Jacobi imaginary transformation counterparts of the mixed-pair circular summation formulas for theta functions (Cai–Bi–Luo Theorems 4 and 5)

The 2014 paper of Cai, Bi and Luo proved mixed-species circular summation formulas for Jacobi theta functions — finite sums over $kn$ regularly shifted arguments of $\theta_1^a\theta_2^b$ and of $\theta_3^a\theta_4^b$ collapsing to a single $\theta_3(z|\tau)$ — and supplied the Jacobi imaginary (modular) transformation $\tau\to-1/\tau$ only for its pure-power theorems, dismissing the transformed counterparts of the two mixed theorems in Remark 1 as obtainable “the same way” and omitting them. Imaginary-transformed circular summations have been published only for the pure-power families, and mixed-product summations only in direct form; no published work supplies the transformed counterparts of the two mixed theorems. The problem is to derive and state explicitly both transformed identities, with the coefficients $\mathscr{G}_{12}$ and $\mathscr{G}_{34}$ given in the source paper’s dual form — as the modular image of the direct coefficients $\mathscr{C}_{12}$ and $\mathscr{C}_{34}$ and as explicit multiple series — together with the parity/divisibility constraints and sign factors under which each identity holds.

Background

Jacobi’s four theta functions, in the notation used throughout this literature, are defined for $q=\mathrm{e}^{\mathrm{i}\pi\tau}$ with $\mathrm{Im}(\tau)>0$ and $z\in\mathbb{C}$ by

$$\theta_1(z|\tau)=-\mathrm{i}q^{1/4}\sum_{m=-\infty}^{\infty}(-1)^m q^{m(m+1)}\mathrm{e}^{(2m+1)\mathrm{i}z},\qquad \theta_2(z|\tau)=q^{1/4}\sum_{m=-\infty}^{\infty}q^{m(m+1)}\mathrm{e}^{(2m+1)\mathrm{i}z},$$ $$\theta_3(z|\tau)=\sum_{m=-\infty}^{\infty}q^{m^2}\mathrm{e}^{2m\mathrm{i}z},\qquad \theta_4(z|\tau)=\sum_{m=-\infty}^{\infty}(-1)^m q^{m^2}\mathrm{e}^{2m\mathrm{i}z}.$$

A circular summation formula expresses the sum of copies of a theta function (or of products of theta functions) evaluated at arguments shifted regularly around a period — “circularly” — as a single theta function times an explicit coefficient. The prototype goes back to Ramanujan; Chan, Liu and Ng (2006) proved the equivalence of the original claim with the theta-function form

$$\sum_{k=0}^{n-1}q^{k^2}\mathrm{e}^{2k\mathrm{i}z}\,\theta_3^{\,n}(z+k\pi\tau\,|\,n\tau)=\theta_3(z|\tau)\,F_n(\tau),$$

with $F_n$ an explicit function of $\tau$ (this is Theorem 1.8 as restated in the 2020 paper of Ge and Luo).

This equivalence triggered a family of generalizations. Zhou and Luo (2014) treated the pure $\theta_4$-power family and also wrote out its image under the Jacobi imaginary transformation. Cai, Chen and Luo (2013) established mixed-product circular summations — products $\theta_i^a\theta_j^b$ of two distinct species — for the pairs $(1,3)$, $(1,4)$, $(2,3)$, $(2,4)$. The source paper, “Some new circular summation formulas of theta functions” (Cai, Bi and Luo, 2014), closed the remaining two pairs with mixed $\theta_1^a\theta_2^b$ (Theorem 4) and $\theta_3^a\theta_4^b$ (Theorem 5) summations; later, Ge and Luo (2021) unified all six mixed pairs with general arguments, in direct form only.

The source paper’s two mixed theorems read as follows (the sums run over $a$ indices $m_1,\ldots,m_a$ and $b$ indices $n_1,\ldots,n_b$, with $\Sigma m=m_1+\cdots+m_a$, $\Sigma n=n_1+\cdots+n_b$, and per-index squares $\Sigma m^2=m_1^2+\cdots+m_a^2$, $\Sigma n^2=n_1^2+\cdots+n_b^2$).

