Commonly used functions in modular equations

Commonly used functions in modular equations


Ramanujan's general two argument symmetric theta function is defined by :
f(a,b) := 1 +(a+b) +(a^2+b^2)(ab) +(a^3+b^3)(ab)^3 +(a^4+b^4)(ab)^6 +... .

Ramanujan's one argument theta functions are defined by :
f(-q)   := f(-q,-q^2) = (1-q)(1-q^2)(1-q^3)(1-q^4)... .         (A010815)
phi(q) := f(q,q) = 1 +2q +2q^4 +2q^9 +2q^16 +... = sum_n q^n^2 .
psi(q) := f(q,q^3) = 1 +q +q^3 +q^6 +q^10 +... = sum_{n=0..} q^(n(n+1)/2) .
chi(q) := (1+q)(1+q^3)(1+q^5)(1+q^7)... [[ not really a theta function ]] .

Dedekind's eta function is defined by :
eta(z) = q^(1/24) f(-q) := exp(pi i z/12) f(-q)  where q = exp(2 pi i z) .

Jacobi's theta null functions are defined by :
theta_2(q) = 2 q^(1/4)(1 + q^2 + q^6 + q^20 + ...) = sum_n q^(n + 1/2)^2 .
theta_3(q) = 1 + 2 q + 2 q^4 + 2 q^9 + 2 q^16 + ... = sum_n q^n^2 .
theta_4(q) = 1 - 2 q + 2 q^4 - 2 q^9 + 2 q^16 + ... = sum_n (-1)^n q^n^2 .

Jacobi/Legendre moduli in powers of the nome q = exp(-pi K'/K) are :
k(q)  = 4 q^(1/2)(1 - 4 q + 14 q^2 - 40 q^3 + 101 q^4 - ...) .
k'(q) = 1 - 8 q + 32 q^2 - 96 q^3 + 256 q^4 - 624 q^5 + ... .

Jacobi/Legendre functions in terms of q^n and f(-q^n) are :
k(q)   := 4 q^(1/2) (f(-q)f^2(-q^4)/f^3(-q^2))^4 .               (A127393)
k(-q)  := 4 (-q)^(1/2) (f(-q^4)/f(-q))^4 .                       (A093160)
k'(q)  := (f^2(-q)f(-q^4)/f^3(-q^2))^4 .                         (A139820)
k'(-q) := (f^3(-q^2)/(f^2(-q)f(-q^4)))^4 .                       (A014969)
K(q)   := pi/2 f(-q^2)^10/(f(-q)f(-q^4))^4 .                     (A004018)
K(-q)  := pi/2 f(-q)^4/f(-q^2)^2 .                               (A104794)

Note: lamba is usually a function of tau and here q = exp(pi i tau).
Elliptic modular lambda function = k(q)^2 in terms of q^n and f(-q^n) :
lambda(q)    := 16 q (f(-q)f^2(-q^4)/f^3(-q^2))^8 .              (A115977)
lambda(-q)   := -16 q (f(-q^4)/f(-q))^8 .                        (A132136)
1-lambda(q)  := (f^2(-q)f(-q^4)/f^3(-q^2))^8 .                   (A128692)
1-lambda(-q) := (f^3(-q^2)/(f^2(-q)f(-q^4)))^8 .                 (A014972)

Ramanujan theta function definitions in terms of f(-q^n) are :
f(q)       := f^3(-q^2)/(f(-q)f(-q^4)) .                         (A121373)
f(q,q^5)   := f^2(-q^2)f(-q^3)f(-q^12)/(f(-q)f(-q^4)f(-q^6)) .   (A089801)
f(-q,-q^5) := f(-q)f^2(-q^6)/(f(-q^2)f(-q^3)) .                  (A089802)
f(q,q^2)   := f(-q^2)f^2(-q^3)/(f(-q)f(-q^6)) .                  (A080995)
f(-q,q^2)  := f(-q)f(-q^4)f^5(-q^6)/(f(-q^2)f(-q^3)f(-q^12))^2 . (A133985)
phi(q)     := f^5(-q^2)/(f^2(-q)f^2(-q^4)) .                     (A000122)
phi(-q)    := f^2(-q)/f(-q^2) .                                  (A002448)
psi(q)     := f^2(-q^2)/f(-q) .                                  (A010054)
psi(-q)    := f(-q)f(-q^4)/f(-q^2) .                             (A106459)
chi(q)     := f^2(-q^2)/(f(-q)f(-q^4)) .                         (A000700)
chi(-q)    := f(-q)/f(-q^2) .                                    (A081362)

Jacobi theta null functions in terms of Ramanujan theta functions are :
theta_2(q) := 2 q^(1/4) psi(q^2) .                               (A089799)
theta_3(q) := phi(q) .                                           (A000122)
theta_4(q) := phi(-q) .                                          (A002448)

Borwein cubic AGM theta functions in terms of q^n and f(-q^n) are :
a(q) := (f^3(-q) + 9 q f^3(-q^9))/f(-q^3) .                      (A004016)
b(q) := f^3(-q)/f(-q^3) .                                        (A005928)
c(q) := 3 q^(1/3) f^3(-q^3)/f(-q) .                              (A005882)

My notation for Dedekind eta functions of multiple arguments is
u1 = eta(tau) , u2 = eta(2 tau) , u3 = eta(3 tau) , etc ...
but *without* q^(n/24) factors which are supplied separately.
Note that u1 = f(-q), u2 = f(-q^2), u3 = f(-q^3), etc ...
and that here q = exp(2 pi i tau) .

Last updated 07 Feb 2017
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