Values of the Rogers–Ramanujan continued fraction $R(e^{-\pi\sqrt r})$

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Numbers

$r$

$R(e^{-\pi\sqrt r})$

0.5114284554037035192946330135425788104157543814174665124187982080507562021874566144382624867458127297

0.4064605812995065920896182374103892775964745719514374630420405047120796284213807241937612229976144283

0.3353422604252457377634254037231980060911401804469839137514494403981197158034830427585877742249190029

0.2840790438404122960282918323931261690910880884457375827591626661550458773514845537303784177522316259

comment: $R(e^{-2\pi})=\sqrt{(5+\sqrt5)/2}-\varphi$, where $\varphi=(1+\sqrt5)/2$.

0.2451580725478960661742616314868277784988095784262814050649070560543180476935191058932106262375090517

0.2144851757810122094843330484296364262159774762955522817042235012528848654272534330601769719470861147

0.1896414433626258810635833635358665334551168078243071176454224940962761159614816852362060618893602495

0.1690955211413606647324925685630614029229222821013405754306635661791216675199939804389706354456199562

0.1518235498934964612889308227856287827262113648732169136778450380297056040232555701504761174628340209

10:

0.1371107725802946953936999666911150326625769604237964781548830393212795425045014729725418859154115993

11:

0.1244406155944850389345146658271114198662517270509441941158684253296282940477084817669991642753205119

12:

0.1134290196256987498645032183805072562556239227331849802867129861912415464701439372773742128680384351

13:

0.1037836618433571538341555035953966017744924423011021010075793509584941643885994770154726600613392364

14:

0.09527760187294194811057419769441411864911856350813810626094230031141715959738319396044541438942549896

15:

0.08773166335795842144575528494303693200453342602152481475771891720081893540895335450675022950524175058

16:

0.08100230967515765130997208783934959185511642003879960863612336429419885436704082489739142811537929868

17:

0.07497308942488644807926852923533727633291941276025841301502134273459658306520694646834856998459361529

18:

0.06954846786635897000329366473882390663577399276122517584028519742150830572432593996832432152217392771

19:

0.06464929303543405613383876419120455844244789022917027481079190067446697971834722547514327827146654697

20:

0.06020940656245182043572967581957492337261937064631186642499705561396077550229157404778396819126206185

comment: $R(e^{-2\pi\sqrt5})=\frac{\sqrt5}{1+(5^{3/4}(\varphi-1)^{5/2}-1)^{1/5}}-\varphi$, where $\varphi=(1+\sqrt5)/2$ and the roots are positive real roots.

21:

0.05617307177318694688930036962423089132354626298667960771464204218178390582565297196694090372182654392

22:

0.05249299550482749878731224588165945228459438465231388342542003021631429046940293099000172290699652041

23:

0.04912878799236763704821133969954768737081142248997483336190931287436007905879859617065661661298093385

24:

0.04604575055945772409276228878245288009776200402395567064240712571631505509867404874199779446014222458

25:

0.04321391175136109512627636216954364995889851512776411554505360353128650494640427090891341122261566302

26:

0.04060725396296401859427765216994914400133354386213344294351559757769546227674620851397991986412863779

27:

0.03820308769222036555134883174771449152228051843062227742572742133126662299133392586827759043052829993

28:

0.03598154131985774060641670212085857014368294363489152763389369609684884522429079370047810986059248572

29:

0.03392514211317227066765247260560945671845402012350759430353072180434769283021960011019256395228973433

30:

0.03201846986591873259999054740086854038034452707341757754216517634514643809178991528684235414193265484

31:

0.03024786882183901665678717265455018739530499021683733437850934768831981376981110038857709639660349639

32:

0.02860120670208507364013960637690325366829664221211978204029123852437190033941455716387394332381295160

33:

0.02706767205672242200644632679203631962024662262896583947614755260235238698892202595428049987573263313

34:

0.02563760299248542922101969328881840547662414834349555033106951891072978233587609357559841690383932323

35:

0.02430234173934742380730549000286557919335623394065430083484969975472434505900932748471103481820729000

36:

0.02305411061296895069837383747683861034437966426766271644949773605834199451369639800560570750495796093

37:

0.02188590578581574586736080319892855399484859727087601701117445093465193257346114467126762201148890447

38:

0.02079140595349735652181717570894802204576686993968603267424409829398913384483015031238542409441332975

39:

0.01976489351688158554620048712662363056172733885910638090960437242911626613646831789849990828382993484

40:

0.01880118632642539943536387209240057739609806441369750353192044662030528107496588998781885757347239065

