Central binomial sums

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Numbers

$p$

$v$

$U_p(v)$

-1/4:

0.8813735870195430252326093249797923090281603282616354107532956086533771842220260878337068919102560429

-1/16:

0.9624236501192068949955178268487368462703686687713210393220363376803277352164435488240188582454469500

1/16:

1.047197551196597746154214461093167628065723133125035273658314864102605468762069666209344941780705689

1/4:

1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534

-1/4:

0.9552018064811796875605004268114980826827146158467533568885747408977197436114529656525358876794355848

-1/16:

0.9869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232

1/16:

1.014941606409653625021202554274520285941689307530299792017489106776597476258244022136470354228256695

1/4:

1.088793045151801065250344449118806973669291850184643147162897626597154274588370993215164480805331513

-1/4:

0.9838252138836616460628098791198589480758749379730996786792853964597012320617297570917297319287997083

-1/16:

0.9955449411856964841153790906857533571143144581374033780919333659230512654627322583057339268206608360

1/16:

1.004833040565271950131176877174582251923967685005535897495424845956688724177129362328550496426945386

1/4:

1.023311012236370323084820504088486738318720976747328130305134276362953343975608668292344233816456263

-1/4:

0.9943271106154154887954544270937045792159540381562342311864655850637674637899765430428815792344467527

-1/16:

0.9984924045556103701808511247846968234045573923714654389211357998163168424037791481666472725830736612

1/16:

1.001582924969814819367464784367211790099493882906594736472677321125251673667133708781270643728176106

1/4:

1.006980484962515587025070688445999330918118384295773651209766835876673837577595969340078471098043616

-1/4:

0.9980473181903847174065170414420028068014868338008833262783208097233892917909706354151969963895727752

-1/16:

0.9994928229150153079200361224722996041992325833394765402161215791879161706982422505470928220812315527

1/16:

1.000522213563204775625654748484651116459535350443596459245900865930804159240153393530373274437162507

1/4:

1.002203585872759559997017624380616213130668888464329679853285683698638721755834273552229858095742778

-1/4:

0.9993358822162093746956385001731891142723242678513204447854330809726415087498500221476422513236505772

-1/16:

0.9998299926173062913482106703349160502459813622588986399277395636816438148066358682357663682804502693

1/16:

1.000173011425460834795874071993273012362074375369178908184741266219207200832958503781878857110818386

1/4:

1.000713262948214430433721594324683604878979177572049772308877373807629083763550958366363095629669028

-1/4:

0.9997758441026676688222478787702458917192455622121885048232668577237749084007152605153780856238464526

-1/16:

0.9999431383614082242315079296965376550260764566746442542294722812539558620859239615135462441478134412

1/16:

1.000057462086994703933896442244001156002741635158095764874926195453616047628019149454967331646328377

1/4:

1.000233878660531615762254662041722728149441939340479880631296151618480646912013248318424457321221914

-1/4:

0.9999247033158900087344870704261393632148508921114989973399036544195076241337226670865388744523731170

-1/16:

0.9999810072029657065547101992999395280318109107597452577540030839770704772911349289926396802836415776

1/16:

1.000019112846796051252646382769157773043007221544373059120541566491888865142883785833062963085452911

1/4:

1.000077230010084646277959187875751642812691368270676540613530845224475295712330764352689218263997627

-1/4:

0.9999747822427304138475401983370559790460978760389979999261179813473985302998233289467208460732168595

-1/16:

0.9999936612237311477051515489379149809299939453066438590039236061683221617327802010037056849924042445

1/16:

1.000006362781793274537476863345951078511714109176293127350518859311849335803536443576823230430509939

1/4:

1.000025603215265060247446005817831074587873015376668825737746291876365450929604868246194867089178728

Definition

The central binomial sum is $U_p(v)=\sum_{n=0}^{\infty}\binom{2n}{n}v^n/(2n+1)^p$, for an integer $p\geq1$ and rational $-1/4\leq v\leq1/4$ [1], Lemma 2.1. The endpoint values are ordinary absolutely convergent sums.

Parameters

$p$

— denominator power ($p\geq1$)

$v$

— series argument ($-1/4\leq v\leq1/4$)

Formulas

(1)

$U_p(v)=\dfrac1{(p-1)!}\int_0^1\dfrac{(-\log x)^{p-1}}{\sqrt{1-4vx^2}}\,dx$, by expanding the binomial kernel and integrating termwise.

