Metadata
bibkey: ulgenes2026a398064
authors: David Ulgenes
year: 2026
title: OEIS A398064, truncated-binomial limit for Hadamard's Gamma function
doi: null
url: https://oeis.org/A398064
claim: "Conjecture: H(x+1) = lim_{n->oo} n^{x+1} * Sum_{k=0..n} (-1)^k/(n+k) * binomial(x, k) such that H(1/4) = 0.79564036... . Changing the upper bound of the sum to oo yields Euler's product representation of the Gamma function."
strata_touched:
- D5/S3/Weil/ZetaGamma/HadamardTruncatedBinomialLimit
license: citation-only
triage: anchor
Verified locator
DOI: null
Source: https://oeis.org/A398064
The unpaginated FORMULA field, revision 39 (2026-08-03 01:00:24), states:
Conjecture: H(x+1) = lim_{n->oo} n^{x+1} * Sum_{k=0..n} (-1)^k/(n+k) * binomial(x, k) such that H(1/4) = 0.79564036… . Changing the upper bound of the sum to oo yields Euler’s product representation of the Gamma function.
The COMMENT field states:
The Hadamard function is an entire extension of the factorial function not equal to the gamma function. In other words, H(n) = (n-1)! for positive integers n.
For the source function, the entire continuation of
(digamma(1-y/2)-digamma((1-y)/2))/(2*Gamma(1-y)) is used, with
factorial values at positive integers. The quotient also follows from equation
(4.1) in Yin–Zhang, On some properties of special functions involving k-gamma
and k-digamma functions, https://arxiv.org/abs/2502.15852v1, by the chain rule.
Ulgenes’s related Gamma-function article is https://arxiv.org/abs/2301.09699v4,
journal DOI 10.5206/mt.v5i1.16090; it does not supply the OEIS conjecture’s
source, and no arXiv identifier for this conjecture was identified.
The formal assertion ranges over all complex parameters. Generalized binomial
coefficients are Ring.choose; the positive-integer power uses
Complex.exp ((x+1) * (Real.log n : Complex)). The finite sum includes both
endpoints, k=0 and k=n. The value at n=0 is irrelevant to the limit.
Existence of the entire source function is a conclusion of the assertion.
The quoted decimal example and the separate infinite-sum sentence are not
additional conclusions of the settling theorem.
The proof combines Euler’s coefficient limit, a continuous weighted Riemann
sum on Re(x)<-1, identification of the Nielsen-beta integral, and an exact
finite-sum shift which extends convergence to every complex parameter.
Literature completeness and worldwide novelty are ASSUMED-UNVERIFIED. The finite source screen and the independent exact-arithmetic experiment are specified in https://github.com/the-omega-institute/trureturing/issues/22706.