Hyperbolic summations derived using the Jacobi functions dc and nc

John M. Campbell

###### Abstract

We introduce a method that is based on Fourier series expansions related to
Jacobi elliptic functions and that we apply to determine new identities for
evaluating hyperbolic infinite sums in terms of the complete elliptic
integrals $K$ and $E$. We apply our method to determine generalizations of a
family of sech-sums given by Ramanujan and generalizations of a family of
csch-sums given by Zucker. Our method has the advantage of producing
evaluations for hyperbolic sums with sign functions that have not previously
appeared in the literature on hyperbolic sums. We apply our method using the
Jacobian elliptic functions dc and nc, together with the elliptic alpha
function, to obtain new closed forms for $q$-digamma expressions, and new
closed forms for series related to discoveries due to Ramanujan, Berndt, and
others.

Keywords: Jacobi elliptic function, elliptic integral, hyperbolic function,
$\Gamma$-function, symbolic evaluation, elliptic alpha function

Mathematics Subject Classification: 42A16, 33E05

##  1 Introduction

The Jacobi elliptic functions are defined via the inversion of the elliptic
integral

$u=F(\phi,k)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-k^{2}\sin^{2}t}}$ (1)

of the first kind, letting $0<k^{2}<1$. The expression $k=\text{mod}\,u$ is
referred to as the _elliptic modulus_ , and $\phi=\text{am}(u,k)=\text{am}(u)$
is referred to as the _Jacobi amplitude_. The _Jacobi elliptic functions_ sn,
cn, and dn may be defined as follows:

$\displaystyle\sin\phi$ $\displaystyle=\text{sn}(u,k),$ (2)
$\displaystyle\cos\phi$ $\displaystyle=\text{cn}(u,k),\ \ \ \text{and}$ (3)
$\displaystyle\sqrt{1-k^{2}\sin^{2}\phi}$ $\displaystyle=\text{dn}(u,k).$ (4)

In this article, we apply Fourier series expansions related to Jacobi elliptic
functions to build on the work on hyperbolic infinite sums due to Zucker [67],
Yakubovich [64], and Xu [63]. The relevance of Zucker’s work in [67] (cf.
[66]) within high-energy physics [17, 18, 23], nuclear physics [22],
electrostatics [38], the study of lattice sums [16], number-theoretic subjects
in the study of Fibonacci sums [24, 25, 26, 27, 28], and number-theoretic
areas concerning Ramanujan’s series for $\frac{1}{\pi}$ [4, 5, 15] and for
constants such as Apéry’s constant [59] serve as a main source of motivation
behind the techniques and results introduced in this article.

###  1.1 Background and further preliminaries

The complete elliptic integral of the first kind $K$ is such that
$K(k)=F\left(\frac{\pi}{2},k\right)$. We may also write $K(k)=K$ and
$K^{\prime}=K^{\prime}(k)=K(k^{\prime})$, with $k^{\prime}=\sqrt{1-k^{2}}$,
and similarly with respect to the complete elliptic integral
$E(k)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^{2}\sin^{2}\theta}\,d\theta$ of the
second kind.

Combinations involving the sn, cn, and dn functions give us the remaining
Jacobi elliptic functions, as listed below:

$\displaystyle\text{cd}(u,k)$
$\displaystyle=\frac{\text{cn}(u,k)}{\text{dn}(u,k)},$ (5)
$\displaystyle\text{cs}(u,k)$
$\displaystyle=\frac{\text{cn}(u,k)}{\text{sn}(u,k)},$
$\displaystyle\text{dc}(u,k)$
$\displaystyle=\frac{\text{dn}(u,k)}{\text{cn}(u,k)},$ (6)
$\displaystyle\text{ds}(u,k)$
$\displaystyle=\frac{\text{dn}(u,k)}{\text{sn}(u,k)},$
$\displaystyle\text{nc}(u,k)$ $\displaystyle=\frac{1}{\text{cn}(u,k)},$
$\displaystyle\text{nd}(u,k)$ $\displaystyle=\frac{1}{\text{dn}(u,k)},$
$\displaystyle\text{ns}(u,k)$ $\displaystyle=\frac{1}{\text{sn}(u,k)},$
$\displaystyle\text{sc}(u,k)$
$\displaystyle=\frac{\text{sn}(u,k)}{\text{cn}(u,k)},\ \ \ \text{and}$
$\displaystyle\text{sd}(u,k)$
$\displaystyle=\frac{\text{sn}(u,k)}{\text{dn}(u,k)}.$