41:

0.01789557837682738779386028185886686123774575331005535802841061103239419626898951056787757884772686429

42:

0.01704378811573806607666810215446368844932268399075664898095793552141375025070564204763727738912234065

43:

0.01624191325380269481427343585263721595570304728258997057953987249639683983679064453455985544794384159

44:

0.01548639114551371674380032861899938540479217138191482842612082842442716599243636875752246744225608974

45:

0.01477396395956972524093501163260669563731236670637551617483454203223530214517163650250676968330273387

46:

0.01410164798020127463171929710751678380722608771953588948679146635113216515707035718713739224572493269

47:

0.01346670648235895975977717233753343896367529157155522203193154413964194432398085423261487583274917913

48:

0.01286662570781773586826611445305896916686201531037480027163653153368491613268053602851783526750317907

49:

0.01229909353935142020918119156279785760635650940579284081680187697643320204270519926300792072000617617

50:

0.01176198052874133656912513293680090347929300124675486872622133226365881700809279913374219017975613560

51:

0.01125332298356321718384754530526296690268359147565052051805318888497524187566348046568692866647799391

52:

0.01077130785910654160191741030803559460362780683829421756895620847819590084231363183069866734175360761

53:

0.01031425923676357922767245726190433202978202898508560891838723798350882423872629335781246274724564335

54:

0.009880626199872929032494132521680086596818931658701419579212978365321932296626825855480721783433528406

55:

0.009468971943204775091309597583417675594478191509427808098163230680945783578409630072307622183503246127

56:

0.009077963973761708067964832224261400676735632729315044620911433957319862971615473540904202566070751731

57:

0.008706365278939338509869076027669745653080688925865514450530672168784354549262593312827351516861686084

58:

0.008353026353839665373341926259801238868613037212722307753996735276186643653740242614711730185859655032

59:

0.008016877993066722358263143619398248148600075056915992495585866461616848612014774599929462694273526695

60:

0.007696924763998489935244432791359697099576804385999482162951114672357172959254382392064610980084884956

61:

0.007392239088604796340879025821925511547587360804120954593106194417018407461896725983049566744007213293

62:

0.007101955869604765430018516679789061061389252748741759241215029928917039373435797009174766542652595445

63:

0.006825267604327826629905430173210763696787729202186542137719539205593528115572487136057289573096134389

64:

0.006561419936226272588622208626907442508981648788371991612737363818672391349702078731653179389751544573

65:

0.006309707599725519228254231013632469475235872119254620048224732169980991581960757888095977960752493279

66:

0.006069470719109503946037580961874862169700402033217543986738771787307662136702836765134681746653812256

67:

0.005840091426523820851669067342401763285906702048266559981848716888672635513028725891290295007647928055

68:

0.005620990768023846478632801640164039744860649435433140913331370363490102210670584720842103347339642725

69:

0.005411625869972161142024765894387070804393981338858929264451168336614713634892122424697746654544994925

70:

0.005211487341061253297339512038765109978292594992296561044587034385803306658142708236134433206277746515

71:

0.005020096887857050990042254729621181108529723164302329137287635108431272357320770752622152657713876559

72:

0.004837005124071909802808507124629690730773706896327513090970772075419756327157026681353052799463088640

73:

0.004661789555821530471842804269292651003984566467381036840377804716665917926207014975149157805103565013

74:

0.004494052726932661389529492083028200705356682528794966659767017412874785885328779069347305557073606534

75:

0.004333420509976507282795463318386785502826150449864450280145590924876028478703520332111017636117633903

76:

0.004179540530131712225820835748057895347481979430152011609445280964386274819963989313313697327093802437

77:

0.004032080710252437238947428554645958079463116295027343479938372322000375694488603773208781008079040918

78:

0.003890727926650348569691886877006682288556188348698372760772885446573715222764901445599002897335507282

79:

0.003755186766110736258110700748970760228937814830462196062951709471077191764811105336803599613639053070

80:

0.003625178375566831550333363087508202624991486517011685665963580763885255101114378341440261772740695507

81:

0.003500439396665193737363193991764638037942477992470799710811739753407351182593205790210433038960852171

82:

0.003380720978179721259574541684923828704043152882085247705255326505700034033807459781548126108471766694

83:

0.003265787859881973916002260429075346899879081619471748586593313547936132045231471842630009764724603278

84:

0.003155417522059457924935334278178212847377193569129481809527614431497849508446858739557240309463500356

85:

0.003049399395398685235375939359539334867238371803357185137508884323536099017057067802917984478517487631

