Values of Ramanujan's signature hypergeometric functions $F_r(x)$

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Numbers

$r$

$x$

$F_r(x)$

1/2:

1.180340599016096226045337940558488587233716634881447299515864399404304180720715794978458616195807954

comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.

1/4:

1.073182007149364375052841707970349795669526818631368625787232257139788343028625222768722717743272285

3/4:

1.372880500618350164697637575007806058094538625348940055855540038870350633240892464904292347403572711

1/3:

1.103845772797212254324837694207631411924760802239391086997330526905118375631119575423168443314817465

2/3:

1.291675482135082644544318504760649923199863516052326580950539239585172786733440152066839813967604411

1/9:

1.029660375463762623523228181255854543570903969011705041891655029152762974930669348678219260552679533

8/9:

1.609773010724046562579262561955524693504290227947052938680848385709682514542937834153084076614908428

1/2:

1.159595266963928365769992051570020881945165263439782855263105059747973757205286258658085251051146071

comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.

1/4:

1.064925231128516498437819437604916718248282322874928859475392443772515623080612622146534197451821389

3/4:

1.328778200157056363878073355047905281137317875982473604891262407713582251790491637816553512224526597

1/3:

1.092060148081664394316757805253605515050933694436801793669960490099214146997645089823341657245948803

2/3:

1.257546863411625467745864646361554193208060216425676017110925090768116861297340574645820318884594719

1/9:

1.026343520841566885711818335651046428274180826014178666730958109479707060905841596720020526615780503

8/9:

1.535812809958194137289311853725147146475663454326775827621496728753103561572492083134212409167445065

1/2:

1.133915559726082732440156510347438791344295356805403611236317061400497913023777214458207113170347034

comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.

1/4:

1.054648614831467047906718149577441840630273364763410414893082861937658163667864603886693740230931228

3/4:

1.274611308136601952102809952021133623397842247250675660982363975799881719700368808191505969508039952

1/3:

1.077415715750701203246257098513742012348092021182187093026426589300763704600445579059592620512413547

2/3:

1.215502554868454386941982217997043941288277531484048915816265650173538426379192234806255033882957850

1/9:

1.022204943866080919860161806857062989134020137789994880270437701856939901361339037391092749731504665

8/9:

1.445616095140239952072718914467638749495268402565399677644614166992051212768192453714749044140042691

1/2:

1.098430696839862068942935161608698822283938481692253723027033945469518521905843821564973446230505442

comment: A Gauss second-summation value, equal to $2/B((r+1)/(2r),(2r-1)/(2r))$.

1/4:

1.040343865043574145416878113346879640594465685268916549682386133655136588848837323908014059028010148

3/4:

1.200547022551705970623606133782633712417243715140519123069736520860927175815592908672157063181494441

1/3:

1.057076261613892128402122962965324995470501889069986725795375244685741777798779952700216143078965700

2/3:

1.157778216364893874950561674893505207806365917352306655090978043246823064194951989915747252390826364

1/9:

1.016424899916147157209316669882466843802873309665909634999036140962440390965478130838565284637450640

8/9:

1.323467526740777845709483066819782686939481648265356144701869051624955722200944233060569225543086875

Definition

This table gives the real values $F_r(x)={}_2F_1(1/r,1-1/r;1;x)$ of Gauss's hypergeometric function [3] for signatures $r\in\{2,3,4,6\}$ and rational arguments $0\leq x<1$. The argument $x$ is Gauss's hypergeometric variable.

Parameters

$r$

— signature ($r\in\{2,3,4,6\}$)

$x$

— hypergeometric argument ($0\leq x<1$)

Formulas

(1)

With $(a)_n$ denoting the Pochhammer symbol, $F_r(x)=\sum_{n=0}^{\infty} ((1/r)_n(1-1/r)_n/(n!)^2)x^n$ for $|x|<1$ [3].

(2)

For $y=1-x$ and $0<x<1$, $F_r(x)=\frac{\sin(\pi/r)}{\pi}\sum_{n=0}^{\infty} \frac{(1/r)_n(1-1/r)_n}{(n!)^2} \bigl(2\psi(n+1)-\psi(n+1/r)-\psi(n+1-1/r)-\log y\bigr)y^n$, where $\psi$ is the digamma function. This is the $c=a+b$ case of the continuation formula [5].

(3)

As $x\to1^-$, $F_r(x)/(-\log(1-x))\to\sin(\pi/r)/\pi$.

(4)

$F_2(x)=2K(x)/\pi$, where $K$ is the complete elliptic integral of the first kind with parameter $m=x$.

