New proofs for three identities of seventh order mock theta functions

Lijun Hao; Lijun Hao

doi:10.3934/math.2023238

AIMS Mathematics

2023, Volume 8, Issue 2: 4806-4813. doi: 10.3934/math.2023238

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New proofs for three identities of seventh order mock theta functions

Lijun Hao ^,

School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received: 19 October 2022 Revised: 29 November 2022 Accepted: 02 December 2022 Published: 08 December 2022
MSC : 11F27, 33D15

Using the three-term Weierstrass relation for theta functions and the properties of Hecke-type double sums and Appell-Lerch sums, we give new and simple proofs for the seventh order mock theta conjectures.

Keywords:

Citation: Lijun Hao. New proofs for three identities of seventh order mock theta functions[J]. AIMS Mathematics, 2023, 8(2): 4806-4813. doi: 10.3934/math.2023238

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Abstract

1. Introduction

The mock theta functions were named and studied by the Indian mathematician Ramanujan. Four months before he died, he summarized his results in a letter to G.H. Hardy ^[7]. He provided four separate classes of mock theta functions: one class of third order, two of fifth order, and one of seventh order together with identities satisfied by them. Watson laid the foundations of this subject in the twentieth century. He showed many elegant and significant theorems. In particular, he made important contributions to the third order mock theta functions, see ^[9,10]. Since then, many great mathematicians made remarkable achievement of mock theta functions, such as Andrews, Hickerson, Ono, Duke, and so on. We first introduce the basic definitions and notation. Then we provide the background of our research.

Throughout this paper, let $q$ denote a complex number with $|q| < 1$ . Here and in what follows, we adopt the standard $q$ -series notation ^[3]. For any positive integer $n$ ,

$\begin{align*} (a;q)_0: = 1, \qquad (a;q)_n: = \prod\limits_{k = 0}^{n-1}(1-aq^{k}), \qquad (a;q)_{\infty}: = \prod\limits_{k = 0}^{\infty}(1-aq^{k}). \end{align*}$

For convenience, we use $(a)_n$ to denote $(a; q)_n$ .

The Jacobi triple product identity is stated as follows.

$\begin{align} j(x;q): = (x)_\infty(q/x)_\infty(q)_\infty = \sum\limits_{n = -\infty}^{\infty}(-1)^nq^{\binom{n}{2}}x^n. \end{align}$

(1.1)

From the definition of $j(x; q)$ , we have

$\begin{align} j(x;q) = j(q/x;q) \end{align}$

(1.2)

and

$\begin{align} j(q^nx;q) = (-1)^nq^{-\binom{n}{2}}x^{-n}j(x;q), \quad n \in \mathbb{Z}. \end{align}$

(1.3)

Let $m$ and $a$ be integers with $m$ positive. Define

$\begin{align*} &J_{a, m}: = j(q^a;q^m), \quad \overline{J}_{a, m}: = j(-q^a;q^m), \\ &J_{m}: = \prod\limits_{i\geq1}(1-q^{mi}), \quad \overline{J}_{m}: = \prod\limits_{i\geq1}(1+q^{mi}), \ {\rm{ and}}\ \\ &{j(b_1, b_2, \ldots, b_m;q): = j(b_1;q)j(b_2;q)\ldots j(b_m;q).} \end{align*}$

Ramanujan's seventh order mock theta functions [7,p.355] are defined by

$\begin{align*} \mathcal{F}_0(q): = &\sum\limits_{n = 0}^\infty\frac{q^{n^2}}{(q^{n+1};q)_n}, \\ \mathcal{F}_1(q): = &\sum\limits_{n = 1}^\infty\frac{q^{n^2}}{(q^n;q)_n}, \\ \mathcal{F}_2(q): = &\sum\limits_{n = 0}^\infty\frac{q^{n(n+1)}}{(q^{n+1};q)_{n+1}}. \end{align*}$

In 1988, by using the constant term method, Hickerson ^[5] proved the seventh order mock theta conjectures which can be written as follows.

