Mock Theta Functions

Author

Eric W. Weisstein
July 18, 2013

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

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

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

Order-3

f(q)

Product[1+q^k,{k,n}]^2

f[q_,s_:∞]:=Sum[q^(n^2)/((1/4) ((QPochhammer[-1,q,1+n])^2)),{n,0,s}]

f[q]

Series[f[q,200],{q,0,10}]

c=CoefficientList[Series[f[q,200],{q,0,100}],q];

a[n_]:=c[[n]]

γ=First[Table[gamma[n],{n,0,#}]/.Solve[Join[Table[a[n]==Sum[PartitionsP[r]gamma[n-r],{r,0,n}],{n,#}],{gamma[0]==0}],Table[gamma[n],{n,0,#}]]&[100]]

g[n_]:=γ[[n+1]]

Table[Sum[PartitionsP[r]g[n- r],{r,0,n}],{n,20}]

Take[c,20]

ϕ(q)

Product[1+q^(2k),{k,n}]

ϕ[q_,s_:∞]:=Sum[q^(n^2)/((1/2) QPochhammer[-1,(q^2),1+n]),{n,0,s}]

ϕ[q]

c=CoefficientList[Series[ϕ[q,50],{q,0,100}],q]

ψ(q)

Product[1-q^(2k-1),{k,n}]//FullSimplify

ψ[q_,s_:∞]:=Sum[((q^(n^2))/((q QPochhammer[(1/q),(q^2),1+n])/(-1+q))),{n,s}]

ψ[q]

c=CoefficientList[Series[ψ[q,50],{q,0,100}],q]

χ(q)

Product[1-q^k+q^(2k),{k,n}]//FullSimplify

χ[q_,s_:∞]:=Sum[((q^(n^2))/(QPochhammer[(1/2) (1-I (Sqrt[3])),q,1+n] QPochhammer[(1/2) (1+I (Sqrt[3])),q,1+n])),{n,0,s}]

χ2[q_,s_:∞]:=Sum[((q^(n^2))/(Product[1-q^k+q^(2k),{k,n}])),{n,0,s}]

c=CoefficientList[Series[χ2[q,50],{q,0,100}],q]

ω(q)

Product[(1-q^(2k+1))^2,{k,0,n}]

ω[q_,s_:∞]:=Sum[((q^(2n(n+1)))/((QPochhammer[q,(q^2),1+n])^2)),{n,0,s}]

ω[q]

c=CoefficientList[Series[ω[q,50],{q,0,100}],q]

ν(q)

Product[(1+q^(2k+1)),{k,0,n}]

ν[q_,s_:∞]:=Sum[((q^(n(n+1)))/(QPochhammer[-q,(q^2),1+n])),{n,0,s}]

c=CoefficientList[Series[ν[q,50],{q,0,100}],q]

ρ(q)

FullSimplify[Product[(1+q^(2k+1)+q^(4k+2)),{k,0,n}],n∈Integers]

ρ[q_,s_:∞]:=Sum[((q^(2n(n+1)))/(QPochhammer[-(((-1))^(1/3)) q,(q^2),1+n] QPochhammer[(((-1))^(2/3)) q,(q^2),1+n])),{n,0,s}]

ρ2[q_,s_:∞]:=Sum[((q^(2n(n+1)))/(Product[(1+q^(2k+1)+q^(4k+2)),{k,0,n}])),{n,0,s}]

c=CoefficientList[Series[ρ2[q,50],{q,0,100}],q]

Ramanujan Order-5

Andrews (1986, p. 125)

$Version

(Subscript[f, 0])(q)

Sum[((q^(n^2))/(QPochhammer[-q,q,n])),{n,0,∞}]//Timing

(Subscript[F, 0])(q)

Sum[((q^(2(n^2)))/(QPochhammer[q,(q^2),n])),{n,0,∞}]//Timing

1+2(Subscript[ψ, 0])(q)

Sum[QPochhammer[-1,q,n](q^(Binomial[n+1,2])),{n,0,∞}]

(Subscript[ϕ, 0])(q)

Sum[QPochhammer[-q,(q^2),n](q^(n^2)),{n,0,∞}]

(Subscript[f, 1])(q)

Sum[((q^((n^2)+n))/(QPochhammer[-q,q,n])),{n,0,∞}]

(Subscript[F, 1])(q)

Sum[((q^(2(n^2)+2n))/(QPochhammer[q,(q^2),n+1])),{n,0,∞}]

(Subscript[ψ, 1])(q)

Sum[QPochhammer[-q,q,n](q^(Binomial[n+1,2])),{n,0,∞}]

(Subscript[ϕ, 1])(q)

Sum[QPochhammer[-q,(q^2),n](q^(((n+1))^2)),{n,0,∞}]

(Subscript[χ, 0])(q)

(Subscript[χ, 1])(q)

Ramanujan Order-6

ϕ(q)

ψ(q)

ρ(q)

σ(q)

λ(q)

μ(q)

γ(q)

Ramanujan Order-7

(Subscript[ℱ, 0])(q)

Sum[((q^(n^2))/(QPochhammer[(q^(n+1)),(q^(n+1)),n])),{n,0,∞}]

(Subscript[ℱ, 1])(q)

Sum[((q^(n^2))/(QPochhammer[(q^n),(q^n),n])),{n,0,∞}]

(Subscript[ℱ, 2])(q)

Sum[((q^((n^2)+n))/(QPochhammer[(q^(n+1)),(q^(n+1)),n+1])),{n,0,∞}]

Gordon Order-8

Gordon

Series[Sum[q^(2n^2)/qSeries[q,q^2,n],{n,50}],{q,0,20}]

Series[Sum[q^(2n(n-1))/qSeries[q,q^2,n],{n,100}],{q,0,20}]