Rogers-Ramanujan Continued Fraction

Author

Eric W. Weisstein
April 29, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/Rogers-RamanujanContinuedFraction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html.

©2013 Wolfram Research, Inc. except for portions noted otherwise

Additional Author

Code and results by 
Michael Trott
as noted April 2003

Code

MakeBoxes[RogersRamanujanR[q_],TraditionalForm]:=RowBox[{"R","(",MakeBoxes[q,TraditionalForm],")"}]
RogersRamanujanR[q_]:=(q^(1/5))((QPochhammer[q,q^5]QPochhammer[q^4,q^5])/(QPochhammer[q^2,q^5]QPochhammer[q^3,q^5]))
RogersRamanujanR[q_,n_]:=RogersRamanujanR[1,q,n]
RogersRamanujanR[a_,q_,n_]:=q^(1/5)/Fold[1+a q^#2/#1&,1,Range[n,1,-1]]

RogersRamanujanF[a_,q_,n_]:=Fold[1-a q^#2/#1&,1,Range[n,1,-1]]

RamanujanTheta[a_,b_]:=Sum[a^(n(n+1)/2)b^(n(n-1)/2),{n,-Infinity,Infinity}]

Michael's

RogersRamanujanℛ[q_,n_]:=(q^(1/5)) Fold[((#2)/(1+#1))&,(q^n),Table[(q^k),{k,n-1,0,-1}]]
RogersRamanujanℜ[q_]:=((((Sqrt[((#)^2)+2#+5])-#-1)/2))&[(( DedekindEta[-((I Log[q])/(10 π))])/( DedekindEta[-((I Log[(q^5)])/(2 π))]))];

Timings

{Timing[RogersRamanujanℛ[q,100];],Timing[RogersRamanujanR[q,100];]}

{Timing[RogersRamanujanℛ[q,1000];],Timing[RogersRamanujanR[q,1000];]}

{Timing[RogersRamanujanℛ[q,2000];],Timing[RogersRamanujanR[q,2000];]}

{Timing[RogersRamanujanℛ[.8,10000];],Timing[RogersRamanujanR[.8,10000];]}

RogersRamanujanℛ[q,2000]-RogersRamanujanR[q,2000]

Plot of R(q)

<<Utilities`Plot`

Plot[Evaluate[RogersRamanujanR[q,100]],{q,0,3},PlotStyle->Red,PlotRange->{{0,3},{0,.9}},
AxesLabel->TraditionalForm/@{q,R[q]},Prolog->{{Dashing[{.02}],Blue,Line[{{0,GoldenRatio-1},{3,GoldenRatio-1}}]}}]

MakeBoxes[R[q_Symbol,n_],TraditionalForm]:=RowBox[{SubscriptBox["R",MakeBoxes[n,TraditionalForm]],"(",MakeBoxes[q,TraditionalForm],")"}]

ComplexPlot3D[Evaluate[RogersRamanujanR[q,100]],{q,-2,2},
PlotPoints->200,ComplexPlotLabel->TraditionalForm[R[q,100]]]//Timing

ComplexPlot3D[Evaluate[RogersRamanujanR[q,101]],{q,-2,2},
PlotPoints->200,ComplexPlotLabel->TraditionalForm[R[q,101]]]//Timing

ComplexContourPlot[RogersRamanujanR[q,100],{q,-2,2},ComplexPlotLabel->TraditionalForm[R[q,100]]]//Timing

ComplexContourPlot[RogersRamanujanR[q,101],{q,-2,2},ComplexPlotLabel->TraditionalForm[R[q,101]]]//Timing

R(q) Special Values

<<NumberTheory`Recognize`

This code by Michael Trott

SetOptions[Roots,Cubics->False,Quartics->False];

findAlgebraicValue[𝒶_,prec_]:=Block[{$MaxExtraPrecision=1000,o=2,polyα,α,β},α=RogersRamanujanR[N[(E^(-𝒶 π)),prec],prec];While[polyα=Recognize[α,o,t];β=Select[t/.Union[{ToRules[Roots[polyα==0,t]]}],Abs[#-α]==0&];If[β==={},True,β[[1]]-RogersRamanujanR[N[(E^(-𝒶 π)),2 prec],2 prec]≠0],o=o+1];β[[1]]]