The $\Gamma$-function [49, §8] is famous and ubiquitous as a special function,
and is to frequently arise in our work. We recall that this special function
is defined for $\Re(x)>0$ with the Euler integral
$\Gamma(x)=\int_{0}^{\infty}u^{x-1}e^{-u}\,du$ [49, §8]. A useful feature
concerning our method in Section 1.2 is given by how this method may be used
to obtain new evaluations for expressions involving the $q$-digamma function
$\psi_{q}(z)$, which may be defined so that

$\psi_{q}(z)=\frac{1}{\Gamma_{q}(z)}\frac{\partial\Gamma_{q}(z)}{\partial z},$

where $\Gamma_{q}$ denotes the $q$-analogue of the $\Gamma$-function.
Equivalently, we may define the $q$-digamma function so that

$\psi_{q}(z)=-\ln(1-q)+\ln q\sum_{n=0}^{\infty}\frac{q^{n+z}}{1-q^{n+z}}.$ (7)

In Ramanujan’s second notebook [12, §17] (cf. [8]), identities for evaluating

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{s}\text{sech}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$
(8)

in terms of $K$ are given, such as the identity [12, p. 134]

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\text{sech}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)=\frac{2kk^{\prime}K^{2}}{\pi^{2}}.$
(9)

In Zucker’s seminal article on hyperbolic sums [67], an identity for
evaluating (8) was also included, and [67] also provided identities for the
sums given by replacing sech with csch in Ramanujan’s sums in (8):

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{s}\text{csch}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)=2J_{s},$
(10)

with, for example,

$4J_{0}=\left(\frac{2K}{\pi}\right)k\ \ \ \text{and}\ \ \
4J_{2}=\left(\frac{2K}{\pi}\right)^{3}k(1-k^{2}).$

Zucker’s methods in [67] mainly relied on double series manipulations together
with expansions such as

$J_{s}(c)=J_{s}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(2n-1)^{s}q^{n-1/2}}{1-q^{2n-1}},$

writing

$q=e^{-\pi\frac{K^{\prime}}{K}}$ (11)

to denote the _nome_ for Jacobian elliptic functions. In this article, we
provide a method that may be used to to evaluate the members of both of the
families of generalizations of (8) and (10) indicated as follows:

  1. 1.

The sums obtained by replacing sech (resp. csch) with the higher power
$\text{sech}^{2}$ (resp. $\text{csch}^{2}$), for all odd powers $s$, in (8)
(resp. (10)); and

  2. 2.

The sums given by replacing the sign function $(-1)^{n}$ with the sign
function $(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}$ within the sums
indicated in the preceding point.

Our evaluation technique may be applied much more broadly, apart from the
families of hyperbolic sums indicated above. For example, our method may also
be applied to produce new summations that resemble and are related to the
summations

$\sum_{n=1}^{\infty}\frac{n}{e^{2\pi n}-1}=\frac{1}{24}-\frac{1}{8\pi}\ \ \
\text{and}\ \ \ \sum_{n=1}^{\infty}\frac{n^{13}}{e^{2n\pi}-1}=\frac{1}{24}$

due to Ramanujan [51, pp. 326, xxvi]. These Ramanujan summations have been
explored by authors such as Nanjundiah [44] and Sandham [53]. As in [44], we
record that Ramanujan had given the first out of the above identities in his
seminal article on modular equations and approximations to $\pi$ [50].

As in Yakubovich’s article [64], our work is a continuation of the methods due
to Ling and Zucker [39, 40, 67]. Yakubovich’s symbolic forms as in

$\frac{\Gamma^{4}\left(\frac{1}{4}\right)}{32\pi^{4}}+\frac{\Gamma^{8}\left(\frac{1}{4}\right)}{512\pi^{6}}-\frac{1}{8\pi^{2}}=\sum_{n=1}^{\infty}n^{2}\cosh(\pi
n)\text{csch}^{2}(\pi n)$

for infinite series involving $\text{csch}^{2}$ were highlighted as
Corollaries in [64] and motivate our symbolic forms for $\text{csch}^{2}$-sums
as in Examples 7 and 8 below.

###  1.2 Main technique

Our main technique may be summarized in the following manner, letting
$\text{j}(u,k)$ denote a Jacobi elliptic function.

