bibkey: abel2013binomial authors: Ulrich Abel, Wolfgang Gawronski, and Thorsten Neuschel year: 2013 title: Binomial Polynomials doi: 10.1007/s40315-013-0013-3 url: https://doras.dcu.ie/31197/1/Binomialpolynomials.pdf claim: Theorem 3.1 with r=l-1 and z=a^l gives the complete weighted binomial power-sum asymptotic for fixed a>0 and integer l>=2; this is a denominator estimate, not a truncated-ratio maximum theorem. strata_touched:

• D5/S3/AnalyticClosure/BinomialPowerNormalization
• D5/S3/AnalyticClosure/BinomialPoweredRatioMaximum license: citation-only triage: anchor

Complete weighted binomial power sums

Theorem 3.1 and its proof apply with the paper’s parameter r=l-1>=1 and z=a^l>0. For fixed a>0 and integer l>=2 they give

[ D_n:=\sum_{i=0}^{n}(\binom ni a^i)^l \sim\frac{(1+a)^{ln+l-1}} {(2\pi n a)^{(l-1)/2}\sqrt l}. ]

For l=1, D_n=(1+a)^n exactly by the binomial theorem; the displayed expression then agrees with that identity.

Put p=a/(1+a). For 0<=i<=n, binomialMass(p,n,i)=choose(n,i) a^i/(1+a)^n. Dividing the complete weighted sum by (1+a)^(ln) gives the equivalent probability normalization

[ \sum_{i=0}^{n}\operatorname{binomialMass}(p,n,i)^l \sim\frac{(2\pi n p(1-p))^{(1-l)/2}}{\sqrt l}. ]

Conversely a=p/(1-p)>0 recovers every 0<p<1. Thus the arbitrary positive weight in the repository normalization is a classical result. Its use at a growing row r requires r to tend to infinity; application along maximizing sequences additionally uses their positive limiting slope.

This source evaluates a complete row. It does not by this estimate alone locate a maximizing truncation index or evaluate the maximum of a ratio whose numerator is truncated in a different row. Those are distinct obligations of Conjecture 1.1(d).

Verified locator

• DOI: 10.1007/s40315-013-0013-3
• https://doras.dcu.ie/31197/1/Binomialpolynomials.pdf, Theorem 3.1 and its proof.

The bounded audit supplied with #9357 read all 18 pages, including the theorem and proof. Crossref bibliographic metadata was consulted on 2026-09-21 UTC for author, title and 2013 publication identity. No new discovery is claimed for this denominator asymptotic.

D5/L/Analytic/luca2012some records a different, unweighted complete-sum supplier. Its restatement of McIntosh is not the evidence for arbitrary positive a. The original McIntosh 1996 body (DOI 10.1006/jnth.1996.0072) and the Binomial mean body (DOI 10.1080/02331888.2026.2631025) were not inspected in that audit; no stronger exclusion claim is made about them.