$\begin{align} \mathcal{F}_0(q) = &2+2qg(q, q^7)-\frac{J_{3, 7}^2}{J_1}, \end{align}$

(1.4)

$\begin{align} \mathcal{F}_1(q) = &2q^2g(q^2, q^7)+\frac{qJ_{1, 7}^2}{J_1}, \end{align}$

(1.5)

$\begin{align} \mathcal{F}_2(q) = &2q^2g(q^3, q^7)+\frac{J_{2, 7}^2}{J_1}, \end{align}$

(1.6)

where $g(z, q)$ is a universal mock theta function introduced by Hickerson ^[4] as

$\begin{align*} g(z;q): = z^{-1}\left(-1+\sum\limits_{n = 0}^\infty\frac{q^{n^2}}{(z)_{n+1}(qz^{-1})_n}\right) = \sum\limits_{n = 0}^\infty\frac{q^{n(n+1)}}{(z)_{n+1}(qz^{-1})_{n+1}}. \end{align*}$

Furthermore, from [6,Proposition 4.2], we have

$\begin{align} g(z;q) = -z^{-1}m(q^2z^{-3}, q^3, z^2)-z^{-2}m(qz^{-3}, q^3, z^2), \end{align}$

(1.7)

where the Appell-Lerch sum $m(x, q, z)$ is defined by Hickerson and Mortenson ^[6] in the following.

Definition 1.1. Let generic $x, z\in\mathbb{C}^*: = \mathbb{C}\backslash \{0\}$ with neither $z$ nor $xz$ an integer power of $q$ . Then

$\begin{align*} m(x, q, z): = \frac{-z}{j(z;q)}\sum\limits_{r = -\infty}^{\infty}\frac{(-1)^rq^{\binom{r+1}{2}}z^r}{1-q^rxz}. \end{align*}$

Following ^[6], the term "generic" means that the parameters do not cause poles in the Appell-Lerch sums or in the quotients of theta functions. In this paper, employing the three-term Weierstrass relation for theta functions and the relationships between Appell-Lerch sums and Hecke -type double sums, we provide new and simple proofs for the above three identities (1.4)–(1.6).

2. Main results

First, we recall some notation and definitions. The Hecke-type double sums are defined as follows.

Definition 2.1. Let $x$ , $y\in \mathbb{C}^*$ and define sg(r) $: = 1$ for ${r}\geq0$ and sg(r) $: = -1$ for ${r} < 0$ . Then

$\begin{align*} f_{a, b, c}(x, y, q): = \sum\limits_{sg(r) = sg(s)}sg(r)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}}. \end{align*}$

The three-term Weierstrass relation for theta functions is stated in the following.

Proposition 2.1. [,Theorem 1.20] For generic $a, b, c, d\in \mathbb{C}^*$ ,

$\begin{align*} j(ac, a/c, bd, b/d;q) = j(ad, a/d, bc, b/c;q)+b/c\cdot j(ab, a/b, cd, c/d;q). \end{align*}$

Next, we review some propositions for Appell-Lerch sums and Hecke-type double sums.

Proposition 2.2. [,Proposition 3.1] For generic $x, z, z_0, z_1\in \mathbb{C}^*$ ,

$\begin{align} &m(x, q, z) = m(x, q, qz), \end{align}$

(2.1)

$\begin{align} &m(x, q, z) = x^{-1}m(x^{-1}, q, z^{-1}), \end{align}$

(2.2)

$\begin{align} &m(qx, q, z) = 1-xm(x, q, z), \end{align}$

(2.3)

$\begin{align} &m(x, q, z) = m(x, q, x^{-1}z^{-1}), \ {{and}} \end{align}$

(2.4)

$\begin{align} &m(x, q, z_1) = m(x, q, z_0)+\frac{z_0J_1^3j(z_1/z_0;q)j(xz_0z_1;q)} {j(z_0;q)j(z_1;q)j(xz_0;q)j(xz_1;q)}. \end{align}$

(2.5)

Lemma 2.1. [,Theorem 1.3] Let $n$ and $p$ be positive integers with $(n, p) = 1$ . For generic $x, y\in \mathbb{C}^*$ ,