  1. 1.

Start with the Fourier series expansion for $\text{j}(u,k)$ or for some
expression involving a Jacobi elliptic function, or some manipulation of such
Fourier series expansions such as a series expansion obtained via term-by-term
applications of a differential operator;

  2. 2.

In order to use built-in CAS algorithms for reducing, if possible, derivatives
of $q$-powers with respect to the elliptic modulus and using hyperbolic
functions, we need to rewrite the summand of the series indicated in the
previous step so that any exponential expressions only appear in the
denominator, and we need to simplify the powers for any such expressions;

  3. 3.

Enforce a substitution such as $u\mapsto 2wK$ for a variable $w$;

  4. 4.

Argue, if possible, that if $w$ were to be set to some special value, the
series obtained from the third step would reduce to a closed-form evaluation;
and

  5. 5.

If the resultant summand is non-vanishing, and if differentiating with respect
to the elliptic modulus yields a summand that may be expressed in terms of
hyperbolic functions, then simplify the resultant summand.

###  1.3 Organization of the article

The above technique may be applied to many out of the 12 of the Jacobian
functions among cd, cn, cs, dc, dn, ds, nc, nd, ns, sc, sd, and sn. However,
we sometimes obtain equivalent results by applying our technique to different
Jacobian elliptic functions. For example, if we apply this technique to ns,
then the results that we would obtain would also be obtainable by applying our
technique to dc. The results in this article in Sections 2 and 3 are devoted
to the application of our technique to dc and and nc, and if we were to
attempt to apply the same method to Jacobi elliptic functions other than dc
and and nc, this method would either not be applicable, or we would obtain the
same results as in Sections 2 and 3 below.

Our method may also be applied to Fourier series expansions for expressions
other than the 12 Jacobi functions we have listed. For example, an equivalent
formulation of the $q$-expansion

$\left(\frac{2K}{\pi}\right)^{2}\text{ns}^{2}(2Kw,k)=\frac{4K(K-E)}{\pi^{2}}+\csc^{2}(\pi
w)-8\sum_{n=1}^{\infty}\frac{nq^{2n}\cos(2n\pi w)}{1-q^{2n}}$ (12)

for $\text{ns}^{2}$ which, as recorded in [67], was included in the classic
text [62, p. 535] and dates back to the work of Jacobi [32], may be applied in
accordance with the technique in Section 1.2. This is briefly considered in
Section 4.

##  2 Applications of the Jacobi elliptic function dc

The connections between the Jacobi elliptic functions and the generalized
Bessel functions, as explored in [20], serve as a further source of motivation
concerning our applications related to series expansions given in [20]. To
begin with, we record the Fourier series expansion

$\text{dc}(u,k)=\frac{\pi}{2K}\sec
v+\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}}{1-q^{2n+1}}\cos((2n+1)v)$
(13)

given in [20] and in classic texts such as [61, pp. 511–512], writing
$v=\frac{\pi u}{2K}$ and recalling (11). Following the evaluation technique in
Section 1.2, we rewrite the above series expansion for dc as follows:

$\text{{dc}}(2Kw,k)=\frac{\pi}{2K}\left(\frac{1}{\cos(\pi
w)}+4\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\pi\frac{K^{\prime}}{K}}-1}\cos((2n+1)\pi
w)\right).$ (14)

Setting $w=0$, (14) this gives us that

$\frac{K}{2\pi}-\frac{1}{4}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\pi\frac{K^{\prime}}{K}}-1},$
(15)

and the formula in (15) was also proved by Bagis in [3] (Corollary 1), using
an identity due to Ramanujan [12, p. 174]. Applying term-by-term
differentiation to (15) with respect to the elliptic modulus, we may obtain
that

$\frac{2K^{2}\left(E+\left(k^{2}-1\right)K\right)}{\pi^{2}(K^{\prime}(E-K)+E^{\prime}K)}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\text{{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right).$
(16)

We intend to generalize (16), by analogy with Zucker’s sum in (10).

###  2.1 Generalizing Zucker’s sum

To generalize Zucker’s sum in (10) by replacing csch with $\text{csch}^{2}$ in
(10), we apply our technique indicated in Section 1.2, as in the below proof.

######  Theorem 1.

The equality

$\frac{1}{4}+\frac{2\left(k^{2}-1\right)K^{3}}{\pi^{3}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n+1)^{2}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}-1}$

holds if the above series converges.