Theorem 4 — for positive integers $a,b$ with $a$ and $b$ even, $k=a+b$, and $n$ a positive integer:

$$\sum_{s=0}^{kn-1}\theta_1^{a}\!\left(\frac{z}{kn}+\frac{y}{a}+\frac{\pi s}{kn}\,\Big|\,\frac{\tau}{kn^{2}}\right)\theta_2^{b}\!\left(\frac{z}{kn}-\frac{y}{b}+\frac{\pi s}{kn}\,\Big|\,\frac{\tau}{kn^{2}}\right)=\mathscr{C}_{12}\!\left(a,b;\frac{y}{ab},\frac{\tau}{kn^{2}}\right)\theta_3(z|\tau),$$

where

$$\mathscr{C}_{12}(a,b;y,\tau)=kn\,q^{k/4}\,\mathrm{i}^{a}\sum_{\substack{m_{1},\ldots,m_{a},\,n_{1},\ldots,n_{b}=-\infty\\2(\Sigma m+\Sigma n)+k=0}}^{\infty}(-1)^{\Sigma m}\,q^{\Sigma m^{2}+\Sigma n^{2}+\Sigma m+\Sigma n}\,\mathrm{e}^{\mathrm{i}y\left(2k\Sigma m+ak\right)}.$$

Theorem 5 — for positive integers $a,b$ with $k=a+b$ and $bn$ even:

$$\sum_{s=0}^{kn-1}\theta_{3}^{a}\!\left(\frac{z}{kn}+\frac{y}{a}+\frac{\pi s}{kn}\,\Big|\,\frac{\tau}{kn^{2}}\right)\theta_{4}^{b}\!\left(\frac{z}{kn}-\frac{y}{b}+\frac{\pi s}{kn}\,\Big|\,\frac{\tau}{kn^{2}}\right)=\mathscr{C}_{34}\!\left(a,b;\frac{y}{ab},\frac{\tau}{kn^{2}}\right)\theta_{3}(z|\tau),$$

where

$$\mathscr{C}_{34}(a,b;y,\tau)=kn\sum_{\substack{m_{1},\ldots,m_{a},\,n_{1},\ldots,n_{b}=-\infty\\\Sigma m+\Sigma n=0}}^{\infty}(-1)^{\Sigma n}\,q^{\Sigma m^{2}+\Sigma n^{2}}\,\mathrm{e}^{2k\mathrm{i}y\,\Sigma m}.$$

The Jacobi imaginary transformation is the modular involution $\tau\mapsto-1/\tau$ acting on theta functions through the four standard identities (as stated in the papers of Zhou–Luo 2014 and Ge–Luo 2020):

$$\theta_1\!\left(\frac{z}{\tau}\,\Big|\,-\frac{1}{\tau}\right)=-\mathrm{i}\,(-\mathrm{i}\tau)^{1/2}\,\mathrm{e}^{\mathrm{i}z^{2}/(\pi\tau)}\,\theta_1(z|\tau),\qquad \theta_2\!\left(\frac{z}{\tau}\,\Big|\,-\frac{1}{\tau}\right)=(-\mathrm{i}\tau)^{1/2}\,\mathrm{e}^{\mathrm{i}z^{2}/(\pi\tau)}\,\theta_4(z|\tau),$$ $$\theta_3\!\left(\frac{z}{\tau}\,\Big|\,-\frac{1}{\tau}\right)=(-\mathrm{i}\tau)^{1/2}\,\mathrm{e}^{\mathrm{i}z^{2}/(\pi\tau)}\,\theta_3(z|\tau),\qquad \theta_4\!\left(\frac{z}{\tau}\,\Big|\,-\frac{1}{\tau}\right)=(-\mathrm{i}\tau)^{1/2}\,\mathrm{e}^{\mathrm{i}z^{2}/(\pi\tau)}\,\theta_2(z|\tau),$$

so $\theta_1$ and $\theta_3$ map to themselves up to a Gaussian prefactor and a phase, while $\theta_2$ and $\theta_4$ are interchanged. Applied to a direct circular summation, it yields a genuinely new identity: in the published pure-family transforms (Theorems 6–8 of the source paper; Theorem 3.1 of Zhou–Luo 2014; Theorem 3.1 of Ge–Luo 2020) the left side becomes a weighted sum with $\tau$-scaled arguments — of the type $q^{\text{quadratic in }k}\,\mathrm{e}^{\text{linear in }k\text{ and }z}\,\theta(mz+y_j\pi\tau+mk\pi\tau\,|\,m^{2}n\tau)$ summed over $k$ — while the right side stays a single theta function of $z$ at modulus $\tau$ (species fixed by parity), and the coefficient appears in dual form: both as the modular image of the direct coefficient and as an explicit multiple series.

For its three pure-power theorems the source paper carried out this program in full (Theorems 6–8, with coefficients $\mathscr{G}_{11}$, $\mathscr{G}_{22}$, $\mathscr{G}_{44}$ in dual form). For the two mixed theorems it did not: Remark 1 states that the imaginary transformation formulas of Theorems 4 and 5 “may be obtained using the same way, we here omit them.” Under the transformation the $\theta_2$-factors of Theorem 4 become $\theta_4$-factors and the $\theta_4$-factors of Theorem 5 become $\theta_2$-factors, so the omitted formulas are mixed identities of a shape absent from the published literature.