R(1)==ϕ-1

RogersRamanujanR[1.,100]

RogersRamanujanR[q]

Limit[RogersRamanujanR[q],q->1]//InputForm

RogersRamanujanR[q]/.q->.9999

N[GoldenRatio - 1]

ContinuedFractionK[1,1,{n,0,∞}]

R((e^(-π)) )==(1/8) ((Sqrt[5])+3) ((5^(1/4))-1) ((Sqrt[2 (Sqrt[5])+10])-(3+(5^(1/4))) ((5^(1/4))-1))

FullSimplify[RogersRamanujanℜ[Exp[-π]]]//N

eqn=Recognize[r2=N[RogersRamanujanℜ[Exp[-π]],100],12,x]

x2=Select[x/.{ToRules[Roots[eqn==0,x]]},Abs[#-r2]<10^-6&][[1]]

NRoots[1-14 x+22 (x^2)-22 (x^3)+30 (x^4)+22 (x^5)+22 (x^6)+14 (x^7)+(x^8)==0,x]

N[%]

RootReduce[((1/8)(3+(Sqrt[5]))((5^(1/4))-1)((Sqrt[10+2(Sqrt[5])])-(3+(5^(1/4)))((5^(1/4))-1)))]

ToRadicals[%]

RogersRamanujanR[Exp[-Pi]]

FullSimplify[%]

N[%]

(1/8)(3+(Sqrt[5]))((5^(1/4))-1)((Sqrt[10+2(Sqrt[5])])-(3+(5^(1/4)))((5^(1/4))-1))//N

N[(1/8)*(3 + Sqrt[5])*(5^(1/4) - 1)*(Sqrt[10 + 2*Sqrt[5]] - 
    (3 + 5^(1/4))*(5^(1/4) - 1))]

ContinuedFractionK[(q^n),1,{n,0,∞}]

ContinuedFractionK[((Exp[-π])^n),1,{n,0,∞}]

With[{q=Exp[-π]//N},(q^(1/5))ContinuedFractionK[(q^n),1,{n,0,20}]]

R((e^(-2 π)))

eqn=Recognize[r2=N[RogersRamanujanℜ[(E^(-2 π))],100],12,x]

x2=Select[x/.{ToRules[Roots[eqn==0,x]]},Abs[#-r2]<10^-6&][[1]]

N[RogersRamanujanℜ[(E^(-2 π))]-x2,200]

N[-GoldenRatio+(Sqrt[(1/2) (5+(Sqrt[5]))])]

With[{q=Exp[-2π]},(q^(1/5))ContinuedFractionK[(q^n),1,{n,0,∞}]]

With[{q=Exp[-2π]//N},(q^(1/5))ContinuedFractionK[(q^n),1,{n,0,20}]]

R((e^(-3 π)))

Root[1-1886*#1+8464*#1^2+20040*#1^3+51730*#1^4-18688*#1^5-327962*#1^6-933792*#1^7-873525*#1^8+2404400*#1^9+5467360*#1^10-6388310*#1^11-14366560*#1^12+17184200*#1^13+38628000*#1^14-21937440*#1^15-35176135*#1^16+21937440*#1^17+38628000*#1^18-17184200*#1^19-14366560*#1^20+6388310*#1^21+5467360*#1^22-2404400*#1^23-873525*#1^24+933792*#1^25-327962*#1^26+18688*#1^27+51730*#1^28-20040*#1^29+8464*#1^30+1886*#1^31+#1^32&,6]

