In [4], Kaneko–Koike studied various solutions for the so-called Kaneko–Zagier equation:
$$\begin{aligned} f^{\prime \prime }(\tau )-\frac{k+1}{6}E_2(\tau )f^\prime (\tau ) +\frac{k(k+1)}{12}E^\prime _2(\tau )f(\tau )=0, \end{aligned}$$
where \(\prime =(2\pi i)^{-1}d/d\tau =qd/dq\), \(q=e^{2\pi i\tau }\), \(\tau \) a variable in the Poincaré upper-half plane, \(k\) a fixed rational number, and \(E_2(\tau )\) is the (quasimodular) Eisenstein series of weight \(2\) for \(\mathrm {SL}_2(\mathbb {Z})\) defined by
$$\begin{aligned} E_2(\tau )=1-24\sum _{n=1}^\infty \Bigl (\sum _{d | n}d\Bigr )q^n. \end{aligned}$$
They gave modular forms expressed in terms of hypergeometric polynomials and quasimodular forms as its solution for weight \(k\), where \(k\) is integer or half-integer. In particular, Kaneko studied in [3] the modular form as its solution for weight one-fifth, which is closely related to certain models in conformal field theory.
From [5, 6] and [11], we know that the Kaneko–Zagier equation for \(\varGamma _0^*(N) (N=2,3)\)
$$\begin{aligned} (\sharp )^{(N)}_k \quad f^{\prime \prime }(\tau )-\frac{k+1}{6-N}E_{NA}(\tau )f^\prime (\tau ) +\frac{k(k+1)}{2(6-N)}E_{NA}^\prime (\tau )f(\tau )=0 \end{aligned}$$
also has modular/quasimodular solutions similar to the case for \(\mathrm {SL}_2(\mathbb {Z})\), where the Fricke group of level \(N\)
\((N=2,3)\) is defined by
$$\begin{aligned}&\varGamma _0^*(N)=\varGamma _0(N)\cup \varGamma _0(N)W_N,\qquad W_N= \left( {\begin{array}{ll} 0&{}-1/\sqrt{N}\\ \sqrt{N}&{}0 \end{array},}\right)&\\&\varGamma _0(N)=\biggl \{ \left( {\begin{array}{ll} a&{}b\\ c&{}d \end{array}}\right) \in \mathrm {SL}_2(\mathbb {Z})\ \biggl |\biggr .\ c\equiv 0\pmod {N}\biggr \},&\end{aligned}$$
and \(E_{NA}(\tau )\), the (quasimodular) Eisenstein series of weight \(2\) for \(\varGamma _0^*(N)\), is defined by
$$\begin{aligned} E_{NA}(\tau )=\frac{NE_2(N\tau )+E_2(\tau )}{N+1}. \end{aligned}$$
In this paper, we give modular forms of a fractional weight as a solution of the Kaneko–Zagier equation for \(\varGamma _0^*(N) (N=2,3)\). Hereafter, \(N\) denotes the level 2 or 3.
For any complex numbers \(v\) and \(s\), we take \(-\pi < \arg (v) \le \pi \) and put \(v^s= |v|^s e^{is\arg (v)}\). Define
$$\begin{aligned} \phi ^{(2)}_1(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (2\tau )^2}\Bigr )^{1/3}\frac{\eta (2\tau )\eta (3\tau )^2}{\eta (\tau )\eta (6\tau )}\\&=1 \!+\! \frac{2}{3}q \!+\! \frac{8}{9}q^2\! -\! \frac{50}{81}q^3\!+\! \frac{74}{243}q^4\!+\! \frac{320}{729}q^5\!+\!\frac{1232}{6561}q^6\!+\!\frac{7012}{19683} q^7\!+\!\cdots ,\\ \phi ^{(2)}_2(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (2\tau )^2}\Bigr )^{1/3}\frac{\eta (6\tau )^2}{\eta (3\tau )}\\&=q^{1/3}\Bigl (\!1 \!-\! \frac{1}{3}q\!+\! \frac{2}{9}q^2\!+\!\frac{40}{81}q^3\!+\!\frac{62}{243}q^4\!-\!\frac{307}{729}q^5\!+\!\frac{458}{6561} q^6\!-\!\frac{4136}{19683}q^7\!+\!\cdots \!\Bigr ),\\ \phi ^{(3)}_1(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (3\tau )^3}\Bigr )^{1/2}\frac{\eta (2\tau )^3\eta (3\tau )^2}{\eta (\tau )^2\eta (6\tau )}\\&=1 + \frac{3}{2}q+\frac{3}{8}q^2+\frac{15}{16}q^3+\frac{3}{128}q^4-\frac{99}{256}q^5 +\frac{1671}{1024}q^6+\frac{1383}{2048}q^7+\cdots ,\\ \phi ^{(3)}_2(\tau )&=\Bigl (\frac{\eta (\tau )}{\eta (3\tau )^3}\Bigr )^{1/2}\frac{\eta (6\tau )^3}{\eta (2\tau )}\\&=q^{1/2}\Bigl (\!1 - \frac{1}{2}q\!+\!\frac{3}{8}q^2\!+\!\frac{11}{16}q^3\!+\!\frac{35}{128}q^4\!-\!\frac{159}{256}q^5\!+\! \frac{359}{1024} q^6\!-\!\frac{573}{2048} q^7\!+\!\cdots \!\Bigr ), \end{aligned}$$
where \(\eta (\tau )=q^{1/24}\prod _{n=1}^{\infty }(1-q^n)\) is the Dedekind eta function. Then, we find that \(\phi ^{(N)}_1(\tau )\) and \(\phi ^{(N)}_2(\tau )\) are holomorphic modular forms of weight \(N/6\) for
$$\begin{aligned} \varGamma (6):=\Bigl \{ \left( {\begin{array}{ll} a&{}b\\ c&{}d \end{array}}\right) \in \varGamma _0(6) \Bigl | \Bigr .\ a\equiv d\equiv 1,\ b\equiv 0\pmod {6}\Bigr \} \end{aligned}$$
using properties of \(\eta (\tau )\) [7, §1.3 Theorem 1.7 and §2.3 Corollary 2.2]. Moreover, \((\phi ^{(N)}_1)^{6/N}\) and \((\phi ^{(N)}_2)^{6/N}\) are modular forms of weight \(1\) with the Legendre character \(\biggl (\dfrac{*}{3}\biggr )\) for \(\varGamma _0(6)\).