Problem Statement

The problem is to write out, completely and explicitly, the two families of identities whose omission Remark 1 of Cai–Bi–Luo (2014) acknowledges: the Jacobi imaginary transformation ($\tau\to-1/\tau$) counterparts of Theorem 4 (mixed $\theta_1^a\theta_2^b$ circular summation) and Theorem 5 (mixed $\theta_3^a\theta_4^b$ circular summation).

Concretely, for each of the two theorems, an answer must provide:

1. The transformed identity itself: an explicit summation formula in which the left side is built from the transformed factors — the $\theta_2$-factors of Theorem 4 appearing as $\theta_4$-factors and the $\theta_4$-factors of Theorem 5 as $\theta_2$-factors, with $\tau$-scaled arguments (of the $mz+y_j\pi\tau+mk\pi\tau$ type at a modulus of $m^2n\tau$ type) and the Gaussian/phase weights the transformation generates — and the right side is a single theta function of $z$ at modulus $\tau$ (species determined by parity) times a coefficient independent of $z$, in the pattern set by the published pure-family transforms (Theorems 6–8 of the source paper; Theorem 3.1 of Zhou–Luo 2014 and of Ge–Luo 2020).

2. The coefficient $\mathscr{G}_{12}(a,b;y,\tau)$ (respectively $\mathscr{G}_{34}$) in dual form: both as the modular image of the direct coefficient $\mathscr{C}_{12}$ (respectively $\mathscr{C}_{34}$) and as an explicit multiple series.

3. The parity/divisibility constraints on $a$, $b$, $n$ under which each transformed identity holds, and the sign and phase factors appearing in each case (including the parity-determined choice of the right-side theta species).

The two identities, each complete in all three items enumerated above — the identity itself, the coefficient in both of its representations, and the parity/divisibility constraints with sign and phase factors — together constitute the completion boundary: an answer covering only one theorem, giving the coefficient in only one representation, omitting the parity/divisibility constraints or the sign and phase factors, or stating that the second follows “analogously” without writing it out, does not complete the problem. Both statements must be self-contained enough to be numerically evaluated from the statement alone.

References

1. Heng Huat Chan, Zhi-Guo Liu, Say Tiong Ng, “Circular summation of theta functions in Ramanujan’s Lost Notebook”, Journal of Mathematical Analysis and Applications 316 (2006) 628–641. DOI: 10.1016/j.jmaa.2005.05.015, https://doi.org/10.1016/j.jmaa.2005.05.015 (its circular summation theorems appear as Theorems 1.7–1.9 of the Ge–Luo 2020 entry below).
2. Jun-Ming Zhu, “An alternate circular summation formula of theta functions and its applications”, Applicable Analysis and Discrete Mathematics 6 (2012) 114–125. DOI: 10.2298/AADM120204004Z, https://doi.org/10.2298/AADM120204004Z.
3. Yi Cai, Si Chen, Qiu-Ming Luo, “Some circular summation formulas for theta functions”, Boundary Value Problems 2013 (2013) 59. DOI: 10.1186/1687-2770-2013-59, https://doi.org/10.1186/1687-2770-2013-59.
4. Yi Cai, Ya-Qing Bi, Qiu-Ming Luo, “Some new circular summation formulas of theta functions”, Integral Transforms and Special Functions 25 (2014) 497–507. DOI: 10.1080/10652469.2014.883388, https://doi.org/10.1080/10652469.2014.883388.
5. Yun Zhou, Qiu-Ming Luo, “Circular summation formulas for theta function ϑ4(z|τ)”, Advances in Difference Equations 2014 (2014) 243. DOI: 10.1186/1687-1847-2014-243, https://doi.org/10.1186/1687-1847-2014-243.
6. Qiu-Ming Luo, “A note for Ramanujan’s circular summation formula”, Integral Transforms and Special Functions 26 (2015) 567–585. DOI: 10.1080/10652469.2015.1028403, https://doi.org/10.1080/10652469.2015.1028403.
7. Ji-Ke Ge, Qiu-Ming Luo, “Some Ramanujan-type circular summation formulas”, Advances in Difference Equations 2020 (2020) 690. DOI: 10.1186/s13662-020-03115-9, https://doi.org/10.1186/s13662-020-03115-9.
8. Ji-Ke Ge, Qiu-Ming Luo, “Some extensions for Ramanujan’s circular summation formulas and applications”, The Ramanujan Journal 56 (2021) 491–518. DOI: 10.1007/s11139-020-00293-8, https://doi.org/10.1007/s11139-020-00293-8.