N[Root[1-1886*#1+8464*#1^2+20040*#1^3+51730*#1^4-18688*#1^5-327962*#1^6-933792*#1^7-873525*#1^8+2404400*#1^9+5467360*#1^10-6388310*#1^11-14366560*#1^12+17184200*#1^13+38628000*#1^14-21937440*#1^15-35176135*#1^16+21937440*#1^17+38628000*#1^18-17184200*#1^19-14366560*#1^20+6388310*#1^21+5467360*#1^22-2404400*#1^23-873525*#1^24+933792*#1^25-327962*#1^26+18688*#1^27+51730*#1^28-20040*#1^29+8464*#1^30+1886*#1^31+#1^32&,6]]

With[{q=Exp[-3π]},(q^(1/5))ContinuedFractionK[(q^n),1,{n,0,∞}]]

With[{q=Exp[-3π]//N},(q^(1/5))ContinuedFractionK[(q^n),1,{n,0,20}]]

R((e^(-4 π)))

findAlgebraicValue[4,200]//Timing

R((e^(-5π)))

findAlgebraicValue[5,200]//Timing

R((e^(-6 π)))

findAlgebraicValue[6,200]//Timing

R((e^(-7 π)))

findAlgebraicValue[7,200]//Timing

R((e^(-8 π)))

findAlgebraicValue[8,200]//Timing

R((e^(-9 π)))

