Inverse binomial sums

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Numbers

$s$

$u$

$F_s(u)$

0.5483113556160754788241383888820083964063166337355994792451860764566691568010669579445429668732529018

-1:

-0.4631296411543887849938581424630655200328127000985977416306024573795906911339236259701090941727767901

1.233700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029

2.193245422464301915296553555528033585625266534942397916980744305826676627204267831778171867493011607

4.934802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116

0.5229461921333351084911851835273035401630445917439778414659410141442073577644132993150426219131503384

-1:

-0.4808227612638377141598952646045799963059945169361995527169086221367352823145252360745823494437341033

comment: This is $-2\zeta(3)/5$, the negative of the usual positive Apéry sum [2].

1.103227363928688220571587832459512954228919266581326961257118255909230019099577148559227854946255627

1.771169181407037230943036872078618205271213420369788871109638152425418642996557316552465917926061220

2.633889302798536545948395588664021103622327024996167429051181404820683518670672911053354548252519978

0.5110970825858152571047795233666626207547435050727321508502943239597236243105130664296517725280249609

-1:

-0.4901504329100475917249234105122572748097435257624383907889806675297793100638272250799253757956519391

1.047777280122816409261760603286409627082771025200759690057933473185174508907716880894051276660984827

1.617611689492181150364716432299309922885052024029682982223263247146071178923259170206953574506230103

2.236996286580134933332549651932203367454564761254972072835817328715041667153286712779398038717478566

0.5054294746835192416424504819084321491886690145682628649826647128757334733761759068271645331815001366

-1:

-0.4949846230206228983815193778955691314921605399050327449064630966822374523851063367840481577116760799

1.022755267803785750376730943273404688354432667068883247310725938853217899379517477521620256067978051

1.554037403737046884947562351938429426947399325371733825493102240586390101142679547263076270772411181

2.102666835144012078235689356487622385299818263615306695829492830865244814552924807845760847212326008

0.5026765214782692864546774599793486396646026000916406614686276523248716150885463121176234157278405277

-1:

-0.4974611657018299216433527284831315831702971040695227167898058934307181493217782967115681780606262935

1.011031199276579007687632475172164427837230644694660073502140394418017001770330122324756990927444156

1.525659197131678634365495146632820810783492284581013269078792526982095226975969936767089371108734787

2.047390002276735258924252608294108832958203469277849180080901523204939272328325324936972276015531316

0.5013258726881788094022967105527494437268783298580456815364512173388874158450606533090311388979439896

-1:

-0.4987199543525685781880163568888465132631258581777355432577882801423253098934833303943273923146759321

1.005407113737221019566043140833801299911265789707118931790502134035681338738502134805375563971838097

1.512422978068136225838809623501238692212223445803527934931731391485909580069422734580699807585588961

2.022598283657701933158632328951172061837238714098248634607028638576263814444354723111091055527271194

0.5006588912976705433145571270829868383840732523404540388886438047662171820334135876545662709081516772

-1:

-0.4993563707008741447136376779463078233267948055648344631578579729527431987380888114212049157383206676

1.002668994997103789243807827115554410255543087023400400326334871131994175817377333296855183159578910

1.506085886705820964281216219973358278716680516232019470842664171838709846213432912906373581952988711

2.010974618473109640062378390184080528850955814680965602754534803042623448281872323281997836604386264

0.5003281177391752247382483935315136287554736542251884687920478714940897528936611019380111637574346821

-1:

-0.4996769668652078688773129922868732106201851408310726974546407622080104688362847321268116368364733461

1.001323350365168650000704199292988946663855170237330038194836631439811941065001962642833790688356551

1.503003281880716477298718932027912008613192890342249428877039071552908087908571344282073442773563284

2.005387562397422748831026227909584257870500922378320305935853404700627478047809811972922874698328867

0.5001636212203339271396019065779786077705578513264313962760629919422764431848008832064633924682447169

-1:

-0.4998380731032202514816335591825282621612818289779626653112364362843572449523320752591490271882424069

1.000658047987443113434047881554549548652124364136637967495107626045145177342392420943460698431866515

1.501488924340201145774763804596872701245391668201325964549486499350479710936285662353543220713992637

2.002662360268212873712345741447323367385785501188768557629846525418279546202860689350980853604487065

Definition

The inverse central binomial sum is $F_s(u)=\sum_{j=1}^{\infty}u^j/(j^s\binom{2j}{j})$, for an integer $s\geq2$ and rational $-4\leq u\leq4$ [1]. The endpoint values are ordinary absolutely convergent series.

Parameters

$s$

— denominator power ($s\geq2$)

$u$

— series argument ($-4\leq u\leq4$)

Formulas

(1)

$F_2(u)=2\arcsin^2(\sqrt{u}/2)$ for $0\leq u\leq4$, and $F_2(u)=-2\operatorname{arsinh}^2(\sqrt{-u}/2)$ for $-4\leq u<0$. The inverse functions are real principal branches.

(2)

$F_s(u)=-\sum_{i=0}^{s-2}\dfrac{(-2)^i(\log u)^{s-2-i}}{i!(s-2-i)!}\operatorname{Ls}_{i+2}^{(1)}(\theta)$ for $0<u\leq4$, with $\theta=2\arcsin(\sqrt{u}/2)$ [1], equation (1.2). Here $\operatorname{Ls}_n^{(k)}(\theta)=-\int_0^\theta v^k\log^{n-1-k}|2\sin(v/2)|\,dv$ is the generalized log-sine integral, and $\operatorname{Ls}_2^{(1)}(\theta)=-\theta^2/2$. A zeroth power is $1$, including when $\log u=0$.

