Ramanujan Theta Functions

Author

Eric W. Weisstein
December 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/RamanujanThetaFunctions.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

f(a,b)-Function

f(a,b)==(∑_(n=-∞)^∞)(a^((1/2) n (n+1))) (b^((1/2) n (n-1)))==(QPochhammer[-a, a b, ∞]) (QPochhammer[-b, a b, ∞]) (QPochhammer[a b, a b, ∞])

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

f[a,b]

f(-1,a)==0

f[-1,a]

Sum[I^(n*(1+n))*a^((1/2)*(-1+n)*n),{n,-Infinity,Infinity},Assumptions->0<a<1]

Series[Sum[I^(n*(1+n))*a^((1/2)*(-1+n)*n),{n,-100,100}],{a,0,10}]

f(q)-Function

f(-q)==(∑_(n=-∞)^∞)(((-q))^((1/2) n (n+1))) (((-(q^2)))^((1/2) (n-1) n))==(QPochhammer1[q])

f[q_,n_:Infinity]:=f[q,-q^2,n];

f[-q]

Series[f[-q,20],{q,0,100}]

Series[QPochhammer[q],{q,0,100}]

Series[f[-q,-q^2,100]-QPochhammer[q],{q,0,10}]

f(-q)==(q^(-1/12))(DedekindEta[-((i log(q))/π)])

Series[Exp[-I π τ/12]DedekindEta[τ]/.τ->Log[q]/(I π),{q,0,20}]

f(-q)==(∑_(k=0)^∞)(((-1))^k) (q^((1/2) k (3 k-1)))+(∑_(k=1)^∞)(((-1))^k) (q^((1/2) k (3 k+1)))

Series[Sum[(((-1))^k)(q^(k(3k-1)/2)),{k,0,∞}]+Sum[(((-1))^k)(q^(k(3k+1)/2)),{k,∞}]/.∞->500,{q,0,100}]

Series[Product[1-(q^k),{k,500}],{q,0,100}]

f(-q)==(∏_k^∞)(1-(q^k))

Product[1-(q^k),{k,∞}]

q f(-(q^24))==(DedekindEta[24 τ])==q (QPochhammer2[(q^24), (q^24)])==(∑_(k=-∞)^∞)(((-1))^k) (q^(((6 k+1))^2))

Sum[(((-1))^k)(q^(((6k+1))^2)),{k,-∞,∞}]

Series[q QPochhammer[(q^24),(q^24)],{q,0,500}]//Normal

Series[q f[-q^24]/.{-∞->-500,∞->500},{q,0,200}]

Series[Sum[(((-1))^k)(q^(((6k+1))^2)),{k,-∞,∞}]/.{-∞->-500,∞->500},{q,0,200}]//Normal

{DedekindEta[24τ],Exp[2π I τ]QPochhammer[Exp[48π I τ]]}/.τ->.1I

Exp[2π I τ]QPochhammer[Exp[48π I τ]]/.τ->-((I Log[q])/(2 π))

Series[q QPochhammer[(q^24),(q^24)],{q,0,100}]

φ-Function

φ(q)==(EllipticTheta[3, 0, q])

φ[q_]:=EllipticTheta[3,0,q]

V6

Sum[x^(m^2),{m,-Infinity,Infinity}]

V7

Sum[x^(m^2),{m,-Infinity,Infinity}]

1+2Sum[(q^(k^2)),{k,∞}]

q-Series

FullSimplify[((QPochhammer[-q,(q^2)]QPochhammer[(q^2)])/(QPochhammer[q,(q^2)]QPochhammer[-(q^2),(q^2)]))]

Series[((QPochhammer[-q,-q])/(QPochhammer[q,-q]))-EllipticTheta[3,0,q],{q,0,100}]

Series

Series[φ[q],{q,0,100}]

φ(exp(-π (Sqrt[2])))==(((Gamma[(9/8)]) (Sqrt[((Gamma[(1/4)])/((2^(1/4)) π))]))/(Gamma[(5/4)]))

φ[Exp[-π(Sqrt[2])]]

FunctionExpand[%]

FullSimplify[%]

N[{%,(((Gamma[(9/8)]) (Sqrt[((Gamma[(1/4)])/((2^(1/4)) π))]))/(Gamma[(5/4)]))}]