$\begin{align*} f_{n, n+p, n}(x, y, q) = g_{n, n+p, n}(x, y, q, -1, -1)+\frac{1} {\overline{J}_{0, np(2n+p)}}\theta_{n, p}(x, y, q), \end{align*}$

where

$\begin{align*} &g_{a, b, c}(x, y, q, z_1, z_0): = \\ &\sum\limits_{t = 0}^{a-1}(-y)^tq^{c\binom{t}{2}}j(q^{bt}x;q^a)m(-q^{a\binom{b+1}{2} -c\binom{a+1}{2}-t(b^2-ac)}(-x)^{-b}(-y)^a, q^{a(b^2-ac)}, z_0)\\ &+\sum\limits_{t = 0}^{c-1}(-x)^tq^{a\binom{t}{2}}j(q^{bt}x;q^c)m(-q^{c\binom{b+1}{2} -a\binom{c+1}{2}-t(b^2-ac)}(-x)^{c}(-y)^b, q^{c(b^2-ac)}, z_1), \end{align*}$

and

$\begin{align*} \theta_{n, p}&(x, y, q): = \sum\limits_{r^* = 0}^{p-1}\sum\limits_{s^* = 0}^{p-1} q^{n\binom{r-(n-1)/2}{2}+(n+p)(r-(n-1)/2)(s-(n+1)/2)n\binom{s+(n+1)/2}{2}} (-x)^{r-(n-1)/2}\\ &\times \frac{(-y)^{s+(n+1)/2}J_{p^2(2n+p)}^3j(-q^{np(s-r)}x^n/y^n;q^{np^2}) j(q^{p(2n+p)(s+r)+p(n+p)}x^py^p;q^{p^2(2n+p)})} {j(q^{p(2n+p)r+p(n+p)/2}(-y)^{n+p}/(-x)^{n}, q^{p(2n+p)s+p(n+p)/2}(-x)^{n+p}/(-y)^{n};q^{p^2(2n+p)})}. \end{align*}$

Here $r: = r^*+\{(n-1)/2\}$ and $s: = s^*+\{(n-1)/2\}$ , with $0\leq\{\alpha\} < 1$ denoting the fractional part of $\alpha$ .

Taking the $n = 3$ , $p = 1$ specialization of the above lemma, we derive the following result.

Proposition 2.3. For generic $x, y\in \mathbb{C}^*$ ,

$\begin{align} f_{3, 4, 3}(x, y, q) = &j(x;q^3)m(q^{12}x^{-4}y^3, q^{21}, -1) -yj(q^4x;q^3)m(q^{5}x^{-4}y^3, q^{21}, -1)\\ &+q^3y^2j(q^8x;q^3)m(q^{-2}x^{-4}y^3, q^{21}, -1)+ j(y;q^3)m(q^{12}x^3y^{-4}, q^{21}, -1)\\ &-xj(q^4y;q^3)m(q^{5}x^3y^{-4}, q^{21}, -1) +q^3x^2j(q^8y;q^3)m(q^{-2}x^3y^{-4}, q^{21}, -1)\\ &-\frac{y^2J_7^3j(-x^3y^{-3};q^3)j(q^4xy;q^7)} {q^2xj(-q^2x^{-3}y^4, -q^2x^4y^{-3};q^7)\overline{J}_{0, 21}}. \end{align}$

(2.6)

Then we have the following lemma.

Lemma 2.2. We have

$\begin{align} f_{3, 4, 3}(q^2, q^2, q)& = 2J_{1, 3}m(q^{10}, q^{21}, -1)-2q^{-1}J_{1, 3}m(q^{4}, q^{21}, -1) +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{\overline{J}_{4, 7}^2 \overline{J}_{0, 21}}, \end{align}$

(2.7)

$\begin{align} f_{3, 4, 3}(q^3, q^3, q) & = -2q^{-2}J_{1, 3}m(q^{2}, q^{21}, -1)-2q^{-1}J_{1, 3}m(q^5, q^{21}, -1) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{\overline{J}_{2, 7}^2 \overline{J}_{0, 21}}, \end{align}$

(2.8)

$\begin{align} f_{3, 4, 3}(q^4, q^4, q) = -2q^{-1}J_{1, 3}m(q^{8}, q^{21}, -1)-2q^{-3}J_{1, 3}m(q, q^{21}, -1) +\frac{q^{-3}J_7^3\overline{J}_{0, 3}J_{2, 7}}{\overline{J}_{1, 7}^2 \overline{J}_{0, 21}}. \end{align}$