###### Proof.

We again begin with the Fourier series expansion for dc in (13). We rewrite
the left-hand side of (13) according to (6), and then apply the operator
$\frac{d^{2}}{du^{2}}$ to both sides of the resultant equality. We then expand
the left-hand according to the following differential equations:

$\displaystyle\frac{d\,\text{sn}u}{du}$
$\displaystyle=\text{cn}u\,\text{dn}u,$ (17) $\displaystyle\frac{d\,\text{cn
}u}{du}$ $\displaystyle=-\text{sn}u\,\text{dn}u,\ \ \ \text{and}$ (18)
$\displaystyle\frac{d\,\text{dn}u}{du}$
$\displaystyle=-k^{2}\text{sn}u\,\text{cn}u,$ (19)

After much simplification and rearrangement using (17)–(19), we can show that

$\frac{\text{dn}(u,k)\left(\text{dn}^{2}(u,k)-k^{2}\text{cn}^{2}(u,k)\right)\left(\text{cn}^{2}(u,k)+2\text{sn}^{2}(u,k)\right)}{\text{cn}^{3}(u,k)}$
(20)

equals

$\frac{\pi^{3}\left(3-\cos\left(\frac{\pi
u}{K}\right)\right)\sec^{3}\left(\frac{\pi
u}{2K}\right)}{16K^{3}}-\frac{\pi^{3}}{2K^{3}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}(2n+1)^{2}\cos\left(\frac{(2n+1)\pi
u}{2K}\right)}{1-q^{2n+1}},$ (21)

reversing the order of the limiting operations $\frac{d^{2}}{du^{2}}$ and
$\sum_{n=0}^{\infty}\cdot$. Following the steps in Section 1.2, we set
$u\mapsto 2wK$. Setting $w=0$ then gives us an equivalent formulation of the
desired result. ∎

We let elliptic singular values be denoted in the usual way [14, p. 298],
writing

$\frac{K^{\prime}(k_{r})}{K(k_{r})}=\sqrt{r},$ (22)

where the elliptic lambda function is such that

$\lambda^{\ast}(r)=k_{r},$ (23)

with expressions of the form $K(k_{r})$ admitting explicit symbolic
evaluations for natural numbers $r\in\mathbb{N}$. Although the result
highlighted as Theorem 1 is to be applied to obtain new hyperbolic summations,
we are to apply this same result using the relations in (22) and (23).

######  Example 1.

Setting $r=1$ in (22), from the symbolic forms

$\lambda^{\ast}(1)=k_{1}=\frac{\sqrt{2}}{2}$ (24)

and

$K(k_{1})=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{4\sqrt{\pi}},$ (25)

this gives us, from Theorem 1, that

$\frac{1}{4}-\frac{\Gamma^{6}\left(\frac{1}{4}\right)}{64\pi^{9/2}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n+1)^{2}}{e^{(2n+1)\pi}-1}.$

######  Example 2.

Setting $r=4$ in (22), the symbolic forms

$\lambda^{\ast}(4)=k_{4}=3-2\sqrt{2},$ (26)

and

$K(k_{4})=\frac{\left(1+\sqrt{2}\right)\Gamma^{2}\left(\frac{1}{4}\right)}{8\sqrt{2\pi}}$
(27)

give us, via Theorem 1, that

$\frac{1}{4}-\frac{\left(1+\sqrt{2}\right)\Gamma^{6}\left(\frac{1}{4}\right)}{128\pi^{9/2}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n+1)^{2}}{e^{2(2n+1)\pi}-1}.$

Theorem 1 may be further applied so as to obtain the following evaluation for
the sum given by setting $s=3$ and replacing csch with $\text{csch}^{2}$ in
Zucker’s sum in (10).

######  Theorem 2.

The identity

$\frac{8\left(k^{2}-1\right)K^{4}\left(3E+\left(k^{2}-3\right)K\right)}{\pi^{4}(K^{\prime}(E-K)+E^{\prime}K)}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{3}\text{\emph{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

holds if the above series converges.