findAlgebraicValue[9,200]//Timing

List

Calculated by Michael Trott

tab={R[E^(-Pi)]==Root[1-14*#1+22*#1^2-22*#1^3+30*#1^4+22*#1^5+22*#1^6+14*#1^7+#1^8&,4],R[E^(-2*Pi)]==Root[1-2*#1-6*#1^2+2*#1^3+#1^4&,3],R[E^(-3*Pi)]==Root[1-1886*#1+8464*#1^2+20040*#1^3+51730*#1^4-18688*#1^5-327962*#1^6-933792*#1^7-873525*#1^8+2404400*#1^9+5467360*#1^10-6388310*#1^11-14366560*#1^12+17184200*#1^13+38628000*#1^14-21937440*#1^15-35176135*#1^16+21937440*#1^17+38628000*#1^18-17184200*#1^19-14366560*#1^20+6388310*#1^21+5467360*#1^22-2404400*#1^23-873525*#1^24+933792*#1^25-327962*#1^26+18688*#1^27+51730*#1^28-20040*#1^29+8464*#1^30+1886*#1^31+#1^32&,6],R[E^(-4*Pi)]==Root[1-14*#1+22*#1^2-22*#1^3+30*#1^4+22*#1^5+22*#1^6+14*#1^7+#1^8&,3],R[E^(-5*Pi)]==Root[1-286770*#1+5358420*#1^2+17950740*#1^3+221643240*#1^4+925781706*#1^5+3167669670*#1^6+10209367380*#1^7+24998928660*#1^8+52347527310*#1^9+103558646722*#1^10+177471853470*#1^11+260180189280*#1^12+346879234260*#1^13+441913352310*#1^14+439113506058*#1^15+431662158900*#1^16+323979470100*#1^17+366428567700*#1^18+109043347950*#1^19+242765765070*#1^20-109043347950*#1^21+366428567700*#1^22-323979470100*#1^23+431662158900*#1^24-439113506058*#1^25+441913352310*#1^26-346879234260*#1^27+260180189280*#1^28-177471853470*#1^29+103558646722*#1^30-52347527310*#1^31+24998928660*#1^32-10209367380*#1^33+3167669670*#1^34-925781706*#1^35+221643240*#1^36-17950740*#1^37+5358420*#1^38+286770*#1^39+#1^40&,4],R[E^(-6*Pi)]==Root[1-38*#1-240*#1^2+300*#1^3-235*#1^4+726*#1^5+92*#1^6+1840*#1^7-675*#1^8-1840*#1^9+92*#1^10-726*#1^11-235*#1^12-300*#1^13-240*#1^14+38*#1^15+#1^16&,5],R[E^(-7*Pi)]==Root[1-43702722*#1+3984771624*#1^2-40275364928*#1^3+460271097876*#1^4-3212938451316*#1^5+11345609479702*#1^6-19074475520184*#1^7+23191571672298*#1^8-53917431812566*#1^9+141518018122684*#1^10-443503163244498*#1^11+1030469984720316*#1^12-1051714902285152*#1^13+2058798193822434*#1^14-3359401712927540*#1^15+6849801752089255*#1^16-19185716643666210*#1^17+19419968502073120*#1^18-36093429620178790*#1^19+47799019315288620*#1^20-56411258934181390*#1^21+164891220639102630*#1^22-154396404022443360*#1^23+303246132471541120*#1^24-365259539413747106*#1^25+262447742713057782*#1^26-726156742176566394*#1^27+371452676656112368*#1^28-1083059284170510606*#1^29+1026220927285031326*#1^30-586854309925331422*#1^31+1506057762402917049*#1^32+586854309925331422*#1^33+1026220927285031326*#1^34+1083059284170510606*#1^35+371452676656112368*#1^36+726156742176566394*#1^37+262447742713057782*#1^38+365259539413747106*#1^39+303246132471541120*#1^40+154396404022443360*#1^41+164891220639102630*#1^42+56411258934181390*#1^43+47799019315288620*#1^44+36093429620178790*#1^45+19419968502073120*#1^46+19185716643666210*#1^47+6849801752089255*#1^48+3359401712927540*#1^49+2058798193822434*#1^50+1051714902285152*#1^51+1030469984720316*#1^52+443503163244498*#1^53+141518018122684*#1^54+53917431812566*#1^55+23191571672298*#1^56+19074475520184*#1^57+11345609479702*#1^58+3212938451316*#1^59+460271097876*#1^60+40275364928*#1^61+3984771624*#1^62+43702722*#1^63+#1^64&,6],R[E^(-8*Pi)]==Root[1-148*#1-670*#1^2-240*#1^3+1570*#1^4+2616*#1^5+302*#1^6-1180*#1^7-1610*#1^8+1180*#1^9+302*#1^10-2616*#1^11+1570*#1^12+240*#1^13-670*#1^14+148*#1^15+#1^16&,5],R[E^(-9*Pi)]==Root[1-6660544968*#1+1744861065300*#1^2+40801903892200*#1^3+1167874116268950*#1^4+17368587394307808*#1^5+215816825313836956*#1^6+1535473868499121800*#1^7+7104486432810185325*#1^8+16701943421213839200*#1^9+23169371030001675804*#1^10-34634150838112194672*#1^11-68121550121580637400*#1^12-586350834197344386600*#1^13-492699921301808544600*#1^14-1942451650662574510408*#1^15+3184887730510133586894*#1^16+110611473044478122400*#1^17+20861144863360295706100*#1^18-4370981368517091296400*#1^19+64040191774208196691926*#1^20-100486436732542379063568*#1^21+28425264965955626253000*#1^22-406575373295077900443000*#1^23+342256286264754720878250*#1^24-984420887362268667817704*#1^25+1308821720457676114514472*#1^26-388916042640613464077200*#1^27+4846021938545899648623450*#1^28-5694908219409776412454800*#1^29+6605849046232373780893468*#1^30-5411151122978716781159424*#1^31+746831535293889395939175*#1^32-36812497951764602093040800*#1^33+46352766519828251267768700*#1^34-5009084197667399095808016*#1^35-26330282701156217001172162*#1^36+3514308056689521885375600*#1^37+174839054486539765542320400*#1^38-179967195643002849661012600*#1^39-158640261584590887689078718*#1^40+261638188177685356842883224*#1^41+111856065337939582332803400*#1^42-483972150177152865011912400*#1^43+128985984909285897635851350*#1^44+677154896614139244681234096*#1^45-416571441253265008822784628*#1^46-734267575400445332221608000*#1^47+561609515280435445214732375*#1^48+734267575400445332221608000*#1^49-416571441253265008822784628*#1^50-677154896614139244681234096*#1^51+128985984909285897635851350*#1^52+483972150177152865011912400*#1^53+111856065337939582332803400*#1^54-261638188177685356842883224*#1^55-158640261584590887689078718*#1^56+179967195643002849661012600*#1^57+174839054486539765542320400*#1^58-3514308056689521885375600*#1^59-26330282701156217001172162*#1^60+5009084197667399095808016*#1^61+46352766519828251267768700*#1^62+36812497951764602093040800*#1^63+746831535293889395939175*#1^64+5411151122978716781159424*#1^65+6605849046232373780893468*#1^66+5694908219409776412454800*#1^67+4846021938545899648623450*#1^68+388916042640613464077200*#1^69+1308821720457676114514472*#1^70+984420887362268667817704*#1^71+342256286264754720878250*#1^72+406575373295077900443000*#1^73+28425264965955626253000*#1^74+100486436732542379063568*#1^75+64040191774208196691926*#1^76+4370981368517091296400*#1^77+20861144863360295706100*#1^78-110611473044478122400*#1^79+3184887730510133586894*#1^80+1942451650662574510408*#1^81-492699921301808544600*#1^82+586350834197344386600*#1^83-68121550121580637400*#1^84+34634150838112194672*#1^85+23169371030001675804*#1^86-16701943421213839200*#1^87+7104486432810185325*#1^88-1535473868499121800*#1^89+215816825313836956*#1^90-17368587394307808*#1^91+1167874116268950*#1^92-40801903892200*#1^93+1744861065300*#1^94+6660544968*#1^95+#1^96&,6],R[E^(-10*Pi)]==Root[1-510*#1-13590*#1^2-32280*#1^3-82230*#1^4-153522*#1^5-302910*#1^6-273540*#1^7-412830*#1^8-268230*#1^9-262006*#1^10+268230*#1^11-412830*#1^12+273540*#1^13-302910*#1^14+153522*#1^15-82230*#1^16+32280*#1^17-13590*#1^18+510*#1^19+#1^20&,3]};