(2)

$U_1(v)=\arcsin(2\sqrt v)/(2\sqrt v)$ for $v>0$, and $U_1(v)=\operatorname{arsinh}(2\sqrt{-v})/(2\sqrt{-v})$ for $v<0$. In particular $U_1(1/4)=\pi/2$, $U_1(1/16)=\pi/3$, and $U_2(1/4)=\pi\log(2)/2$.

(3)

$\sum_{m=0}^{\infty}U_{m+1}(1/4)h^m=\tfrac12 B((1-h)/2,1/2)$ for $|h|<1$, where $B$ is the beta function. This follows from the moment integral and Euler's beta integral [3].

(4)

$U_p(1/16)=(-1)^p\operatorname{Ls}_p(\pi/3)/(p-1)!$, where $\operatorname{Ls}_p(\theta)=-\int_0^\theta\log^{p-1}|2\sin(t/2)|\,dt$ is the basic log-sine integral [1], Lemma 2.1.

Comments

(5)

The initial term is $1$. A negative $v$ supplies the sign $(-1)^n$; no factor of $\pi$ or factorial is divided out. In [1], equations (2.2) and (2.3), $v=\pm z^2$ and the series is $zU_{p+1}(\pm z^2)$ in the notation used there.

(6)

At $|v|=1/4$, the absolute summands are $O(n^{-p-1/2})$. At $v=0$, $U_p(0)=1$. Natural logarithms, positive real square roots, and the real principal inverse sine and inverse hyperbolic sine are used.

(7)

The arguments $v=\pm1/16$ occur in the Apéry-like identities in [1], equations (1.1) and (1.2). Their moment integrals have endpoints $\pi/6$ and $\operatorname{arsinh}(1/2)=\log((1+\sqrt5)/2)$. The arguments $v=\pm1/4$ are the two real convergence boundaries.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130

def central_binomial_sum(p, v):
    # Hypergeometric form of the defining series.
    half = mp.mpf(1) / 2
    return mp.hyper([half] * (p + 1), [half + 1] * p, 4 * v)

central_binomial_sum(4, mp.mpf(1) / 16)

References

[1]

S. Charlton, H. Gangl, L. Lai, C. Xu and J. Zhao, On two conjectures of Sun concerning Apéry-like series, Forum Mathematicum 35 (2023), 1533–1547. (arXiv) (doi)

Links

[2]

Charlton et al.: integral representations in Lemma 2.1

[3]

DLMF 5.12.1: Euler's beta integral

Similar tables

Generalized log-sine integrals — The basic log-sine moments give $U_p(1/16)$ by the factorial relation above; other positive arguments give linear combinations of these moments.

Values of the beta function — The complete beta function generates the positive-endpoint values as Taylor coefficients.

Inverse binomial sums — These sums place $\binom{2n}{n}$ in the denominator and use powers of $n$, defining a different family.

Generalized log-sinh integrals — The basic log-sinh moments give $U_p(-1/16)=(-1)^p\operatorname{Lsh}_p^{(0)}(2\log\varphi)/(p-1)!$, where $\varphi=(1+\sqrt5)/2$, by [1], Lemma 2.1. The stored $k=0$ moments at $x=2\log\varphi$ cover $2\leq p\leq9$ here; the $p=1$ case uses the elementary boundary $\operatorname{Lsh}_1^{(0)}(x)=-x$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every $1\leq p\leq9$ at $v=-1/4,-1/16,1/16,1/4$. The interior pair comes from the Apéry-like identities and the outer pair consists of the real convergence boundaries. The noncanonical order cutoff covers both parities and successive moments, including the unweighted orders needed through order eight in the harmonic-index shift. The attached argument-selection.md explains each parameter family)

How they were obtained:

All 36 entries carry 100 significant digits. The generator uses 944-bit real ball arithmetic. Interior sums have an explicit geometric remainder; the negative endpoint uses the Euler transform of a Hausdorff moment sequence, with remainder at most $2^{-432}$. The positive endpoint uses finite Taylor coefficients of the beta-function generating series, expressed through enclosed values of $pi$, $log2$ and $zeta(k)$. The derivations and cancellation safeguards are in computation.md.

Every value was independently enclosed by 450-bit analytic quadrature of a transformed logarithmic-moment integral, with an explicit exponential tail. The whole independent enclosure must lie inside the stored decimal's last-place interval. Known first-order values and the second-order positive endpoint are controls; missing-initial-term, wrong-sign, last-place perturbation, vacuous-radius and NaN controls must fail. The two implementations share Arb arithmetic but no numerical generating code. Executable checks, numerical results and the generating source are attached.