The classical Jacobi four theta functions \(\vartheta _{i} (z|\tau )\), \(i=1,2,3,4\), with the notation of Tannery and Molk, are defined as follows.
Definition 1.1
(see, e.g., [3, 12, 25])
For \(q=e^{\pi {i}\tau }\), \(\operatorname{Im} (\tau )>0\), \(z \in \mathbb{C}\).
$$\begin{aligned}& \vartheta _{1} (z|\tau ) =-iq^{\frac{1}{4}}\sum _{n=-\infty }^{\infty } (-1 )^{n}q^{n ( n+1 )}e^{ (2n+1 )iz}, \end{aligned}$$
(1.1)
$$\begin{aligned}& \vartheta _{2} (z|\tau ) =q^{\frac{1}{4}}\sum _{n=-\infty }^{ \infty } q^{n (n+1 )}e^{ (2n+1 )iz}, \end{aligned}$$
(1.2)
$$\begin{aligned}& \vartheta _{3} (z|\tau ) =\sum _{n=-\infty }^{\infty }q^{n^{2}}e^{2niz}, \end{aligned}$$
(1.3)
$$\begin{aligned}& \vartheta _{4} (z|\tau ) =\sum _{n=-\infty }^{\infty } (-1 )^{n}q^{n^{2}}e^{2niz}. \end{aligned}$$
(1.4)
From the Jacobi theta functions (1.1)–(1.4), via the direct calculation, we have the following properties, respectively.
Proposition 1.2
$$\begin{aligned} & \vartheta _{1} (z+\pi |\tau )=-\vartheta _{1} (z| \tau ),\qquad \vartheta _{1} (z+\pi \tau |\tau )=-q^{-1}e^{-2iz} \vartheta _{1} (z|\tau ), \end{aligned}$$
(1.5)
$$\begin{aligned} & \vartheta _{2} (z+\pi |\tau )=-\vartheta _{2} (z| \tau ),\qquad \vartheta _{2} (z+\pi \tau |\tau )=q^{-1}e^{-2iz} \vartheta _{2} (z|\tau ), \end{aligned}$$
(1.6)
$$\begin{aligned} & \vartheta _{3} (z+\pi |\tau )=\vartheta _{3} (z| \tau ),\qquad \vartheta _{3} (z+\pi \tau |\tau )=q^{-1}e^{-2iz} \vartheta _{3} (z|\tau ), \end{aligned}$$
(1.7)
$$\begin{aligned} & \vartheta _{4} (z+\pi |\tau )=\vartheta _{4} (z| \tau ),\qquad \vartheta _{4} (z+\pi \tau |\tau )=-q^{-1}e^{-2iz} \vartheta _{4} (z|\tau ). \end{aligned}$$
(1.8)
Proposition 1.3
$$\begin{aligned} & \vartheta _{1} \biggl(z+{\frac{\pi }{2}}\Big| \tau \biggr)=\vartheta _{2} (z|\tau ),\qquad \vartheta _{1} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=iq^{-\frac{1}{4}}e^{-iz} \vartheta _{4} (z| \tau ), \end{aligned}$$
(1.9)
$$\begin{aligned} & \vartheta _{2} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=-\vartheta _{1}(z| \tau ),\qquad \vartheta _{2} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=q^{-\frac{1 }{4}}e^{-iz} \vartheta _{3} (z|\tau ), \end{aligned}$$
(1.10)
$$\begin{aligned} & \vartheta _{3} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=\vartheta _{4}(z| \tau ),\qquad \vartheta _{3} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=q^{-\frac{1 }{4}}e^{-iz} \vartheta _{2} (z|\tau ), \end{aligned}$$
(1.11)
$$\begin{aligned} & \vartheta _{4} \biggl(z+\frac{\pi }{2}\Big| \tau \biggr)=\vartheta _{3}(z| \tau ),\qquad \vartheta _{4} \biggl(z+\frac{\pi \tau }{2}\Big|\tau \biggr)=iq^{-\frac{1 }{4}}e^{-iz} \vartheta _{1} (z|\tau ). \end{aligned}$$
(1.12)
From (1.5)–(1.8), by applying induction, we easily obtain the following.