(3)

$F_3(-1)=-2\zeta(3)/5$ and $F_4(1)=17\zeta(4)/36$ [2], with $\zeta(3)$ and $\zeta(4)$ in the Riemann zeta table.

(4)

$F_3(1)=\dfrac{2\pi}{3}\operatorname{Cl}_2(\pi/3)-\dfrac43\zeta(3)$ and $F_5(1)=2\pi\operatorname{Cl}_4(\pi/3)-\dfrac{19}{3}\zeta(5)+\dfrac23\zeta(3)\zeta(2)$ [2], Theorem 1. The Clausen functions of even order are $\operatorname{Cl}_{2m}(\theta)=\sum_{j=1}^\infty\sin(j\theta)/j^{2m}$.

Comments

(5)

The power $u^j$ appears in the numerator, with no factor of $4$ absorbed into $u$. In [1], the hypergeometric argument is $z=u/4$; the table stores the unscaled sum.

(6)

At $u=1$, $F_s(1)$ is the unweighted central binomial sum $S(s)$. At $u=-1$, $F_s(-1)=-A(s)$, where the usual positive Apéry sum is $A(s)=\sum_{j\geq1}(-1)^{j+1}/(j^s\binom{2j}{j})$ [2].

(7)

The series converges absolutely throughout the stated domain, including both endpoints $u=\pm4$, because its terms in absolute value are asymptotic to $\sqrt{\pi}\,j^{1/2-s}$ there. In particular, $F_s(0)=0$.

(8)

Inverse binomial sums occur in expansions of massive Feynman diagrams [1]. The slices $u=1,3$ are associated with the odd arithmetic basis discussed there, and $u=2$ with the even basis. These application families are retained through the same finite prefix of denominator powers.

(9)

For $0<u\leq4$, put $\theta=2\arcsin(\sqrt{u}/2)$, using the real principal inverse sine. The arguments $u=1,2,3,4$ correspond to $\theta/\pi=1/3,1/2,2/3,1$ in the generalized log-sine integrals. Natural logarithms and radian angles are used throughout.

Programs

(P1)

Python (mpmath)

import mpmath as mp

def inverse_binomial(s, u, dps=120):
    # Numerical approximation; the attached generator supplies rigorous bounds.
    with mp.workdps(dps):
        u = mp.mpf(u)
        def order_two(z):
            if z >= 0:
                return 2*mp.asin(mp.sqrt(z)/2)**2
            return -2*mp.asinh(mp.sqrt(-z)/2)**2
        if s == 2:
            return order_two(u)
        return mp.quad(lambda y:
            2*y**(2*s-5)*order_two(u*mp.exp(-y*y))/mp.factorial(s-3),
            [0, 1, 2, 4, 8, 16, mp.inf])

inverse_binomial(3, -1)

References

[1]

A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums. Introduction, equations (1.1)–(1.4), and section 4. (arXiv)

[2]

Jonathan M. Borwein, David J. Broadhurst and Joel Kamnitzer, Central Binomial Sums, Multiple Clausen Values and Zeta Values. Experimental Mathematics 10 (2001), 25–34. Introduction, Lemma 1, Theorem 1 and section 6. (arXiv) (doi)

Links

[3]

DLMF 4.22.1: Euler product for sine

Similar tables

Generalized log-sine integrals — The positive slices are finite triangular combinations of angular moments with $k=1$, using $u=4\sin^2(\theta/2)$.

Riemann zeta values — The classical evaluations $F_3(-1)=-2\zeta(3)/5$ and $F_4(1)=17\zeta(4)/36$ relate these series to zeta values.

Clausen values — The values $\operatorname{Cl}_2(\pi/3)$ and $\operatorname{Cl}_4(\pi/3)$ enter the evaluations of $F_3(1)$ and $F_5(1)$.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (it holds every $2\leq s\leq10$ at $u=1,-1,2,3,4$. The arguments $1$ and $-1$ select the unweighted and alternating families, $4$ is the positive convergence boundary, and $2,3$ are the distinct arithmetic application families identified in [1]. The noncanonical order cutoff covers both parities and higher weights at all five slices before extending one slice more deeply. This is a selection of families, not a rational mesh; the attached argument-selection.md explains the bounds and omitted arguments)

How they were obtained:

Arb ball arithmetic encloses the defining series for $|u|<4$, with a geometric bound for its entire omitted tail. At $u=4$, the log-sine integral is evaluated by the convergent series from Euler's sine product [3]; polynomial products and logarithmic endpoint moments are computed explicitly, and a positive-coefficient majorant bounds the remainder. The attached certification.md derives both bounds. Computation uses 525 working bits and endpoint degree 532; the smallest measured relative accuracy is 513 bits. Every entry is recomputed at 140 requested digits, and the complete enclosure must fit inside the interval represented by the stored 100-significant-digit decimal.

Independent mpmath quadrature at 150 and 200 decimal digits evaluates a repeated integral of the elementary $F_2$, using a different representation from the generator. Nonzero known-answer integrals and deliberate wrong-answer controls are tested first. All 45 values are compared with this quadrature; ten closed-form cases and forty relations to stored zeta, Clausen and generalized log-sine values are also checked. The attached checks.py repeats the applicable identities on the entries read back from this draft and runs verify() on every entry. The companion T697 is itself a draft, so its agreement is supplementary evidence; the series remainder bounds certify this table independently.