###### Proof.

This follows in a direct way by applying term-by-term differentiation with
respect to the elliptic modulus in Theorem 1. ∎

######  Example 3.

Using the elliptic alpha function

$\alpha(r)=\frac{\pi}{4K^{2}(k_{r})}+\sqrt{r}-\frac{E(k_{r})\sqrt{r}}{K(k_{r})}$
(28)

together with the elliptic function identity

$E^{\prime}=\frac{\pi}{4K}+\alpha(r)K$ (29)

and the known formula $\alpha(1)=\frac{1}{2}$, we may obtain symbolic forms
for $E(k_{1})$ and $E^{\prime}(k_{1})$. So, in Theorem 2, we set $k=k_{1}$ as
the value in (24), so as to obtain that

$\frac{\Gamma^{10}\left(\frac{1}{4}\right)}{128\pi^{15/2}}-\frac{3\Gamma^{6}\left(\frac{1}{4}\right)}{32\pi^{11/2}}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{3}\text{csch}^{2}\left(\frac{2n+1}{2}\pi\right).$

######  Example 4.

Setting $k=k_{4}$ as the value in (26), from the known valuation
$\alpha(4)=2(\sqrt{2}-1)^{2}$, this, together with the elliptic alpha function
identities in (28) and (29), allows us to obtain, via Theorem 2, that

$\frac{\left(2+\sqrt{2}\right)\Gamma^{10}\left(\frac{1}{4}\right)}{1024\pi^{15/2}}-\frac{3\left(1+\sqrt{2}\right)\Gamma^{6}\left(\frac{1}{4}\right)}{128\pi^{11/2}}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{3}\text{csch}^{2}((2n+1)\pi).$

Relative to our proof of Theorem 2, we may similarly evaluate

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{s}\text{{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$
(30)

for odd natural numbers $s\in\mathbb{N}_{\geq 5}$. It would be desirable to
explicitly evaluate (30) for odd $s\in\mathbb{N}$. This is nontrivial, and may
require combinatorial formulas for higher derivatives for $\text{dc}(u,k)$
resulting from repeated applications of the differential equation system given
by (17)–(19). We leave this, together with some other problems considered in
Section 5, for a separate project.

###  2.2 A new sign function

Our identities for alternating sums involving factors of the form

$\text{csch}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right),$ (31)

as in Theorem 4 below, are of interest in part because these identities cannot
be derived from previously known results on sums involving expressions such as

$\text{csch}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right),$

such as the relation whereby

$\frac{kK}{\pi}=\sum_{n=0}^{\infty}(-1)^{n}\text{csch}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

given as (5.3.4.1) in [48] and employed in [64] via term-by-term
differentiation with respect to the elliptic modulus. To obtain new sums
involving (31), we are to interchange the limiting operations given by the
application of $\frac{d}{dk}$ and the application of
$\sum_{n=0}^{\infty}\cdot$ in the new result highlighted as Theorem 3 below.
To the best of our knowledge, hyperbolic sums involving sign functions such as
$n\mapsto(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}$ have not previously
appeared in the relevant literature on hyperbolic sums, which further
motivates the research interest in results as in Theorems 3 and 4 below.

######  Theorem 3.

The identity

$\frac{\left(\sqrt{1-k}+\sqrt{1+k}\right)K}{2\pi}-\frac{1}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}-1}$

holds if the above series converges.

###### Proof.

From the identity in (14), we find that the quotient

$\frac{\text{dn}(2Kw,k)}{\text{cn}(2Kw,k)}$

admits the same expansion as in (14). So, setting $w=\frac{1}{4}$ and applying
the half-$K$ formulas

$\text{cn}\left(\frac{1}{2}K,k\right)=\frac{\sqrt{2}\sqrt[4]{1-k^{2}}}{\sqrt{1+k}+\sqrt{1-k}}$
(32)

and

$\text{dn}\left(\frac{1}{2}K,k\right)=\sqrt[4]{1-k^{2}},$ (33)

we obtain the expansion

$\frac{\sqrt{1-k}+\sqrt{k+1}}{\sqrt{2}}=\frac{\pi}{2K}\left(\sqrt{2}+4\sum_{n=0}^{\infty}\frac{(-1)^{n}\cos\left(\frac{2n+1}{4}\pi\right)}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}-1}\right),$

which is equivalent to the desired result. ∎

######  Example 5.

For the elliptic singular value corresponding to the $r=1$ case in Theorem 3,
we obtain that

$\frac{\sqrt{2+\sqrt{2}}\Gamma^{2}\left(\frac{1}{4}\right)}{8\pi^{3/2}}-\frac{1}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\pi}-1}.$