Exponent[#[[2,1]][x],x]&/@tab

%/4

ListPlot[Log@Abs@CoefficientList[#[[2,1]][z],z],Joined->True,AspectRatio->Automatic]&/@tab

R(q)

RogersRamanujanR[q,5]

Short[r=Series[RogersRamanujanR[q,50],{q,0,30}]/(q^(1/5))]

CoefficientList[r,q]

q^(-4/5) ContinuedFractionK[1,q^(n-1),{n,0,Infinity}]

ContinuedFractionK[1,q^(n-1),{n,0,3}]//Factor

Table[{1,q^(n-1)},{n,0,3}]

Table[{q^n,1},{n,0,3}]

ContinuedFractionK[q^n,1,{n,0,3}]//Factor

ContinuedFractionK[(q^(n-1)),1,{n,4}]//Factor

(1/((q^(-1))+(1/((q^0)+(1/((q^1)+(1/(q^2))))))))//Factor

(q/(1+(q/((q^0)+(1/((q^1)+(1/(q^2))))))))//Factor

q^(-4/5) ContinuedFractionK[1,q^(n-2),{n,3}]//FullSimplify

((q^(1/5))/(ContinuedFractionK[1,q^(n-1),{n,3}]))//FullSimplify

FullSimplify[%-RogersRamanujanR[q]]

V6

r2=Series[qInf[q,q^5]qInf[q^4,q^5]/qInf[q^2,q^5]/qInf[q^3,q^5],{q,0,20}]

CoefficientList[r2,q]

Series[Product[(1-q^(5k-1))(1-q^(5k-4))/(1-q^(5k-2))/(1-q^(5k-3)),{k,100}],{q,0,20}]

Series[Product[(1-q^(5k+1))(1-q^(5k+4))/(1-q^(5k+2))/(1-q^(5k+3)),{k,0,100}],{q,0,20}]