Lemma 1.4
For n is a nonnegative integer, we have
$$\begin{aligned} & \vartheta _{1} (z+n \pi |\tau )=(-1)^{n} \vartheta _{1} (z|\tau ),\qquad \vartheta _{1} (z+n \pi \tau | \tau )=(-1)^{n}q^{-n^{2}}e^{-2niz} \vartheta _{1} (z|\tau ), \end{aligned}$$
(1.13)
$$\begin{aligned} & \vartheta _{2} (z+n \pi |\tau )=(-1)^{n} \vartheta _{2} (z|\tau ),\qquad \vartheta _{2} (z+n\pi \tau |\tau )=q^{-n^{2}}e^{-2niz} \vartheta _{2} (z|\tau ), \end{aligned}$$
(1.14)
$$\begin{aligned} & \vartheta _{3} (z+n \pi |\tau )=\vartheta _{3} (z| \tau ),\qquad \vartheta _{3} (z+n\pi \tau | \tau )=q^{-n^{2}}e^{-2niz}\vartheta _{3} (z|\tau ), \end{aligned}$$
(1.15)
$$\begin{aligned} & \vartheta _{4} (z+n \pi |\tau )=\vartheta _{4} (z| \tau ),\qquad \vartheta _{4} (z+n\pi \tau | \tau )=(-1)^{n}q^{-n^{2}}e^{-2niz}\vartheta _{4} (z|\tau ). \end{aligned}$$
(1.16)
From (1.9)–(1.16), we have the following lemmas.
Lemma 1.5
For n is any positive integer, we have
$$\begin{aligned} & \vartheta _{1} \biggl(z+\frac{n \pi }{2}\Big| \tau \biggr)= \textstyle\begin{cases} i^{n} \vartheta _{1} (z|\tau ), & n \textit{ is even}, \\ i^{n-1} \vartheta _{2} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.17)
$$\begin{aligned} & \vartheta _{2} \biggl(z+\frac{n \pi }{2}\Big| \tau \biggr)= \textstyle\begin{cases} i^{n} \vartheta _{2} (z|\tau ), & n \textit{ is even}, \\ -i^{n-1} \vartheta _{1} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.18)
$$\begin{aligned} & \vartheta _{3} \biggl(z+\frac{n \pi }{2}\Big| \tau \biggr)= \textstyle\begin{cases} \vartheta _{3} (z|\tau ), & n \textit{ is even}, \\ \vartheta _{4} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.19)
$$\begin{aligned} & \vartheta _{4} \biggl(z+\frac{n \pi }{2}\Big| \tau \biggr)= \textstyle\begin{cases} \vartheta _{4} (z|\tau ), & n \textit{ is even}, \\ \vartheta _{3} (z|\tau ), & n \textit{ is odd}. \end{cases}\displaystyle \end{aligned}$$
(1.20)
Lemma 1.6
For n is any positive integer, we have
$$\begin{aligned} & \vartheta _{1} \biggl(z+\frac{n \pi \tau }{2}\Big| \tau \biggr)= \textstyle\begin{cases} i^{n} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{1} (z|\tau ), & n \textit{ is even}, \\ i^{n} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{4} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.21)
$$\begin{aligned} & \vartheta _{2} \biggl(z+\frac{n \pi \tau }{2}\Big| \tau \biggr)= \textstyle\begin{cases} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{2} (z|\tau ), & n \textit{ is even}, \\ q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{3} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.22)
$$\begin{aligned} & \vartheta _{3} \biggl(z+\frac{n \pi \tau }{2}\Big| \tau \biggr)= \textstyle\begin{cases} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{3} (z|\tau ), & n \textit{ is even}, \\ q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{2} (z|\tau ), & n \textit{ is odd}, \end{cases}\displaystyle \end{aligned}$$
(1.23)
$$\begin{aligned} & \vartheta _{4} \biggl(z+\frac{n \pi \tau }{2}\Big| \tau \biggr)= \textstyle\begin{cases} i^{n} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{4} (z|\tau ), & n \textit{ is even}, \\ i^{n} q^{-\frac{n^{2}}{4}}e^{-niz}\vartheta _{1} (z|\tau ), & n \textit{ is odd}. \end{cases}\displaystyle \end{aligned}$$
(1.24)
On page 54 in Ramanujan’s lost notebook (see [21, p. 54, Entry 9.1.1], [2, p. 337]), Ramanujan recorded the following claim (without proof), which is now well known as Ramanujan’s circular summation. The appellation circular summation was initiated by Son (see [2, p. 338]).