(2.9)

Proof. Based on the definition of $j(x; q)$ , for an integer $m$ , we arrive at

$\begin{align} j(q^m;q) = j(q^{-m+1};q) = 0. \end{align}$

(2.10)

From (2.6) and (2.10), we have

$\begin{align*} f_{3, 4, 3}(q^2, q^2, q)& = 2J_{2, 3}m(q^{10}, q^{21}, -1)+2q^{7}J_{10, 3}m(q^{-4}, q^{21}, -1) -\frac{J_7^3\overline{J}_{0, 3}J_{8, 7}} {\overline{J}_{4, 7}^2\overline{J}_{0, 21}}. \end{align*}$

Then applying (1.2), (1.3) and (2.2), we derive (2.7). The proofs of (2.8) and (2.9) are similar.

In the following, we begin to prove (1.4)–(1.6).

From [,Theorem 2.0], we obtain the Hecke-type double sums for $\mathcal{F}_0(q)$ , $\mathcal{F}_1(q)$ , and $\mathcal{F}_2(q)$ .

$\begin{align} J_1\mathcal{F}_0(q) = &f_{3, 4, 3}(q^2, q^2, q), \end{align}$

(2.11)

$\begin{align} J_1\mathcal{F}_1(q) = &qf_{3, 4, 3}(q^4, q^4, q), \end{align}$

(2.12)

$\begin{align} J_1\mathcal{F}_2(q) = &f_{3, 4, 3}(q^3, q^3, q). \end{align}$

(2.13)

Therefore, combining Lemma 2.2, (2.11)–(2.13) and the fact that $J_1$ and $J_{1, 3}$ are the same from the definitions of them, we have

$\begin{align} \mathcal{F}_0(q) = &2m(q^{10}, q^{21}, -1)-2q^{-1}m(q^{4}, q^{21}, -1) +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &2m(q^{10}, q^{21}, q^{-2})-2q^{-1}m(q^{4}, q^{21}, q^2)\\ &-\frac{2J_{21}^3j(-q^2, -q^8;q^{21})} {j(-1, q^2, q^8, -q^{10};q^{21})}-\frac{2q^{-1}J_{21}^3j(-q^2, -q^6;q^{21})} {j(-1, q^2, -q^{4}, q^6;q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &2m(q^{10}, q^{21}, q^{-2})-2q^{-1}m(q^{4}, q^{21}, q^2)\\ &-\frac{2J_{21}^3j(-q^2;q^{21})} {j(-1, q^2;q^{21})}\left(\frac{j(-q^8;q^{21})}{j(q^8, -q^{10};q^{21})} +\frac{q^{-1}j(-q^6;q^{21})}{j(-q^4, q^{6};q^{21})}\right) +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &2m(q^{10}, q^{21}, q^{-2})-2q^{-1}m(q^{4}, q^{21}, q^2)\\ &-\frac{2J_{21}^3j(-q^2;q^{21})} {j(-1, q^2;q^{21})}\left(\frac{j(q^{-2}, -q^5, q^7, -q^9;q^{21})}{j(q^{-1}, -q^4, q^6, q^8, -q^{10};q^{21})} \right) +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &2m(q^{10}, q^{21}, q^{-2})-2q^{-1}m(q^{4}, q^{21}, q^2)\\ &-\frac{2q^{-1}J_{21}^3j(-q^{2}, -q^5, q^7, -q^9;q^{21})}{j(-1, q, -q^4, q^6, q^8, -q^{10};q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &2m(q^{10}, q^{21}, q^{-2})-2q^{-1}m(q^{4}, q^{21}, q^2)\\ &-\frac{2q^{-1}J_7^2j(-q^{2};q^{7})j(-q^{3};q^{21})}{j(q, -q^3;q^{7})j(-1;q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}\\ = &{2q^{-10}m(q^{-10}, q^{21}, q^{2})-2q^{-1}m(q^{4}, q^{21}, q^2)}\\ &{-\frac{2q^{-1}J_7^2j(-q^{2};q^{7})j(-q^{3};q^{21})}{j(q, -q^3;q^{7})j(-1;q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}}\\ = &{2(1-m(q^{11}, q^{21}, q^{-2}))-2q^{-1}m(q^{4}, q^{21}, q^2)}\\ &{-\frac{2q^{-1}J_7^2j(-q^{2};q^{7})j(-q^{3};q^{21})}{j(q, -q^3;q^{7})j(-1;q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}}\\ = &{2+2qg(q, q^7)-\frac{2q^{-1}J_7^2j(-q^{2};q^{7})j(-q^{3};q^{21})}{j(q, -q^3;q^{7})j(-1;q^{21})} +\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}}, } \end{align}$

