• Any modular form of weight
k (possibly only meromorphic at cusps) is a mock modular form of weight
k with shadow 0. • The quasimodular Eisenstein series ::\displaystyle E_2(\tau) = 1-24\sum_{n>0}\sigma_1(n)q^n :of weight 2 and level 1 is a mock modular form of weight 2, with shadow a constant. This means that ::\displaystyle E_2(\tau) -\frac{3}{\pi y} :transforms like a modular form of weight 2 (where =
x +
iy). • The function studied by Don Zagier with Fourier coefficients that are Hurwitz class numbers
H(
N) of imaginary quadratic fields is a mock modular form of weight , level 4 and shadow Σ
q n2. The corresponding weak Maass wave form is ::F(\tau) = \sum_NH(N)q^n + y^{-1/2}\sum_{n\in Z}\beta(4\pi n^2y)q^{-n^2} :where ::\beta(x) = \frac{1}{16\pi}\int_1^\infty u^{-3/2}e^{-xu}du :and
y = Im(),
q = e2
i . Mock theta functions are mock modular forms of weight whose shadow is a unary theta function, multiplied by a rational power of
q (for historical reasons). Before the work of Zwegers led to a general method for constructing them, most examples were given as
basic hypergeometric functions, but this is largely a historical accident, and most mock theta functions have no known simple expression in terms of such functions. The "trivial" mock theta functions are the (holomorphic) modular forms of weight , which were classified by Serre and Stark, who showed that they could all be written in terms of theta functions of 1-dimensional lattices. The following examples use the
q-Pochhammer symbols (
a;
q)
n which are defined as: :(a;q)_n = \prod_{0\le j
Order 2 Some order 2 mock theta functions were studied by McIntosh. :A(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2}(-q;q^2)_n}{(q;q^2)^2_{n+1}} = \sum_{n\ge 0} \frac{q^{n+1}(-q^2;q^2)_n}{(q;q^2)_{n+1}} :B(q) = \sum_{n\ge 0} \frac{q^{n(n+1)}(-q^2;q^2)_n}{(q;q^2)^2_{n+1}} = \sum_{n\ge 0} \frac{q^{n}(-q;q^2)_n}{(q;q^2)_{n+1}} :\mu(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q^2;q^2)^2_{n}} The function
μ was found by Ramanujan in his lost notebook. These are related to the functions listed in the section on order-8 functions by : U_0(q) - 2U_1(q) = \mu(q) : V_0(q) - V_0(-q) = 4qB(q^2) : V_1(q) + V_1(-q) = 2A(q^2)
Order 3 Ramanujan mentioned four order-3 mock theta functions in his letter to Hardy, and listed a further three in his lost notebook, which were rediscovered by
G. N. Watson. The latter proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appell–Lerch sums. Dragonette described the asymptotic expansion of their coefficients. Zwegers related them to harmonic weak Maass forms. See also the monograph by Nathan Fine. The seven order-3 mock theta functions given by Ramanujan are : f(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q; q)_n^2} = \frac{2}{\prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^nq^{n(3n+1)/2}}{1+q^n} , . : \phi(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q^2;q^2)_n} = \frac{1}{\prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^n(1+q^n)q^{n(3n+1)/2}}{1+q^{2n}} . : \psi(q) = \sum_{n > 0} \frac{q^{n^2}}{(q;q^2)_n} = \frac{q}{\prod_{n>0}(1-q^{4n})}\sum_{n\in \mathbf{Z}}\frac{(-1)^nq^{6n(n+1)}}{1-q^{4n+1}} . : \chi(q) = \sum_{n\ge 0} \frac{q^{n^2}}{\prod_{1\le i\le n}(1-q^i+q^{2i})} = \frac{1}{2 \prod_{n>0}(1-q^n)}\sum_{n\in \mathbf{Z}}\frac{(-1)^n(1+q^n)q^{n(3n+1)/2}}{1-q^n+q^{2n}} . : \omega(q) = \sum_{n\ge 0} \frac{q^{2n(n+1)}}{(q;q^2)^2_{n+1}} = \frac{1}{\prod_{n>0}(1-q^{2n})}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)} \frac{1+q^{2n+1}}{1-q^{2n+1}}} . : \nu(q) = \sum_{n\ge 