Theorem 1.7
(Ramanujan’s circular summation)
For each positive integer n and \(\vert ab \vert < 1\),
$$ \sum_{-n/2< r \leq n/2} \Biggl(\sum _{ \substack{k=-\infty \\ k \equiv r (\operatorname{mod} n)}}^{\infty } a^{{k(k+1)}/{(2n)}}b^{{k(k-1)}/{(2n)}} \Biggr)^{n}=f(a,b)F_{n}(ab), $$
(1.25)
where
$$ F_{n}(q):=1+2nq^{(n-1)/2}+\cdots , \quad n \geq 3. $$
Ramanujan’s theta function \(f(a,b)\) is defined by
$$ f(a,b)=\sum_{n=-\infty }^{\infty }a^{{n(n+1)}/{2}}b^{{n(n-1)}/{2}},\quad \vert ab \vert < 1. $$
Chan, Liu, and Ng [10] proved that Theorem 1.7 is equivalent to the following form.
Theorem 1.8
(Ramanujan’s circular summation)
For each positive integer n,
$$ \sum_{k=0}^{n-1}q^{k^{2}}e^{2k iz} \vartheta ^{n}_{3}(z+k\pi \tau |n \tau )=\vartheta _{3}(z|\tau )F_{n}(\tau ), $$
(1.26)
where for \(n \geq 3\),
$$ F_{n}(\tau )=1+2nq^{n-1}+\cdots . $$
Chan, Liu, and Ng [10] also showed that Theorem 1.8 is an equivalent of the theorem below by applying the Jacobi imaginary transformation formulas [25, p. 475]. They also proved that Theorem 1.9 is equivalent to Theorem 1.7.
Theorem 1.9
(Ramanujan’s circular summation)
For any positive integer n, there exists a quantity \(G_{n}(\tau )\) such that
$$ \sum_{k=0}^{n-1} \vartheta ^{n}_{3} \biggl(z+\frac{k\pi }{n}\Big|\tau \biggr)=G_{n}(\tau )\vartheta _{3}(nz|n\tau ), $$
(1.27)
where
$$ G_{n}(\tau )=\sqrt{n} (-i \tau )^{(1-n)/2}F_{n} \biggl(- \frac{1}{n \tau } \biggr). $$
(1.28)
Ramanujan’s circular summation is an interesting subject in his notebook. On the subject of Ramanujan’s circular summation and related theta function identities and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, Andrews, Berndt, Rangachari, Ono, Ahlgren, Chua, Murayama, Son, Chan, Liu, Ng, Chan, Shen, Cai, Zhu, and Xu et al. [1, 2, 4–11, 13, 14, 16, 18–20, 22–24, 26, 27, 29, 30]).
Recently, Liu and Luo [15] obtained the alternating circular summation formulas of theta function \(\vartheta _{3} (z|\tau )\). Luo [17] further generalized the results of Chan and Liu on Ramanujan’s circular summation formula for theta functions \(\vartheta _{3} (z|\tau )\) and deduced some alternating summation formulas of theta functions \(\vartheta _{1} (z|\tau )\) and \(\vartheta _{2} (z|\tau )\). Zhou and Luo [28] studied a variation for Ramanujan’s circular summation of theta function \(\vartheta _{4} (z|\tau )\), which here we call the Ramanujan-type circular summation.
Motivated by [10, 11], and [15, 17, 28], by applying the theory of elliptic functions, we further investigate other two Ramanujan-type circular summations for theta functions \(\vartheta _{1} (z|\tau )\) and \(\vartheta _{2} (z|\tau )\), which are two variations of Ramanujan’s circular summations (noting that is not alternating).
The paper is organized as follows: In the first section we display the definitions and properties of four theta functions. In the second section we show and prove two Ramanujan-type circular summation formulas for theta functions \(\vartheta _{1}(z|\tau )\) and \(\vartheta _{2}(z|\tau )\) based on the theory and method of elliptic functions and properties of theta functions. In the third section we derive the corresponding imaginary transformation formulas of circular summation formulas by using the imaginary transformation formulas of \(\vartheta _{1}(z|\tau )\) and \(\vartheta _{2}(z|\tau )\). In the fourth section we give some further results and remarks.