This is equivalent to the $q$-digamma evaluation shown below:

$4\pi+\sqrt{\frac{2+\sqrt{2}}{\pi}}\Gamma^{2}\left(\frac{1}{4}\right)=-\psi_{e^{8\pi}}\left(\frac{1}{8}\right)-\psi_{e^{8\pi}}\left(\frac{3}{8}\right)+\psi_{e^{8\pi}}\left(\frac{5}{8}\right)+\psi_{e^{8\pi}}\left(\frac{7}{8}\right).$

######  Example 6.

Setting $k=k_{4}$ in Theorem 3, we may obtain that

$\frac{\sqrt{3+2\sqrt{2}+2\sqrt{4+3\sqrt{2}}}\Gamma^{2}\left(\frac{1}{4}\right)}{16\pi^{3/2}}-\frac{1}{2}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{2(2n+1)\pi}-1}.$

This is equivalent to the $q$-digamma evaluation shown below:

$\displaystyle
8\pi+\sqrt{\frac{3+2\sqrt{2}+2\sqrt{4+3\sqrt{2}}}{\pi}}\Gamma^{2}\left(\frac{1}{4}\right)=$
$\displaystyle-\psi_{e^{16\pi}}\left(\frac{1}{8}\right)-\psi_{e^{16\pi}}\left(\frac{3}{8}\right)+\psi_{e^{16\pi}}\left(\frac{5}{8}\right)+\psi_{e^{16\pi}}\left(\frac{7}{8}\right).$

######  Theorem 4.

The identity

$\displaystyle\frac{2k\left(k^{2}-1\right)K^{2}\left(\frac{\left(\sqrt{1-k}+\sqrt{1+k}\right)\left(E+\left(k^{2}-1\right)K\right)}{k\left(k^{2}-1\right)}-\frac{1}{2}\left(\frac{1}{\sqrt{1+k}}-\frac{1}{\sqrt{1-k}}\right)K\right)}{\pi^{2}(E^{\prime}K+(E-K)K^{\prime})}$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)\text{\emph{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

holds for suitably bounded $k$.

###### Proof.

This follows in a direct way by differentiating both sides of the identity in
Theorem 3 with respect to the elliptic modulus, and then reversing the order
of differentiation and infinite summation, and then applying much
simplification. ∎

Our series as in Example 8 are motivated by series evaluations of a similar
appearance recorded in [7]. For example, the series evaluation

$\sum_{n=1}^{\infty}(-1)^{n+1}n\text{csch}(\pi n)=\frac{1}{4\pi}$

recorded [7] was, as noted in [7], previously proved by many difference
authors in [19, 35, 44, 45, 52, 54, 67]. To obtain new closed forms from
Theorem 4, we are to make use of the elliptic alpha function [14, §5]

######  Example 7.

Using the elliptic singular value $k_{1}$, a special case of Theorem 4 gives
us that

$\frac{\sqrt{2+\sqrt{2}}\Gamma^{2}\left(\frac{1}{4}\right)}{4\pi^{5/2}}-\frac{\sqrt{2-\sqrt{2}}\Gamma^{6}\left(\frac{1}{4}\right)}{64\pi^{9/2}}=\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)\text{csch}^{2}\left(\frac{2n+1}{2}\pi\right).$

######  Example 8.

Using the elliptic singular value $k_{4}$, a special case of Theorem 4 gives
us that

$\displaystyle\frac{\left(\sqrt{1+\sqrt{2}}+\sqrt{2+\sqrt{2}}\right)\Gamma^{2}\left(\frac{1}{4}\right)}{16\pi^{5/2}}-\frac{\sqrt{4+\sqrt{2}+2^{7/4}}\Gamma^{6}\left(\frac{1}{4}\right)}{256\pi^{9/2}}=$
$\displaystyle\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)\text{csch}^{2}((2n+1)\pi).$

As in [13], we record that Nanjundiah’s formula

$\sum_{n=1}^{\infty}\text{csch}^{2}(n\pi)=\frac{1}{6}-\frac{1}{2\pi}$

has, subsequent to Nanjundiah 1951 proof [44], been proved by many different
authors, including Berndt [9], Ling [40], Kiyek and Schmidt [33], Muckenhoupt
[43], and Shafer [55]. Nanjundiah’s formula, together with our new sums
involving $\text{csch}^{2}$ given in Examples 7 and 8, inspire us to further
apply our method in the evaluation of sums involving $\text{csch}^{2}$.