FullSimplify[EllipticTheta[3,0,(E^(-(Sqrt[2])π))]-(((Gamma[(9/8)])(Sqrt[((Gamma[(1/4)])/((2^(1/4))π))]))/(Gamma[(5/4)]))]

φ(exp(-π))==((π^(1/4))/(Gamma[(3/4)]))

φ[Exp[-π]]

FunctionExpand[%]

FullSimplify[%]

N[{%,((π^(1/4))/(Gamma[(3/4)]))}]

FullSimplify[EllipticTheta[3,0,(E^(-π))]-((π^(1/4))/(Gamma[(3/4)]))]

ψ-Function

f(q,(q^3))==(∑_(k=0)^∞)(q^((1/2) k (k+1)))==((EllipticTheta[2, 0, (Sqrt[q])])/(2 (q^(1/8))))

V6

Sum[(q^(k(k+1)/2)),{k,0,∞}]

V7

Sum[(q^(k(k+1)/2)),{k,0,∞}]

Series[f[q,q^3,100],{q,0,100}]

Series[((EllipticTheta[2,0,(Sqrt[q])])/(2 (q^(1/8)))),{q,0,100}]

CoefficientList[Series[((EllipticTheta[2,0,(Sqrt[q])])/(2 (q^(1/8)))),{q,0,100}],q]

f(q,(q^3))==(∑_(n=-∞)^∞)(q^((1/2) n (n+1))) ((((q^3)))^((1/2) (n-1) n))

f[q,q^3]

f(q,(q^3))==(QPochhammer1[(q^2)]) (QPochhammer2[-q, q])

FullSimplify[QPochhammer[-q,q]QPochhammer[(q^2)]]

FunctionExpand[((EllipticTheta[2,0,(Sqrt[q])])/(2 (q^(1/8))))]

FullSimplify[QPochhammer[-q,q]QPochhammer[(q^2)]-((EllipticTheta[2,0,(Sqrt[q])])/(2 (q^(1/8))))]

ByteCount/@{QPochhammer[-q,q]QPochhammer[(q^2)],((EllipticTheta[2,0,(Sqrt[q])])/(2 (q^(1/8))))}

f(q,(q^3))==((QPochhammer1[(q^2)])/(QPochhammer2[q, (q^2)]))

FullSimplify[((QPochhammer[(q^2)])/(QPochhammer[q,(q^2)]))]

FullSimplify[((QPochhammer[(q^2)])/(QPochhammer[q,(q^2)]))-QPochhammer[-q,q]QPochhammer[(q^2)]]

ByteCount/@{((QPochhammer[(q^2)])/(QPochhammer[q,(q^2)])),QPochhammer[-q,q]QPochhammer[(q^2)]}

χ-Function

Product[1+(q^(2k+1)),{k,0,∞}]

Series[QPochhammer[-q,q^2],{q,0,20}]

Series[Product[1+(q^(2k+1)),{k,0,100}],{q,0,20}]

CoefficientList[Series[QSeries[-q,q^2,1000],{q,0,100}],q]

Different Definition of the φ-Function

(Subscript[φ, 2])[z_]:=Sqrt[EllipticTheta[2,0,z]/EllipticTheta[3,0,z]]

args={-Exp[-Pi Sqrt[3]]};
vals={(4Sqrt[3]-7)^(1/8)};

FullSimplify[(Subscript[φ, 2])/@args]

N[Transpose[{(Subscript[φ, 2])/@args,vals}],20]

FullSimplify[(Sqrt[((EllipticTheta[2,0,-(E^(-(Sqrt[3]) π))])/(EllipticTheta[3,0,-(E^(-(Sqrt[3]) π))]))])-(4Sqrt[3]-7)^(1/8)]

FullSimplify[(Sqrt[((EllipticTheta[2,0,-(E^(-(Sqrt[3]) π))])/(EllipticTheta[3,0,-(E^(-(Sqrt[3]) π))]))])]

FunctionExpand[%]

FullSimplify[(((((-1))^(1/8)) (Sqrt[2]) (E^(-(((Sqrt[3]) π)/8))) QPochhammer[-(E^(-2 (Sqrt[3]) π)),(E^(-2 (Sqrt[3]) π))])/(QPochhammer[(E^(-(Sqrt[3]) π)),(E^(-2 (Sqrt[3]) π))]))-(4Sqrt[3]-7)^(1/8)]

N[%,1000]