V7

Series[((QPochhammer[q,q^5]QPochhammer[q^4,q^5])/(QPochhammer[q^2,q^5]QPochhammer[q^3,q^5])),{q,0,20}]

Product[(((1-(q^(5k+1)))(1-(q^(5k+4))))/((1-(q^(5k+2)))(1-(q^(5k+3))))),{k,0,∞}]

Product[(((1-(q^(5k-1)))(1-(q^(5k-4))))/((1-(q^(5k-2)))(1-(q^(5k-3))))),{k,∞}]

FullSimplify[%]

FullSimplify[(((1+q+(q^2)) QPochhammer[(1/(q^4)),(q^5)] QPochhammer[(1/q),(q^5)])/((1+(q^2)) QPochhammer[(1/(q^3)),(q^5)] QPochhammer[(1/(q^2)),(q^5)]))-((QPochhammer[q,(q^5)] QPochhammer[(q^4),(q^5)])/(QPochhammer[(q^2),(q^5)] QPochhammer[(q^3),(q^5)]))]

Series[%,{q,0,20}]

(q^(1/5))/ContinuedFractionK[q^n,{n,20}]/.q->.2

1/Fold[1+ q^#2/#1&,1,Range[3,1,-1]]//Together

(q^(-4/5))ContinuedFractionK[1,(q^(n-1)),{n,0,∞}]

RogersRamanujanR[q]

Table[ContinuedFractionK[(q^n),1,{n,0,k}],{k,4}]//Together

Together[{((q^(1/5))/(1+(q/(1+(q^2))))),((q^(1/5))/(1+(q/(1+((q^2)/(1+(q^3))))))),((q^(1/5))/(1+(q/(1+((q^2)/(1+((q^3)/(1+(q^4)))))))))}]

(q^(-1/5))R(q) Convergents

Table[(q^(-1/5))RogersRamanujanℛ[q,n],{n,4}]//Together

{numer,denom}=Through[{Numerator,Denominator}[Together[Table[(q^(-1/5))RogersRamanujanℛ[q,n],{n,5}]]]]

Numerator

numer

nc=CoefficientList[#,q]&/@numer

Flatten[nc]

https://oeis.org/A128915

Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^n*P_{n-2}(x).

RSolve[{P[0,x]==P[1,x]==1,P[n,x]==P[n-1,x]+x^n P[n-2,x]},P[n,x],n]

RSolve[{P[0]==1,P[1]==1,P[n]==P[n-1]+x^n P[n-2]},P[n],n]

A[0,x_]=A[1,x_]=1;
A[n_,x_]:=A[n,x]=A[n-1,x]+x^n A[n-2,x]

Expand[Table[A[n,q],{n,5}]-numer]

Denominator

nd=CoefficientList[#,q]&/@d

Flatten[nd]

https://oeis.org/A127836

Triangle read by rows: row n gives coefficients (lowest degree first) of P_n(x), where P_0(x) = P_1(x) = 1; P_n(x) = P_{n-1}(x) + x^(n-1)*P_{n-2}(x).

RSolve[{P[0]==1,P[1]==1,P[n]==P[n-1]+(x^(n-1)) P[n-2]},P[n],n]

Clear[B]

B[-1,x_]=B[0,x_]=1;
B[n_,x_]:=B[n,x]=B[n-1,x]+(x^n) B[n-2,x]

Expand[Table[B[n,q],{n,5}]-denom]

Both

res1=Table[((A[n,q])/(B[n,q])),{n,5}]

res2=Table[(q^(-1/5 ))RogersRamanujanℛ[q,n],{n,5}]

Together[res1-res2]

R(q) Identities

-R(q)+(1/(R(q)))-1==((QPochhammer1[(q^(1/5))])/((q^(1/5)) (QPochhammer1[(q^5)])))