86:

0.002947534126422645995452140040991326216954681984210647664475644117501777565577052519264497180439533556

87:

0.002849632894098528459235569271615533027254294458383162539062675686384096031917526410127303546004865378

88:

0.002755516773616110295795565977551596796882759524052086391963216497003910294113797099068585968060059938

89:

0.002665016143684667904343106566107072202793240007475801506644948402518151643354020652910910454149370861

90:

0.002577970134010430881887070115645458570890547962012608241047495225204301899218086827207185523465199077

91:

0.002494226109901022551322461138661229127574357228686939437109259078174938471563618190273376903834326800

92:

0.002413639191201055638964307321000003447720808580595294639675794438145118452237038719688563013107886580

93:

0.002336071802996830044880115417948755892475782314672773914517052848155038101784731454215487616482937789

94:

0.002261393255740338343917698408586486409061326308533800830936907963649513772553885239930001470745074387

95:

0.002189479352635687703071990005672102261562217305990366596245748418342718077648521697401407934338999078

96:

0.002120212022306521916197924207005679925446370077770809007393236301678885890887648489880659594957477709

97:

0.002053478974922793674109658353869029489818579297190141279660920092209877754888698168840212880238056189

98:

0.001989173380110830488772222582676916047924689577121231864886653295542617528147730893253954213380944438

99:

0.001927193565103430611263820060957977387723617964156479622773335505217959186502287824605689934946594523

100:

0.001867442731707946402918382775014798517045926243423716872473730757485005391632170538360673564420818617

Definition

The Rogers–Ramanujan continued fraction is $R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}$, with $q^{1/5}$ the positive real fifth root [2]. Each entry is $R(q)$ at $q=e^{-\pi\sqrt r}$ for a positive integer $r$.

Parameters

$r$

— the parameter in the nome $e^{-\pi\sqrt r}$ (positive integer)

Formulas

(1)

$R(q)=q^{1/5}\prod_{k=0}^{\infty} \frac{(1-q^{5k+1})(1-q^{5k+4})} {(1-q^{5k+2})(1-q^{5k+3})}$ [2].

(2)

If $u=R(q)$ and $z=i\sqrt r/2$, so that $q=e^{2\pi i z}$, then $j(z)=-\frac{(u^{20}-228u^{15}+494u^{10}+228u^5+1)^3} {u^5(u^{10}+11u^5-1)^5}$ [1] [2].

(3)

If $u_r=R(e^{-\pi\sqrt r})$, then $u_r u_{4r}^2=(u_{4r}-u_r^2)/(u_{4r}+u_r^2)$ [2].

Comments

(4)

With $z=i\sqrt r/2$, $q=e^{2\pi i z}=e^{-\pi\sqrt r}$. This is also the nome $e^{\pi i\tau}$ for $\tau=i\sqrt r$, the convention used for the singular moduli $k_r$.

(5)

The continued fraction converges for $|q|<1$, and therefore at $q=e^{-\pi\sqrt r}$ for each positive integer $r$ [2].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField

r = 2
terms = 80
RBF = RealBallField(numberdb.bits(100, losing=256))
q = (-RBF.pi() * RBF(ZZ(r)).sqrt()).exp()
value = q ** (ZZ(1) / ZZ(5))
for k in range(terms):
    value *= (1 - q ** (5 * k + 1)) * (1 - q ** (5 * k + 4))
    value /= (1 - q ** (5 * k + 2)) * (1 - q ** (5 * k + 3))
value

References

[1]

W. Duke, Continued fractions and modular functions, Bulletin of the American Mathematical Society 42 (2005), no. 2, 137-162. (doi)

Links

[2]

MathWorld: Rogers-Ramanujan Continued Fraction

Similar tables

Singular values $k_r$ of the elliptic modulus — both tables use the CM point $\tau=i\sqrt r$ and the nome $q=e^{-\pi\sqrt r}$

Ramanujan's class invariants $G_n$ — both are Ramanujan modular functions evaluated at the same real nomes

Rational singular moduli — Formula (2) relates these values to singular values of the modular $j$-invariant

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $R(e^{-\pi\sqrt r})$ for every integer $1\leq r\leq100$, matching the range of the singular moduli, $K(k_r)$, and Ramanujan's class invariants)

How they were obtained:

The generator computes the product in (1) in Sage's arb-backed RealBallField, using a finite product with an explicit positive tail bound.

Before the values are sent, the generator checks the continued-fraction recurrence, (2), and (3) on the computed range.