0} \frac{q^{n(n+1)}}{(-q;q^2)_{n+1}} = \frac{1}{\prod_{n>0}(1-q^n)}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)/2}\frac{1-q^{2n+1}}{1+q^{2n+1}}} . : \rho(q) = \sum_{n\ge 0} \frac{q^{2n(n+1)}}{\prod_{0\le i\le n}(1+q^{2i+1}+q^{4i+2})} = \frac{1}{\prod_{n>0}(1-q^{2n})}\sum_{n\ge 0}{(-1)^nq^{3n(n+1)} \frac{1-q^{4n+2}}{1+q^{2n+1}+q^{4n+2}}} . The first four of these form a group with the same shadow (up to a constant), and so do the last three. More precisely, the functions satisfy the following relations (found by Ramanujan and proved by Watson): :\begin{align} 2\phi(-q) - f(q) &= f(q) + 4\psi(-q) = \theta_4(0,q)\prod_{r > 0}\left(1 + q^r\right)^{-1} \\ 4\chi(q) - f(q) &= 3\theta_4^2(0, q^3)\prod_{r > 0}\left(1 - q^r\right)^{-1} \\ 2\rho(q) + \omega(q) &= 3\left(\frac{1}{2}q^{-\frac{3}{8}}\theta_2(0, q^\frac{3}{2})\right)^2\prod_{r > 0}\left(1 - q^{2r}\right)^{-1} \\ \nu(\pm q) \pm q\omega\left(q^2\right) &= \frac{1}{2}q^{-\frac{1}{4}}\theta_2(0, q)\prod_{r > 0}\left(1 + q^{2r}\right) \\ f\left(q^8\right) \pm 2q\omega(\pm q) \pm 2q^3\omega\left(-q^4\right) &= \theta_3(0, \pm q)\theta_3^2\left(0, q^2\right)\prod_{r > 0}\left(1 - q^{4r}\right)^{-2} \\ f(q^8) + q\omega(q) - q\omega(-q) &= \theta_3(0, q^4) \theta_3^2(0, q^2)\prod_{r > 0}\left(1 - q^{4r}\right)^{-2} \end{align}
Order 5 Ramanujan wrote down ten mock theta functions of order 5 in his 1920 letter to Hardy, and stated some relations between them that were proved by Watson. In his lost notebook he stated some further identities relating these functions, equivalent to the
mock theta conjectures, that were proved by Hickerson. Andrews found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight . :f_0(q) = \sum_{n\ge 0} \frac{q^{n^2}}{(-q;q)_{n}} :f_1(q) = \sum_{n\ge 0} \frac{q^{n^2+n}}{(-q;q)_{n}} :\phi_0(q) = \sum_{n\ge 0} {q^{n^2}(-q;q^2)_{n}} :\phi_1(q) = \sum_{n\ge 0} {q^{(n+1)^2}(-q;q^2)_{n}} :\psi_0(q) = \sum_{n\ge 0} {q^{(n+1)(n+2)/2}(-q;q)_{n}} :\psi_1(q) = \sum_{n\ge 0} {q^{n(n+1)/2}(-q;q)_{n}} :\chi_0(q) = \sum_{n\ge 0} \frac{q^{n}}{(q^{n+1};q)_{n}} = 2F_0(q)-\phi_0(-q) :\chi_1(q) = \sum_{n\ge 0} \frac{q^{n}}{(q^{n+1};q)_{n+1}} = 2F_1(q)+q^{-1}\phi_1(-q) :F_0(q) = \sum_{n\ge 0} \frac{q^{2n^2}}{(q;q^2)_{n}} :F_1(q) = \sum_{n\ge 0} \frac{q^{2n^2+2n}}{(q;q^2)_{n+1}} :\Psi_0(q) = -1 + \sum_{n \ge 0} \frac{ q^{5n^2}}{(1-q)(1-q^4)(1-q^6)(1-q^9)...(1-q^{5n-1})(1-q^{5n+1})} :\Psi_1(q) = -1 + \sum_{n \ge 0} \frac{ q^{5n^2}}{(1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^{5n-2})(1-q^{5n+2}) }
Order 6 Ramanujan wrote down seven mock theta functions of order 6 in his lost notebook, and stated 11 identities between them, which were proved by Andrews and Hickerson. Two of Ramanujan's identities relate
φ and
ψ at various arguments, four of them express
φ and
ψ in terms of Appell–Lerch series, and the last five identities express the remaining five sixth-order mock theta functions in terms of
φ and
ψ. Berndt and Chan discovered two more sixth-order functions. The order 6 mock theta functions are: :\phi(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n^2}(q;q^2)_n}{(-q;q)_{2n}} :\psi(q) = \sum_{n\ge 0} \frac{(-1)^nq^{(n+1)^2}(q;q^2)_n}{(-q;q)_{2n+1}} :\rho(q) = \sum_{n\ge 0} \frac{q^{n(n+1)/2}(-q;q)_n}{(q;q^2)_{n+1}} :\sigma(q) = \sum_{n\ge 0} \frac{q^{(n+1)(n+2)/2}(-q;q)_n}{(q;q^2)_{n+1}} :\lambda(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n}(q;q^2)_n}{(-q;q)_{n}} :2\mu(q) = \sum_{n\ge 0} \frac{(-1)^nq^{n+1}(1+q^n)(q;q^2)_n}{(-q;q)_{n+1}} :\gamma(q) = \sum_{n\ge 0} \frac{q^{n^2}(q;q)_n}{(q^3;q^3)_{n}} :\phi_{-}(q) = \sum_{n\ge 1} \frac{q^{n}(-q;q)_{2n-1}}{(q;q^2)_{n}} :\psi_{-}(q) = \sum_{n\ge 1} \frac{q^{n}(-q;q)_{2n-2}}{(q;q^2)_{n}}
