OEIS/Eta products: Difference between revisions
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===Jacobi theta_3 θ_3, Ramanujan phi φ === | ===Jacobi theta_3 θ_3, Ramanujan phi φ === | ||
[https://oeis.org/A000122 A000122] 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2 | [https://oeis.org/A000122 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 | eta(q^2)^5 / (eta(q)*eta(q^4))^2 | ||
Euler transform of period 4: [2,-3,2,-1] | Euler transform of period 4: [2,-3,2,-1] | ||
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===Jacobi theta_4 θ_4=== | ===Jacobi theta_4 θ_4=== | ||
[https://oeis.org/A002448 A002448] 1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2 | [https://oeis.org/A002448 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) | eta(q)^2 / eta(q^2) | ||
Euler transform of period 2: [2,-1] | Euler transform of period 2: [2,-1] | ||
eps P="[1,2;2,-1]" | eps P="[1,2;2,-1]" | ||
===Ramanujan psi ψ=== | ===Ramanujan psi ψ=== | ||
* [https//:oeis.org/ | * [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=== | ===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 | [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] | Euler transform of period 4: [1,-1,1,0] | ||
eps P="[2,2;1,-1;4,-1]" | eps P="[2,2;1,-1;4,-1]" | ||
===Ramanujan f=== | ===Ramanujan f=== | ||
[https//:oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1 | [https//:oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1 | ||
q^(1/24) * eta(q^2)^ | a(n) = (-1)^n * [https://oeis.org/A010815 A010815](n) | ||
Euler transform of period 4: [1,-1,1 | q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4)) | ||
eps P="[2, | Euler transform of period 4: [[1,-2,1,-1] | ||
eps P="[2,3;1,-1;4,-1]" | |||
===Cookbook=== | ===Cookbook=== | ||
* For all tuples [m,k] add m*k. | * 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. | * The resulting integer part gives the q-shift, while the fractional part is noted before the eta product. | ||
Revision as of 15:54, 23 January 2023
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.