Series[Subtract@@{(1/(RogersRamanujanR[q]))-1-RogersRamanujanR[q],((QPochhammer[(q^(1/5))])/((q^(1/5))QPochhammer[(q^5)]))},{q,0,20}]

{(1/(RogersRamanujanR[q]))-1-RogersRamanujanR[q],((QPochhammer[(q^(1/5))])/((q^(1/5))QPochhammer[(q^5)]))}

Subtract@@@Table[{-1+((QPochhammer[(q^2),(q^5)] QPochhammer[(q^3),(q^5)])/((q^(1/5)) QPochhammer[q,(q^5)] QPochhammer[(q^4),(q^5)]))-(((q^(1/5)) QPochhammer[q,(q^5)] QPochhammer[(q^4),(q^5)])/(QPochhammer[(q^2),(q^5)] QPochhammer[(q^3),(q^5)])),((QPochhammer[(q^(1/5))])/((q^(1/5)) QPochhammer[(q^5)]))}/.q->RandomReal[],{100}]//Chop//Union

FullSimplify[-1+((QPochhammer[(q^2),(q^5)] QPochhammer[(q^3),(q^5)])/((q^(1/5)) QPochhammer[q,(q^5)] QPochhammer[(q^4),(q^5)]))-(((q^(1/5)) QPochhammer[q,(q^5)] QPochhammer[(q^4),(q^5)])/(QPochhammer[(q^2),(q^5)] QPochhammer[(q^3),(q^5)]))]

-((((((QPochhammer1[q])))^6) ((R(q))^5))/(q ((((QPochhammer1[(q^5)])))^6)))+(1/((R(q))^5))-11

{(1/((RogersRamanujanR[q])^5))-11-((RogersRamanujanR[q])^5),(((QPochhammer[q])^6)/(q ((QPochhammer[(q^5)])^6)))}

Series[Subtract@@{(1/((RogersRamanujanR[q])^5))-11-((RogersRamanujanR[q])^5),(((QPochhammer[q])^6)/(q ((QPochhammer[(q^5)])^6)))},{q,0,50}]//Timing

Subtract@@@Table[{-11+((((QPochhammer[(q^2),(q^5)])^5) ((QPochhammer[(q^3),(q^5)])^5))/(q ((QPochhammer[q,(q^5)])^5) ((QPochhammer[(q^4),(q^5)])^5)))-((q ((QPochhammer[q,(q^5)])^5) ((QPochhammer[(q^4),(q^5)])^5))/(((QPochhammer[(q^2),(q^5)])^5) ((QPochhammer[(q^3),(q^5)])^5))),(((QPochhammer[q])^6)/(q ((QPochhammer[(q^5)])^6)))}/.q->RandomReal[],{100}]//Chop//Union

FullSimplify[-11+((((QPochhammer[(q^2),(q^5)])^5) ((QPochhammer[(q^3),(q^5)])^5))/(q ((QPochhammer[q,(q^5)])^5) ((QPochhammer[(q^4),(q^5)])^5)))-((q ((QPochhammer[q,(q^5)])^5) ((QPochhammer[(q^4),(q^5)])^5))/(((QPochhammer[(q^2),(q^5)])^5) ((QPochhammer[(q^3),(q^5)])^5)))]

R(q) Series

(∑_(n=-∞)^∞)(((-1))^n) (10 n+3) (q^((1/2) (5 n+3) n))== ((QPochhammer1[(q^5)])^3) ((((q^(1/5)) ((QPochhammer2[q, (q^5)])^3) ((QPochhammer2[(q^4), (q^5)])^3))/(((QPochhammer2[(q^2), (q^5)])^3) ((QPochhammer2[(q^3), (q^5)])^3)))+((3 ((QPochhammer2[(q^2), (q^5)])^2) ((QPochhammer2[(q^3), (q^5)])^2))/( ((QPochhammer2[q, (q^5)])^2) ((QPochhammer2[(q^4), (q^5)])^2))))

