Dedekind eta η
- [https:oeis.org/A000700 A000700]
Jacobi theta_2 θ_2
A089800 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2 ... = theta_2(q)/q^(1/4)
Jacobi theta_3 θ_3, Ramanujan phi φ
A000122 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2
a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.
eta(q^2)^5 / (eta(q)*eta(q^4))^2
Euler transform of period 4: [2,-3,2,-1]
eps P="[2,5;1,-2;4,-2]"
Jacobi theta_4 θ_4
A002448 1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2
a(0) = 1, for n >= 1: a(n) = 2 * (-1)^sqrt(n) if n is a square, otherwise 0.
eta(q)^2 / eta(q^2)
Euler transform of period 2: [2,-1]
eps P="[1,2;2,-1]"
Ramanujan psi ψ
- [https//:oeis.org/A010054 A010054] 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
a(n) = 1 if n is a triangular number, otherwise 0.
q^(-1/8) * eta(q^2)^2 / eta(q)
Euler transform of period 2: [1,-1]
eps P="[2,2;1,-1]"
Ramanujan chi χ
[https//:oeis.org/A000700 A000700] 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5
q^(1/24) * eta(q^2)^2 /(eta(q) * eta(q^4))
Euler transform of period 4: [1,-1,1,0]
eps P="[2,2;1,-1;4,-1]"
Ramanujan f
[https//:oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1
a(n) = (-1)^n * A010815(n)
q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4))
Euler transform of period 4: [[1,-2,1,-1]
eps P="[2,3;1,-1;4,-1]"
Cookbook
- For all tuples [m,k] add m*k.
- The resulting integer part gives the q-shift, while the fractional part is noted before the eta product.