OEIS/Eta products

Eta product signatures (EPSIG)

The coefficient sequences for generating functions that are products of Dedkind's eta function can be computed conveniently by an Euler transform of some periodic integer sequence. Following Michael Somos, we describe an eta product by a matrix resp. a list of pairs (q_i, e_i) of the form

[q1,e1;q2,e2;q3,e3;...]. 

The pairs are separated by ";".

• The q_i are the powers of the argument q inside the eta function , and
• the e_i are the powers of the eta functions. Only these can be negative.

We call such a list the signature ("EPSIG") of the eta product. The following sections contain a number of examples.

The order of the pairs is irrelevant for the computation of the period, but see the section about normalization, below. Once the signatures are normalized, they signatures can be sorted and stored in a database table. The original expression of eta functions can be reconstructed from them, while in the generated periods the original structure is lost.

The generation of a period from an EPSIG, the following Euler transform and the generation of the coefficients are implemented in transform/EtaProductSequence.java as part of the project jOEIS of Sean A. Irvine.

Common generating functions

Partition numbers

A000041 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231
Euler transform of period 1: [1]
G.f.: Product_{k>0} 1/(1-x^k)
EPSIG="[1,-1]" Q="1/24"

Dedekind eta η (without the q^(1/24) factor), Ramanujan f(-q)

A010815 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1
G.f.: eta(q)
Euler transform of period 1: [-1]
EPSIG="[1,1]"

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.
G.f.: eta(q^2)^5/(eta(q)*eta(q^4))^2
G.f.: Product_{k>0} (1 - x^(2*k))*(1 + x^(2*k-1))^2.
Euler transform of period 4: [2,-3,2,-1]
EPSIG="[2,5;4,-2;1,-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.
G.f.: eta(q)^2/eta(q^2)
Euler transform of period 2: [2,-1]
EPSIG="[1,2;2,-1]"

Ramanujan psi ψ

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.
G.f.: q^(-1/8) * eta(q^2)^2/eta(q)
G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1))
Euler transform of period 2: [1,-1]
EPSIG="[2,2;1,-1]"

Ramanujan chi χ

A000700 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5
G.f.: q^(1/24) * eta(q^2)^2/(eta(q)*eta(q^4))
G.f.: Product_{k>=1} (1 + x^(2*k-1))
Euler transform of period 4: [1,-1,1,0]
EPSIG="[2,2;4,-1;1,-1]"

Ramanujan f

A121373 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1
a(n) = (-1)^n * A010815(n)
G.f.: q^(-1/24) * eta(q^2)^3/(eta(q)*eta(q^4))
Euler transform of period 4: [1,-2,1,-1]
EPSIG="[2,3;4,-1;1,-1]"

Ramanujan tau τ

A000594 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612
G.f.: eta(q)^24
G.f.: x * Product_{k>=1} (1 - x^k)^24
Euler transform of period 1: [-24]
EPSIG="[1,24]"

Ramanujan a cubic AGM theta

A004016 1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6
G.f.: theta_3(q)*theta_3(q^3) + theta_2(q)*theta_2(q^3)
G.f.: phi(x)*phi(x^3) + 4*x*psi(x^2)*psi(x^6)

Ramanujan b cubic AGM theta

A005928 1, -3, 0, 6, -3, 0, 0, -6, 0, 6, 0, 0, 6, -6, 0, 0, -3
G.f.: eta(q)^3/(eta(q^3)
Euler transform of period 3: [-3,-3,-2]
EPSIG="[1,3;3,-1]"

Ramanujan c cubic AGM theta

A005882 3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0
G.f.: 3*eta(q^3)^3/eta(q)
3 * (Euler transform of period 3: [-3,-3,-2])
EPSIG="[3,3;1,-1]" Q="-1/3" F=3

Eisenstein series

E_2 = Ramanujan P(q)

A006352 1, -24, -72, -96, -168, -144, -288, -192, -360
a(n) = -24*sigma(n) 
G.f.: 1 + 24 *q*deriv(eta(q))/eta(q)

E_4 = Ramanujan Q(q)

A004009 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400
G.f.: (eta(q)^24 + 256*eta(q^2)^24) / (eta(q)*eta(q^2))^8 
EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"

E_6 = Ramanujan R(q)

A013973 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232
a(0) = 1, for n >= 1: a(n) = -504*sigma_5(n)
G.f.: (eta(q)^8 + 32*eta(q^4)^8) * (eta(q)^16 - 512*eta(q)^8*eta(q^4)^8 - 8192*eta(q^4)^16) / eta(q^2)^12 
"[1,8]+32*[4,8]" convolved with "[1,16;2,-12]-512*[1,8;4,8;2,-12]-8192*[4,16;2,-12]"

E_8

A008410 1, 480, 61920, 1050240, 7926240, 37500480, 135480960
a(0) = 1, for n >= 1: a(n) = 480*sigma_7(n)
G.f.: (theta2(q)^16 + theta3(q)^16 + theta4(q)^16)/2.
G.f.: ((eta(q)^24 + 256*eta(q^2)^24) / (eta(q)*eta(q^2))^8)^2
"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared

E_2k, k=1..16

E_2:  A006352 a(0) =             1, for n >= 1: a(n) =     -24*sigma_1(n) 
E_4:  A004009 a(0) =             1, for n >= 1: a(n) =     240*sigma_3(n) 
E_6:  A013973 a(0) =             1, for n >= 1: a(n) =    -504*sigma_5(n) 
E_8:  A008410 a(0) =             1, for n >= 1: a(n) =     480*sigma_7(n) 
E_10: A013974 a(0) =             1, for n >= 1: a(n) =    -264*sigma_9(n) 
E_12: A029828 a(0) =           691, for n >= 1: a(n) =   65520*sigma_11(n) 
E_14: A058550 a(0) =             1, for n >= 1: a(n) =     -24*sigma_13(n) 
E_16: A029829 a(0) =          3617, for n >= 1: a(n) =   16320*sigma_15(n) 
E_18: A279892 a(0) =         43867, for n >= 1: a(n) =  -28728*sigma_17(n) 
E_20: A029830 a(0) =        174611, for n >= 1: a(n) =   13200*sigma_19(n) 
E_22: A279893 a(0) =         77683, for n >= 1: a(n) =    -552*sigma_21(n) 
E_24: A029831 a(0) =     236364091, for n >= 1: a(n) =  131040*sigma_23(n)
E_26: A282356 a(0) =        657931, for n >= 1: a(n) =     -24*sigma_25(n)
E_28: A282401 a(0) =    3392780147, for n >= 1: a(n) =    6960*sigma_27(n)
E_30: A282182 a(0) = 1723168255201, for n >= 1: a(n) = -171864*sigma_29(n)
E_32: A282540 a(0) = 7709321041217, for n >= 1: a(n) =   32640*sigma_31(n)

The a(0) constants are given by abs(A001067: Numerator of Bernoulli(2*n)/(2*n)), while the factors of sigma_xx(n) are found in A006863: Denominator of B_{2*n}/(-4*n

Leading power of q factor (pqf)

Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to q^0.

• Example A226862:

G.f.: q^(-1/6) * eta(q^4) * eta(q^6)^5 / (eta(q^3) * eta(q^12))^2
EPSIG="[4,1;6,5;3,-2;12,-2]"

For all pairs we multiply the q power with the eta power and add the results, giving 4*1 + 6*5 - 3*2 - 12*2 = 4 + 30 - 6 - 24 = 4. This is the negative numerator of the pqf (expanded for denominator 24).

• Example A286813:

G.f.: q^(-9/8) * (eta(q^2) * eta(q^16))^2 / (eta(q) * eta(q^8))
EPSIG="[2,2;16,2;1,-1;8,-1]" gives 2*2 * 16*2  - 1*1 - 8*1 = 36 - 9 = 27 and pqf = q^(-27/24)

• Example A286134 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, -1, -1

G.f.: q^(-1/2) * eta(q^5) * eta(q^6) * eta(q^7) * eta(q^210)
EPSIG="[5,1;6,1;7,1;210,1]" gives 18 + 210 = 228 and pqf = q^(-228/24) = -9 -1/2. 

The sequence is shifted right by 9 positions in this case.

Computation of the period from an EPSIG

This algorithm is rather straightforward. It uses an array of integers that are usually small. The length of the array is the least common multiple (LCM) of all q_i. The array with indexes [0..lcm-1] is initialized with zeros.

1. For all pairs (q_i, e_i)
   1. Build a block of q_i - 1 zeros, followed by -1 * e_i.
   2. Add a sufficient concatenation of such blocks onto the whole array.
2. The resulting array is the period for the Euler transform.

As an example, we derive the period of the theta_3 function above from its EPSIG [2,5;4,-2;1,-2]. The period length is LCM(2,1,4) = 4.

 pair 2,5   ->   0 -5  0 -5 
 pair 4,-2  ->   0  0  0  2
 pair 1,-2  ->   2  2  2  2
 --------------------------
 sum = period = [2,-3, 2,-1] 

This algorithm for expanding a list of powers of theta_3 functions is realized in theta3_epsig.pl.

Normalization of an EPSIG

For long lists of generating functions it is convenient to require all EPSIGs to be normalized, such that they can be tested for equality and can be sorted. We use the following rules:

1. The sublist of pairs with positive e_i comes before the sublist with negative e_i.
2. Inside a sublist the first components q_i are in descending order.
3. Successive pairs with the same q_i are combined into a single pair with the sum of the e_i as second component.

The normalization rules are implemented in norm_epsig.pl.

Multiplication of two EPSIGs

Two EPSIGs could be multiplied by concatenating their list of pairs. It is convenient, however, to combine all occurrences of specific q_i by adding the corresponding eta exponents e.

Exponentiation of an EPSIG

An EPSIG is taken to a power m by multiplying all e_i (the second components in the pairs) by m. The exponent may be 1/2 for the square root, or any other fraction m_n/m_d as long as the q_i are all divisible by m_d.

Eta polynomials

In a number of OEIS sequences the eta products occur as components of a polynomial (cf. for example E_4, E_6, E_8 above).