###  2.3 Higher powers of $2n+1$

We are to again make use of the equality of (20) and (21), to prove the
following companion to Theorem 1.

######  Theorem 5.

The equality

$\frac{3}{2}+\frac{4\left(k^{2}\left(3+k^{\prime}\right)-3\left(1+k^{\prime}\right)\right)K^{3}}{\left(\sqrt{1-k}+\sqrt{1+k}\right)\pi^{3}}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{2}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}-1}$

holds if the above series converges.

###### Proof.

Starting with the equality of (20) and (21), we again set $u\mapsto 2wK$.
Using the half-$K$ formulas in (32) and (33), together with the half-$K$
formula

$\text{sn}\left(\frac{K}{2},k\right)=\frac{\sqrt{2}}{\sqrt{1+k}+\sqrt{1-k}},$

we may obtain that

$\displaystyle\frac{\sqrt{2}\left(k^{\prime}+2\right)\left(k^{\prime}+1-k^{2}\right)}{\sqrt{1-k}+\sqrt{k+1}}=\frac{3\pi^{3}}{4\sqrt{2}K^{3}}-\frac{\pi^{3}}{2\sqrt{2}K^{3}}\sum_{n=0}^{\infty}\frac{(2n+1)^{2}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}-1},$

and this is equivalent to the desired result. ∎

######  Example 9.

Theorem 5, together with the elliptic integral singular value $k_{1}$, give us
that

$\frac{3}{2}-\frac{\sqrt{26+17\sqrt{2}}\Gamma^{6}\left(\frac{1}{4}\right)}{64\pi^{9/2}}=\sum_{n=0}^{\infty}\frac{(2n+1)^{2}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\pi}-1}.$

######  Example 10.

Theorem 5, together with the elliptic integral singular value $k_{4}$, give us
that

$\frac{3}{2}-\frac{\sqrt{54+37\sqrt{2}+4\sqrt{352+249\sqrt{2}}}\Gamma^{6}\left(\frac{1}{4}\right)}{128\pi^{9/2}}=\sum_{n=0}^{\infty}\frac{(2n+1)^{2}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{2(2n+1)\pi}-1}.$

As below, Theorem 5 may be used to evaluate the $s=3$ case for the sum
obtained from Zucker’s sum in (10) by replacing $(-1)^{n}$ with
$(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}$ and by replacing csch with
$\text{csch}^{2}$.

######  Theorem 6.

The series

$\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{3}\text{\emph{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

may be evaluated as

$\displaystyle\Big{(}4k^{\prime}K^{4}\big{(}6\big{(}k\big{(}k\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}-3\sqrt{1-k}+3\sqrt{1+k}\big{)}-6\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}\big{)}E+$
$\displaystyle\big{(}36\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}+k\big{(}18\big{(}\sqrt{1-k}-\sqrt{1+k}\big{)}+k\big{(}k\big{(}k\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}-8\sqrt{1-k}+$
$\displaystyle
8\sqrt{1+k}\big{)}-23\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}\big{)}\big{)}\big{)}K\big{)}\Big{)}\Big{/}\left(\pi^{4}\big{(}k^{\prime}+1\big{)}(K^{\prime}(E-K)+E^{\prime}K)\right),$

if the above series is convergent.

###### Proof.

This may be proved by applying term-by-term differentiation, with respect to
the elliptic modulus, in Theorem 5. ∎

######  Example 11.

Setting $k=k_{1}$, we obtain that

$\frac{\sqrt{218+151\sqrt{2}}\Gamma^{10}\left(\frac{1}{4}\right)}{512\pi^{15/2}}-\frac{3\sqrt{26+17\sqrt{2}}\Gamma^{6}\left(\frac{1}{4}\right)}{32\pi^{11/2}}=\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{3}\text{csch}^{2}\left(\frac{2n+1}{2}\pi\right).$

######  Example 12.

Setting $k=k_{4}$, we obtain that

$\frac{\sqrt{402+287\sqrt{2}+4\sqrt{20296+14358\sqrt{2}}}\Gamma^{10}\left(\frac{1}{4}\right)}{2048\pi^{15/2}}-\frac{3\sqrt{54+37\sqrt{2}+4\sqrt{352+249\sqrt{2}}}\Gamma^{6}\left(\frac{1}{4}\right)}{128\pi^{11/2}}$

equals

$\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{3}\text{csch}^{2}((2n+1)\pi).$

We may mimic the above proof to evaluate

$\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{s}\text{{csch}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

for odd $s$.