Order 7 Ramanujan gave three mock theta functions of order 7 in his 1920 letter to Hardy. They were studied by Selberg, who found asymptotic expansion for their coefficients, and by Andrews. Hickerson found representations of many of these functions as the quotients of indefinite theta series by modular forms of weight . Zwegers described their modular transformation properties. • \displaystyle F_0(q) = \sum_{n\ge 0}\frac{q^{n^2}}{(q^{n+1};q)_n} • \displaystyle F_1(q) = \sum_{n\ge 0}\frac{q^{n^2}}{(q^{n};q)_n} • \displaystyle F_2(q) = \sum_{n\ge 0}\frac{q^{n(n+1)}}{(q^{n+1};q)_{n+1}} These three mock theta functions have different shadows, so unlike the case of Ramanujan's order-3 and order-5 functions, there are no linear relations between them and ordinary modular forms. The corresponding weak Maass forms are : \begin{align} M_1(\tau) & = q^{-1/168}F_1(q) + R_{7,1}(\tau) \\[4pt] M_2(\tau) & = -q^{-25/168}F_2(q) + R_{7,2}(\tau) \\[4pt] M_3(\tau) & = q^{47/168}F_3(q) + R_{7,3}(\tau) \end{align} where :R_{p,j}(\tau) = \!\!\!\! \sum_{n\equiv j\bmod p}\binom{12}{n}\sgn(n)\beta\left(\frac{n^2y}{6p}\right)q^{-n^2/24p} and :\beta(x) = \int_x^\infty u^{-1/2}e^{-\pi u} \, du is more or less the complementary
error function. Under the metaplectic group, these three functions transform according to a certain 3-dimensional representation of the metaplectic group as follows : \begin{align} M_j\left(-\frac{1}{\tau}\right) & = \sqrt\frac{\tau}{7i}\,\sum_{k=1}^32\sin\left(\frac{6\pi jk}{7}\right)M_k(\tau) \\[6pt] M_1(\tau+1) & = e^{-2\pi i/168} M_1(\tau), \\[6pt] M_2(\tau+1) & = e^{-2\times 25\pi i/168} M_2(\tau), \\[6pt] M_3(\tau+1) & = e^{-2\times 121\pi i/168} M_3(\tau). \end{align} In other words, they are the components of a level 1 vector-valued harmonic weak Maass form of weight .
Order 8 Gordon and McIntosh found eight mock theta functions of order 8. They found five linear relations involving them, and expressed four of the functions as Appell–Lerch sums, and described their transformations under the modular group. The two functions
V1 and
U0 were found earlier by Ramanujan in his lost notebook. :S_0(q) = \sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (-q^2;q^2)_n} :S_1(q) = \sum_{n\ge 0} \frac{q^{n(n+2)} (-q;q^2)_n }{ (-q^2;q^2)_n} :T_0(q) = \sum_{n\ge 0} \frac{q^{(n+1)(n+2)} (-q^2;q^2)_n }{ (-q;q^2)_{n+1}} :T_1(q) = \sum_{n\ge 0} \frac{q^{n(n+1)} (-q^2;q^2)_n }{ (-q;q^2)_{n+1}} :U_0(q) = \sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (-q^4;q^4)_n} :U_1(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2} (-q;q^2)_n }{ (-q^2;q^4)_{n+1}} :V_0(q) = -1+2\sum_{n\ge 0} \frac{q^{n^2} (-q;q^2)_n }{ (q;q^2)_n} = -1+2\sum_{n\ge 0} \frac{q^{2n^2} (-q^2;q^4)_n}{(q;q^2)_{2n+1}} :V_1(q) = \sum_{n\ge 0} \frac{q^{(n+1)^2} (-q;q^2)_n }{ (q;q^2)_{n+1}} = \sum_{n\ge 0} \frac{q^{2n^2+2n+1} (-q^4;q^4)_n}{(q;q^2)_{2n+2}}
Order 10 Ramanujan listed four order-10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by Choi. • \phi(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(q;q^2)_{n+1}} • \psi(q)=\sum_{n\ge 0}\frac{q^{(n+1)(n+2)/2}}{(q;q^2)_{n+1}} • \Chi(q)=\sum_{n\ge 0}\frac{(-1)^nq^{n^2}}{(-q;q)_{2n}} • \chi(q)=\sum_{n\ge 0}\frac{(-1)^nq^{(n+1)^2}}{(-q;q)_{2n+1}} ==Notes==