slug: karp-2016-incomplete-beta-first-parameter-coefficient-signs
bibkey: karp2016incompletebeta
doi: 10.1007/s10958-016-2958-z
triage: theorem
motivation_gids:
  - D5/S3/Analytic/IncompleteBetaShape/Result.result
url: https://arxiv.org/abs/1509.05120v1

First-parameter coefficient signs of the normalized incomplete beta function

Problem

Karp, Normalized incomplete beta function: log-concavity in parameters and other properties, §3, Conjecture 2:

Conjecture 2 The coefficients ψ_k are negative for all a, α, β > 0 and x ∈ (0, 1) if 0 < b < 1 and positive if b > 1, so that a → x^{−a}(1 − x)^{−b} I_x(a, b) is strictly coefficient-wise Wright log-convex on (0, ∞) for α, β > 0, x ∈ (0, 1) and 0 < b < 1; and a → x^{−a}(1 − x)^{−b} I_x(a, b) is strictly coefficient-wise Wright log-concave on (0, ∞) for α, β > 0, x ∈ (0, 1) and b > 1.

Equations (1) and (6) give

$$c_j(t,b)=\frac{\Gamma (t+b+j)}{\Gamma (b)\Gamma (t+j+1)},{\psi }_n=\sum _{k=0}^n[c_k(a+\alpha ,b)c_{n-k}(a+\beta ,b)-c_k(a,b)c_{n-k}(a+\alpha +\beta ,b)],$$

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

Formal reading

Unless otherwise indicated, Lean names have prefix D5.S3.Analytic.IncompleteBetaShape.

Motivation

The settling declaration is D5/S3/Analytic/IncompleteBetaShape/Result.result. The source note is D5/L/Asymptotics/karp2016incompletebeta. Coefficient signs constrain each term of the normalized series separately, strengthening the corresponding pointwise Wright inequalities.

Gap

Status: Proved for all positive real $a,b,\alpha ,\beta$ and every natural $n$, including zero. Both $0<b<1$ and $b>1$ are included; neither shift is restricted to integers.

Route

For $p,q,b>0$, put $c=p+q$ and

$$K(c,n,b)=\frac{(n+1)\Gamma (c+n+2b)}{\Gamma (b{)}^2\Gamma (c+n+2)},w_b(y)=[y(1-y){]}^{b-1}.$$

Termwise beta integration gives

$$S_n(p,q;b)=K(c,n,b){\int }_0^1{w}_b(y){\rho }_{p,q,n}(y)dy,$$

where ${\rho }_{p,q,n}$ is the uniform mixture of the beta densities with shapes $p+k+1,q+n-k+1$. This mixture is the law of $sv+(1-s)u$ for independent $v\sim \mathrm{Beta}(p,q)$, $s\sim \mathrm{Beta}(c,n+2)$ and uniform $u$. The map $(v,s)\mapsto (sv,s(1-v))$ has Jacobian $s$. At fixed output $y$, its domain becomes $0<x<y$, $0<z<1-y$; expanding $(1-x-z{)}^n=((y-x)+(1-y-z){)}^n$ and evaluating the resulting beta integrals proves the affine identity.

Define

$$F_s(v)={\int }_0^1{w}_b(sv+(1-s)u)du,H_{c,n,b}(v)=\frac{1}{B(c,n+2)}{\int }_0^1{s}^{c-1}(1-s{)}^{n+1}{F}_s(v)ds.$$

The kernel $H$ is continuous and bounded on $[0,1]$ and symmetric about $1/2$. For $0<s<1$ and $0<v<1$,

$$F_s^{\prime }(v)=\frac{s}{1-s}(w_b(r)-w_b(\ell )),\ell =sv,r=sv+1-s,$$

$$r(1-r)-\ell (1-\ell )=s(1-s)(1-2v).$$

Consequently $H$ strictly increases on $(0,1/2]$ for $b>1$ and strictly decreases there for $0<b<1$. The bound $|F_s(v)|\le B(b,b)/(1-s)$ gives an integrable majorant for the $s$ average, including the singular regime $0<b<1$.