(2.14)

where in the fourth equality, we replace $q$ , $a$ , $b$ , $c$ , and $d$ by $q^{21}$ , $-q^{6}$ , $-q^{7}$ , $-q^{8}$ , and $q^{2}$ , respectively, in Proposition 2.1.

For $\mathcal{F}_1(q)$ , we have

$\begin{align} \mathcal{F}_1(q) = &-2m(q^{8}, q^{21}, -1)-2q^{-2}m(q, q^{21}, -1) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &-2m(q^{8}, q^{21}, q^{4})-2q^{-2}m(q, q^{21}, q^4)\\ &-\frac{2J_{21}^3j(-q^4, -q^{12};q^{21})} {j(-1, q^4, -q^8, q^{12};q^{21})}-\frac{2q^{-2}J_{21}^3j(-q^4, -q^5;q^{21})} {j(-1, -q, q^4, q^5;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &-2m(q^{8}, q^{21}, q^{4})-2q^{-2}m(q, q^{21}, q^4)\\ &-\frac{2J_{21}^3j(-q^4;q^{21})} {j(-1, q^4;q^{21})}\left(\frac{j(-q^{9};q^{21})}{j(-q^{12}, q^{13};q^{21})} +\frac{q^{-2}j(-q^5;q^{21})}{j(-q, q^{16};q^{21})}\right) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &-2m(q^{8}, q^{21}, q^{4})-2q^{-2}m(q, q^{21}, q^4)\\ &-\frac{2J_{21}^3j(-q^4;q^{21})} {j(-1, q^4;q^{21})}\left(\frac{j(-q^{3}, q^{-4}, -q^{11}, q^{14};q^{21})} {j(q^{-2}, -q, q^{16}, q^{12}, -q^{13};q^{21})}\right) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &-2m(q^{8}, q^{21}, q^{4})-2q^{-2}m(q, q^{21}, q^4)\\ &-\frac{2q^{-2}J_{21}^3j(-q^{3}, -q^4, q^7, -q^{11};q^{21})} {j(-1, -q, q^2, q^5, -q^8, q^{9};q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &-2m(q^{8}, q^{21}, q^{4})-2q^{-2}m(q, q^{21}, q^4)\\ &-\frac{2q^{-2}J_7^2j(-q^{3};q^{7})j(-q^{6};q^{21})}{j(-q, q^2;q^{7})j(-1;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}\\ = &{2q^2g(q^2, q^7)-\frac{2q^{-2}J_7^2j(-q^{3};q^{7})j(-q^{6};q^{21})}{j(-q, q^2;q^{7})j(-1;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}}, } \end{align}$

(2.15)

where in the fourth equality, we replace $q$ , $a$ , $b$ , $c$ , and $d$ by $q^{21}$ , $-q^{5}$ , $-q^{7}$ , $-q^{9}$ , and $q^{4}$ , respectively, in Proposition 2.1.