Remark 1
The following can be expressed in terms of theta series:
$$\begin{aligned} \frac{\eta (2\tau )\eta (3\tau )^2}{\eta (\tau )\eta (6\tau )}&=\sum _{n\in \mathbb {Z}}q^{(6n+1)^2/24},&\quad \frac{\eta (6\tau )^2}{\eta (3\tau )} \quad&=\sum _{n\in \mathbb {Z}}q^{3(4n+1)^2/8}. \end{aligned}$$
Note that if you find a solution \(F(\phi ^{(N)}_1,\ \phi ^{(N)}_2)\) of weight \(k\) for Eq. \((\sharp )^{(N)}_k\), you can get another solution \(F(\phi ^{(N)}_2,\ -\phi ^{(N)}_1/N)\) immediately because the group \(\varGamma _0^*(N)\) acts on the space of solutions as follows:
$$\begin{aligned} \left( {\begin{array}{c}\phi ^{(2)}_1\\ \phi ^{(2)}_2\end{array}}\right) \Biggl |_{\frac{1}{3}}\Biggr . \Biggl [ \begin{array}{ll} 2\sqrt{2}&{} \quad -3/\sqrt{2}\\ 3\sqrt{2}&{} \quad -2\sqrt{2} \end{array} \Biggr ]&= \sqrt{2}e^{-\frac{2}{3}\pi i} \left( {\begin{array}{c}\phi ^{(2)}_2\\ -\frac{1}{2}\phi ^{(2)}_1\end{array}}\right) ,\\ \left( {\begin{array}{c}\phi ^{(3)}_1\\ \phi ^{(3)}_2\end{array}}\right) \Biggl |_{\frac{1}{2}}\Biggr . \Biggl [ \begin{array}{ll} -\sqrt{3}&{} \quad -4/\sqrt{3}\\ 2\sqrt{3}&{} \quad -3\sqrt{3} \end{array} \Biggr ]&= \sqrt{3}e^{-\frac{1}{4}\pi i} \left( {\begin{array}{c}\phi ^{(3)}_2\\ -\frac{1}{3}\phi ^{(3)}_1\end{array}}\right) , \end{aligned}$$
where \(F(X,Y)\) is a homogenous polynomial of two variables, and \(|_k[\cdot ]\) is a slash operator of weight \(k\).
Finally, Heun’s local series \(Hl\) is defined by
$$\begin{aligned} Hl(a,w;\alpha ,\beta ,\gamma ,\delta ;x)={\displaystyle \sum _{n=0}^\infty c_n x^n}, \end{aligned}$$
where the coefficients satisfy the recursion; \(c_0=1,\ \ c_1=\frac{w}{a\gamma }c_0,\) and
$$\begin{aligned} c_{n\!+\!1}\!=\!\frac{(n[(n\!-\!1\!+\!\gamma )(1\!+\!a)\!+\!a\delta \!+\!\varepsilon ]\!+\!w)}{(n\!+\!1)(n\!+\!\gamma )a}c_n \!-\!\frac{(n\!-\!1\!+\!\alpha )(n\!-\!1\!+\!\beta )}{(n\!+\!1)(n\!+\!\gamma )a}c_{n\!-\!1}\qquad (n\ge 1), \end{aligned}$$
where \(\gamma +\delta +\varepsilon =\alpha +\beta +1\). This is a solution of Heun’s equation, which is the canonical form of a second-order linear differential equation with four regular singularities (cf. [10]):
$$\begin{aligned} \frac{d^2y}{dx^2}+\biggl (\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{\varepsilon }{x-a}\biggr ) \frac{dy}{dx}+\frac{\alpha \beta x-w}{x(x-1)(x-a)}y=0. \end{aligned}$$
(1)
In particular, \(Hl\) is a polynomial when \(\alpha \ \text {or}\ \beta \in -\mathbb {N}\).