Sum[(((-1))^n)(10n+3)(q^((5n+3)n/2)),{n,-∞,∞}]

Series[Sum[(((-1))^n)(10n+3)(q^((5n+3)n/2)),{n,-10,10}],{q,0,10}]

Sum[(((-1))^n)(10n+3)(q^((5n+3)n/2)),{n,-30,30}]/.q->.2

((3/((RogersRamanujanR[q])^2))+((RogersRamanujanR[q])^3))(q^(2/5))((QPochhammer[(q^5)])^3)

(q^(2/5)) ((QPochhammer[(q^5)])^3)(((3 ((QPochhammer[(q^2),(q^5)])^2) ((QPochhammer[(q^3),(q^5)])^2))/((q^(2/5)) ((QPochhammer[q,(q^5)])^2) ((QPochhammer[(q^4),(q^5)])^2)))+(((q^(3/5)) ((QPochhammer[q,(q^5)])^3) ((QPochhammer[(q^4),(q^5)])^3))/(((QPochhammer[(q^2),(q^5)])^3) ((QPochhammer[(q^3),(q^5)])^3))))/.q->.2

Series[(q^(2/5)) ((QPochhammer[(q^5)])^5) (((3 ((QPochhammer[(q^2),(q^5)])^2) ((QPochhammer[(q^3),(q^5)])^2))/((q^(2/5)) ((QPochhammer[q,(q^5)])^2) ((QPochhammer[(q^4),(q^5)])^2)))+(((q^(3/5)) ((QPochhammer[q,(q^5)])^3) ((QPochhammer[(q^4),(q^5)])^3))/(((QPochhammer[(q^2),(q^5)])^3) ((QPochhammer[(q^3),(q^5)])^3)))),{q,0,10}]

Limit[((QPochhammer[q+h]-QPochhammer[q])/h),h->0]

Series[Product[1-(q^k),{k,50}],{q,0,20}]

CoefficientList[%,q]

Series[QPochhammer[q],{q,0,2}]

Sum

Sum[(((-1))^n)(10n+1)(q^((5n+1)n/2)),{n,-∞,∞}]//InputForm

Sum[(((-1))^n)(10n+1)(q^((5n+1)n/2)),{n,-30,30}]/.q->.2

((1/((RogersRamanujanR[q])^3))-3((RogersRamanujanR[q])^2))(q^(3/5))((QPochhammer[(q^5)])^3)

%/.q->.2

Plot of F(a,q)

<<Utilities`Plot`

RogersRamanujanF[a,q,4]

Off[Plot::"plnr"];
Plot[If[Abs[#]<5,#]&@RogersRamanujanF[1,q,50],{q,-1,1},PlotRange->{{-1,1},{-3,3}},PlotPoints->100,PlotStyle->Red,
AxesLabel->TraditionalForm/@{q,F[a,q]}]

ComplexPlot3D[Evaluate[RogersRamanujanF[1,q,100]],{q,-2,2},ComplexPlotLabel->TraditionalForm[F[1,q]]]//Timing

ComplexContourPlot[RogersRamanujanR[1,q,100],{q,-2,2},ComplexPlotLabel->TraditionalForm[F[1,q]]]//Timing

F(a,q) Roots

Flatten[q/.PlotRoots[RogersRamanujanF[1,q,200],{q,-1,1},MaxIterations->100,PlotPoints->100,PlotStyle->Red]]

FindRoot[RogersRamanujanF[1,q,100]==0,{q,.6},WorkingPrecision->20]

SmallestRootF1[a_]:=2/(a-1+Sqrt[(a+1)(a+5)])
SmallestRootF2[a_]:=1/(a-1+Sqrt[(a+1)(a+5)]/2+((a+3-Sqrt[(a+1)(a+5)])/(a-1+Sqrt[(a+1)(a+5)]))^3)

{#,N[#]}&[SmallestRootF1[1]]

{FullSimplify[#],N[#]}&[SmallestRootF2[1]]