Moreover, we have

$\begin{align} \mathcal{F}_2(q) = &-2q^{-1}m(q^{5}, q^{21}, -1)-2q^{-2}m(q^2, q^{21}, -1) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &-2q^{-1}m(q^{5}, q^{21}, q^{6})-2q^{-4}m(q^{-2}, q^{21}, q^{6})\\ &-\frac{2q^{-1}J_{21}^3j(-q^6, -q^{11};q^{21})} {j(-1, -q^5, q^6, q^{11};q^{21})}-\frac{2q^{-4}J_{21}^3j(-q^4, -q^6;q^{21})} {j(-1, -q^{-2}, q^4, q^6;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &-2q^{-1}m(q^{5}, q^{21}, q^{6})-2q^{-4}m(q^{-2}, q^{21}, q^{6})\\ &-\frac{2q^{-1}J_{21}^3j(-q^6;q^{21})} {j(-1, q^6;q^{21})}\left(\frac{j(-q^{11};q^{21})}{j(-q^5, q^{11};q^{21})} +\frac{q^{-1}j(-q^4;q^{21})}{j(-q^2, q^{4};q^{21})}\right) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &-2q^{-1}m(q^{5}, q^{21}, q^{6})-2q^{-4}m(q^{-2}, q^{21}, q^{6})\\ &-\frac{2q^{-1}J_{21}^3j(-q^6;q^{21})} {j(-1, q^6;q^{21})}\left(\frac{q^{-1}j(-q, q^6, q^{7}, -q^{8};q^{21})} {j(-q^{2}, q^{3}, q^4, -q^{5}, q^{11};q^{21})}\right) +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &-2q^{-1}m(q^{5}, q^{21}, q^{6})-2q^{-4}m(q^{-2}, q^{21}, q^{6})\\ &-\frac{2q^{-2}J_{21}^3j(-q, -q^6, -q^{8}, q^7;q^{21})} {j(-1, -q^{2}, q^3, q^4, -q^{5}, q^{11};q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &-2q^{-1}m(q^{5}, q^{21}, q^{6})-2q^{-4}m(q^{-2}, q^{21}, q^{6})\\ &-\frac{2q^{-2}J_7^2j(-q;q^{7})j(-q^{9};q^{21})}{j(-q^2, q^3;q^{7})j(-1;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}\\ = &{2q^2g(q^3, q^7)-\frac{2q^{-2}J_7^2j(-q;q^{7})j(-q^{9};q^{21})}{j(-q^2, q^3;q^{7})j(-1;q^{21})} +\frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}}, } \end{align}$

(2.16)

where in the fourth equality, we replace $q$ , $a$ , $b$ , $c$ , and $d$ by $q^{21}$ , $-q^{15}$ , $q^{10}$ , $q^{7}$ , and $q^{4}$ , respectively, in Proposition 2.1.

Furthermore, applying the standard computational techniques from the theory of modular forms, we can prove the following identities. Those unfamiliar with this method might consult the work of Garvan and Liang ^[2] and Robins ^[8].

$\begin{align*} \frac{\eta_{21, 0}(\tau)\eta_{14, 4}(\tau)\eta_{42, 6}(\tau)} {\eta_{14, 6}(\tau)\eta_{21, 3}(\tau)\eta_{42, 0}(\tau)}-\frac{\eta_{6, 0}(\tau) \eta_{7, 1}(\tau)\eta_{21, 0}^{\frac{1}{2}}(\tau)} {\eta_{3, 0}^{\frac{1}{2}}(\tau)\eta_{14, 6}^2(\tau)\eta_{42, 0}(\tau)} = 1, \end{align*}$

$\begin{align*} \frac{\eta_{6, 0}(\tau)\eta_{7, 2}(\tau)\eta_{21, 0}^{\frac{1}{2}}(\tau)} {\eta_{3, 0}^{\frac{1}{2}}(\tau)\eta_{14, 2}^2(\tau)\eta_{42, 0}(\tau)}- \frac{\eta_{14, 6}(\tau)\eta_{21, 0}(\tau)\eta_{42, 12}(\tau)} {\eta_{14, 2}(\tau)\eta_{21, 6}(\tau)\eta_{42, 0}(\tau)} = 1, \end{align*}$

$\begin{align*} \frac{\eta_{6, 0}(\tau)\eta_{7, 3}(\tau)\eta_{21, 0}^{\frac{1}{2}}(\tau)} {\eta_{3, 0}^{\frac{1}{2}}(\tau)\eta_{14, 4}^2(\tau)\eta_{42, 0}(\tau)}- \frac{\eta_{14, 2}(\tau)\eta_{21, 0}(\tau)\eta_{42, 18}(\tau)} {\eta_{14, 4}(\tau)\eta_{21, 9}(\tau)\eta_{42, 0}(\tau)} = 1. \end{align*}$

Here, $\eta_{\delta, g}(\tau)$ is the generalized Dedekind $\eta$ -function defined by

$\begin{align*} \eta_{\delta, g}(\tau) = q^{P_2(g/\delta)\delta/2}\prod\limits_{n > 0\atop n\equiv g\pmod\delta}(1-q^n)\prod\limits_{n > 0\atop n\equiv -g\pmod\delta}(1-q^n), \end{align*}$

where $\tau\in \mathcal{H}: = \{\tau\in \mathcal{C}:Im \tau > 0\}$ , $q = e^{2\pi i\tau}$ , $P_2(t) = \{t\}^2-\{t\}+\frac{1}{6}$ is the second Bernoulli function, and $\{t\}$ is the fractional part of $t$ .

