OEIS/Eta products: Difference between revisions
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imported>Gfis Created page with "===Dedekind eta η=== * [https:oeis.org/A000700 A000700] ===Jacobi theta_2 === ===Jacobi θ_3, Ramanujan phi φ === [https://oeis.org/A000122 A000122] 1, 2, 0, 0,..." |
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===Dedekind eta η=== | ===Dedekind eta η=== | ||
* [https:oeis.org/A000700 A000700] | * [https:oeis.org/A000700 A000700] | ||
===Jacobi theta_2 === | ===Jacobi theta_2 θ_2=== | ||
===Jacobi θ_3, Ramanujan phi φ === | [https://oeis.org/A089800 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 === | |||
[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 | ||
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] | ||
eps P="[2,5;1,-2;4,-2]" | eps P="[2,5;1,-2;4,-2]" | ||
===Jacobi 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 | ||
eta(q)^2 / eta(q^2) | eta(q)^2 / eta(q^2) | ||
| Line 13: | Line 14: | ||
eps P="[1,2;2,-1]" | eps P="[1,2;2,-1]" | ||
===Ramanujan psi ψ=== | ===Ramanujan psi ψ=== | ||
* [https:oeis.org/A000700 A000700] | * [https//:oeis.org/A000700 A000700] | ||
===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 | |||
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=== | ===Ramanujan f=== | ||
[https//:oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0 | |||
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]" | |||
===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 14:42, 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 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 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/A000700 A000700]
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, 0 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]"
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.