(5)

$F_2(1-b^2)=1/\operatorname{AGM}(1,b)$. The cubic analogue is $F_3(1-b^3)=1/\operatorname{AGM}_3(1,b)$, where $\operatorname{AGM}_3$ is the common limit of $a_{n+1}=(a_n+2b_n)/3$ and $b_{n+1}=\sqrt[3]{b_n(a_n^2+a_nb_n+b_n^2)/3}$ [2].

(6)

$F_4(x)=\sqrt{1+z}\,F_2(z)$ with $z=(1-\sqrt{1-x})/(1+\sqrt{1-x})$ [5].

(7)

For $r\in\{2,3,4,6\}$, $F_r(1/2)=2/B((r+1)/(2r),(2r-1)/(2r))= \sqrt{\pi}/(\Gamma((r+1)/(2r))\Gamma((2r-1)/(2r)))$ [4], with $B(a,b)$ as in [6].

Comments

(8)

Signature $2$ is the classical elliptic-integral case, related to $K(m)$ by (4). Signatures $3$, $4$ and $6$ are the alternative-base theories of Ramanujan developed by Berndt, Bhargava and Garvan [1]; in these four cases the two upper hypergeometric parameters are $a$ and $1-a$, with $a=1/r$.

(9)

$F_r(0)=1$, and $F_r(x)$ diverges logarithmically as $x\to1^-$ by (3).

(10)

At $x=1/2$, Gauss's second summation theorem gives the Gamma and Beta quotients in (7). The signature-$2$ entry agrees with [7]. The OEIS entry [8] writes the signature-$3$ value as $2^{-2/3}/L$, with $L=\Gamma(1/3)\Gamma(5/6)/\Gamma(1/6)$.

(11)

The signatures $r=2,3,4,6$ retain their alternative-base elliptic meaning. The half argument has the gamma quotient from Gauss's second summation theorem [10]. Complementary pairs $(x,1-x)$ with $x=1/4,1/3,1/9$ provide a bounded low-complexity reference set on both sides of the complementary centre; this list is our exploratory choice and is not asserted to consist of singular moduli. The origin supplies the exact hypergeometric normalization.

Programs

(P1)

Python

from balanced_values import BalancedValues
from numberdb._write import to_text
reference = BalancedValues()
for parameters in reference.enumerate():
    print(parameters, to_text(reference.value(parameters, reference.digits), reference.digits))

References

[1]

Bruce C. Berndt, S. Bhargava and Frank G. Garvan, Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4163-4244. (doi) (zbMATH) (MR)

[2]

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc. 323 (1991), no. 2, 691-701. (doi) (MR)

Links

[3]

DLMF 15.2: Hypergeometric Function

[4]

DLMF 15.4: Special Cases

[5]

DLMF 15.8: Transformations of Variable

[6]

DLMF 5.12: Beta Function

[7]

OEIS A175574: ${}_2F_1(1/2,1/2;1;1/2)$

[8]

OEIS A355178: ${}_2F_1(1/3,2/3;1;1/2)$

[9]

DLMF 15.4.28: Gauss's second summation theorem

[10]

DLMF 15.4: hypergeometric special cases

Similar tables

Complete elliptic integral of the first kind $K(m)$ — signature $2$ is related to $K(m)$ by (4)

Values of the arithmetic-geometric mean — signature $2$ has the AGM expression and signature $3$ has the cubic analogue in (5)

Values of the Beta function $B(a,b)$ at pairs of rational numbers — the entries with $x=1/2$ are reciprocal Beta values by (7)

Values of the Gamma function at rational numbers — the entries with $x=1/2$ are Gamma quotients by (7)

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

All 32 stored entries are recomputed from exact rational or enclosed square-root inputs using Arb at 100 decimal digits plus 160 guard bits. Existing numeric strings and row annotations are retained unchanged; their written intervals strictly contain the fresh enclosures. Exact anchors and rational polynomial integrals use exact arithmetic.

Every inexact result is finite, with at least 482 relative accuracy bits. Independent mpmath computations at 150 and 200 decimal digits check every value; the AGM check uses its elliptic-integral expression, and the incomplete-gamma check uses integer/half-integer closed forms. Known nonzero and exact controls, shifted-last-place rejection controls, enumeration checks and applicable identities pass. The attached balanced-evidence.json records the checks, and balanced_checks.py repeats them. The current generate.py and balanced-selection.json reproduce the full retained-plus-added selection; older auxiliary attachments are preserved as historical sources, not claimed as the default enumeration.

Table is complete: no (it holds $F_r(x)$ for each signature $r=2,3,4,6$ at $x=0,1/2,1/4,3/4,1/3,2/3,1/9,8/9$; the complementary pairs are exploratory references, not a catalogue of arithmetic moduli, and the origin is exact)