriction on x imposes no further condition on the coefficient.

Definition 1.2 (Strict signs for positive real shifts).

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

Formalization. D5/S3/Analytic/IncompleteBetaShape/Defs.claim (✓ std3).

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 every positive a, b, alpha and beta and every natural n, the coefficient difference is negative when b is less than one and positive when b is greater than one.

References