The affine identity yields

$$S_n(p,q;b)=K(c,n,b)\frac{1}{B(p,q)}{\int }_0^1{v}^{p-1}(1-v{)}^{q-1}{H}_{c,n,b}(v)dv.$$

Apply the existing strict beta-mean comparison to $H$ for $b>1$ and to $-H$ for $0<b<1$. At fixed $p+q$, the convolution strictly decreases with $|p-q|$ in the first case and strictly increases in the second. The pairs $(a+\alpha ,a+\beta )$ and $(a,a+\alpha +\beta )$ have the same sum and half imbalances $|\alpha -\beta |/2<(\alpha +\beta )/2$. Their comparison gives both signs.

Falsifier

Positive real $a,b,\alpha ,\beta$ and natural $n$ with ${\psi }_n\ge 0$ when $0<b<1$, or ${\psi }_n\le 0$ when $b>1$, would contradict the statement. Zero parameters and zero shifts are outside its hypotheses; $b=1$ is outside both strict branches.

Evidence

D5/S3/Analytic/IncompleteBetaShape/Result.result proves exactly Defs.claim, with no additional hypotheses. Its principal dependencies are Result.convolution_strictAnti_imbalance and Result.convolution_strictMono_imbalance, which consume Mixture.convolution_eq_betaMean_H, Mixture.convolution_eq_mixtureMean, Mixture.mixture_eq_affine, Kernel.H_continuousOn, Kernel.H_bounded, Kernel.H_symm, Kernel.H_strictMonoOn_of_one_lt and Kernel.H_strictAntiOn_of_lt_one. The comparison is reused from D5.S3.Analytic.IncompleteBeta.Comparison.betaMean_strictAntiOn. Beta integration is reused from D5.S3.Analytic.KummerRatio.BetaIntegral and D5.S3.Analytic.KilbasSaigo.Bounds.FractionalCertificate. The geometric integral identity is reused from D5.S3.Quantum.Measurement.QubitIncompatibilityProbability.triangle_change_lintegral, restricted to a rectangle and with the two input coordinates exchanged. The axiom closure of Result.result is {propext, Classical.choice, Quot.sound}.

The proof route was derived with the assistance of the ChatGPT Pro model. The source’s Corollary 1 proves $0<b<1$ for all positive shifts and $b>1$ for $\alpha \in \mathbb{N}$, $\beta \ge \alpha -1$; it states the general case as Conjecture 2. No settlement was found in the supplied bounded literature scope: the source identifier and title with coefficient-wise log-concavity, log-convexity and incomplete beta functions; the thirteen citing works listed by OpenAlex, with the two concerning coefficient-wise log-concavity read (arXiv:1906.04975 and arXiv:1611.06724); and Kalmykov–Karp arXiv:1211.2882, Karp–Zhang arXiv:2305.09029 and Askitis arXiv:1710.09708. This scope does not establish exhaustive citation-index coverage or priority.

Triage

• PROVED. For every $a,\alpha ,\beta >0$ and natural $n$, ${\psi }_n<0$ if $0<b<1$, and ${\psi }_n>0$ if $b>1$, by Result.result.
• PROVED. The beta-mixture representation and the affine mixture identity, by Mixture.convolution_eq_mixtureMean and Mixture.mixture_eq_affine.
• PROVED. For $c,b>0$, $H_{c,n,b}$ is bounded, continuous and symmetric on $[0,1]$; it strictly increases on $(0,1/2]$ for $b>1$ and strictly decreases there for $0<b<1$, by Kernel.H_bounded, Kernel.H_continuousOn, Kernel.H_symm, Kernel.H_strictMonoOn_of_one_lt and Kernel.H_strictAntiOn_of_lt_one.
• PROVED in prose. For $b=1$, the Gamma quotient gives $c_j(t,1)=1$ for $t>0$, hence ${\psi }_n=0$. This boundary case is not asserted as an additional Lean theorem.
• PROVED in prose. At fixed $p+q$ with $p,q>0$, the uniform mixture of Beta$(p+k+1,q+n-k+1)$ has strictly smaller expectation of every continuous symmetric function strictly increasing toward $1/2$ as $|p-q|$ grows. Its affine average has the same strict midpoint order: the moving-window derivative compares the two endpoint values, whose distances from $1/2$ are strictly ordered before the midpoint. Beta averaging preserves strictness, and the existing beta-mean comparison applies. The affine identity is formalized; this general test-function conclusion is not an additional Lean theorem.
• OPEN. Nonuniform mixture weights are an open direction.

ASSUMED-UNVERIFIED

Exhaustive literature coverage, citation-index completeness and priority are unverified. Independent external-source confirmation of the supplied bounded literature findings is unverified.