The above three identities can be rewritten as

$\begin{align} \frac{2q^{-1}J_7^2\overline{J}_{2, 7}\overline{J}_{3, 21}}{J_{1, 7}\overline{J}_{3, 7} \overline{J}_{0, 21}} -\frac{q^{-1}J_7^3\overline{J}_{0, 3}J_{1, 7}}{J_1\overline{J}_{4, 7}^2\overline{J}_{0, 21}} = \frac{J_{3, 7}^2}{J_1}, \end{align}$

(2.17)

$\begin{align} \frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{2, 7}}{J_1\overline{J}_{1, 7}^2\overline{J}_{0, 21}} -\frac{2q^{-2}J_7^2\overline{J}_{3, 7}\overline{J}_{6, 21}}{\overline{J}_{1, 7}J_{2, 7} \overline{J}_{0, 21}} = \frac{qJ_{1, 7}^2}{J_1}, \end{align}$

(2.18)

$\begin{align} \frac{q^{-2}J_7^3\overline{J}_{0, 3}J_{3, 7}}{J_1\overline{J}_{2, 7}^2\overline{J}_{0, 21}} -\frac{2q^{-2}J_7^2\overline{J}_{1, 7}\overline{J}_{9, 21}}{\overline{J}_{2, 7}J_{3, 7} \overline{J}_{0, 21}} = \frac{J_{2, 7}^2}{J_1}. \end{align}$

(2.19)

Then combining (2.14) (resp. (2.15) and (2.16)), (2.17) (resp. (2.18) and (2.19)) and (1.7), we arrive at (1.4) (resp. (1.5) and (1.6)).

References

[1]	S. Cooper, Ramanujan's theta functions, Cham: Springer, 2017. http://dx.doi.org/10.1007/978-3-319-56172-1
[2]	F. Garvan, J. Liang, Automatic proof of theta-function identities, arXiv: 1807.08051, 2016.
[3]	G. Gasper, M. Rahman, Basic hypergeometric series, 2Eds., Cambridge: Cambridge University Press, 2004. http://dx.doi.org/10.1017/CBO9780511526251
[4]	D. Hickerson, A proof of the mock theta conjectures, Invent. Math., 94 (1988), 639–660. http://dx.doi.org/10.1007/BF01394279 doi: 10.1007/BF01394279
[5]	D. Hickerson, On the seventh order mock theta functions, Invent. Math., 94 (1988), 661–677. http://dx.doi.org/10.1007/BF01394280 doi: 10.1007/BF01394280
[6]	D. Hickerson, E. Mortenson, Hecke-type double sums, Appell-Lerch sums, and mock theta functions, Ⅰ, Proc. Lond. Math. Soc., 109 (2014), 382–422. http://dx.doi.org/10.1112/plms/pdu007 doi: 10.1112/plms/pdu007
[7]	S. Ramanujan, Collected papers, London: Cambrige University Press, 1927.
[8]	S. Robins, Generalized Dedekind $\eta$ -products, In: The Rademacher legacy to mathematics, New York: Amer Mathematical Society, 1994,119–128.
[9]	G. Watson, The final problem: an account of the mock theta functions, J. Lond. Math. Soc., 1 (1936), 55–80. http://dx.doi.org/10.1112/jlms/s1-11.1.55 doi: 10.1112/jlms/s1-11.1.55
[10]	G. Watson, The mock theta functions (2), Proc. Lond. Math. Soc., 2 (1937), 274–304. http://dx.doi.org/10.1112/plms/s2-42.1.274 doi: 10.1112/plms/s2-42.1.274

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AIMS Mathematics

New proofs for three identities of seventh order mock theta functions

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New proofs for three identities of seventh order mock theta functions

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