##  3 Applications of the Jacobi elliptic function nc

The results introduced in this section on symbolic forms for summations
involving the sech function are inspired by the formula

$\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{4\pi^{3/2}}-\frac{1}{2}=\sum_{n=1}^{\infty}\text{sech}(\pi
n)$

due to Ramanujan, which is highlighted as an especially amazing formula in the
Wolfram Mathworld entry on the hyperbolic secant [60]. A Ramanujan summation
involving sech more closely related to our work is Ramanujan’s formula (cf.
[9])

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{4m-1}\text{sech}\left(\frac{2n+1}{2}\pi\right)=0.$
(34)

Our applications of identities as in (15) are also inspired by well known
Ramanujan formulas such as the following [11, §14]:

$\sum_{n=1}^{\infty}\frac{n}{e^{2\pi n}-1}=\frac{1}{24}-\frac{1}{8\pi}.$

We begin with the following Fourier series expansion [61, pp. 511–512]:

$\text{nc}(u,k)=\frac{\pi}{2Kk^{\prime}}\sec
v-\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{2n+1}}{1+q^{2n+1}}\cos\left((2n+1)v\right).$

Following the steps given in Section 1.2, we obtain that

$\text{nc}(2wK,k)=\frac{\pi}{2k^{\prime}K}\sec(\pi
w)-\frac{2\pi}{k^{\prime}K}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}+1}\cos((2n+1)\pi
w).$

Setting $w=0$, we can show that the left-hand side of the above equality
equals $1$. This gives us an equivalent formulation of the following result:

$\frac{1}{4}-\frac{\sqrt{1-k^{2}}K}{2\pi}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}+1}.$
(35)

An equivalent result is given as Theorem 6 by Bagis in [3]. From this
previously known formula, by differentiating with respect to the elliptic
modulus, we may obtain the identity

$\frac{2k^{\prime}K^{2}(K-E)}{\pi^{2}(K^{\prime}(E-K)+E^{\prime}K)}=\sum_{n=0}^{\infty}(-1)^{n}(2n+1)\text{{sech}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right).$
(36)

By direct analogy with the material in Section 2.1, we may generalize (36) so
as to evaluate

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{s}\text{{sech}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$
(37)

for odd $s$. For the sake of brevity, we omit a detailed examination of this,
and we leave it do a separate project to obtain an explicit combinatorial
formula for (37) based on repeated applications of the differential equations
among (17)–(19).

A direct application of the formula in (35) that we have proved is given by
how this formula allows us to prove a stronger version of the formula

$2\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\pi}+1}+\sum_{n=0}^{\infty}\text{sech}\left(\frac{2n+1}{2}\pi\right)=\frac{1}{2}$
(38)

highlighted as Corollary 4.22 in [7]. This was proved using Euler polynomials
in [7].

######  Theorem 7.

The following stronger version of Berndt’s identity in (38) holds:

$\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{4\sqrt{2}\pi^{3/2}}=\sum_{n=0}^{\infty}\text{\emph{sech}}\left(\frac{2n+1}{2}\pi\right).$

###### Proof.

Using the value for $k_{1}$ in (24) together with the value for $K(k_{1})$
shown in (25), the identity in (35) gives us that

$\frac{1}{4}-\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{8\sqrt{2}\pi^{3/2}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{e^{(2n+1)\pi}+1},$

so that the desired result then follows from (38). ∎

Our strengthening of Corollary 4.22 in [7], as in Theorem 7, has not appeared
in past publications influenced by [7], which have been based in areas such as
number theory [2, 21, 31, 34, 41, 42, 46, 47, 57, 65] mathematical physics
[58], high energy physics [29], and nuclear physics [22]. The many areas in
mathematics and physics related to the discoveries on hyperbolic infinite sums
given by Berndt [7] inspire the development of further results as in Theorem
7. Since Berndt’s formula in (38) is highlighted as a Corollary in [7], we
find it appropriate to highlight the following equivalent formulation of
Theorem 7 as a Corollary.

######  Corollary 1.

The $q$-digamma evaluation

$\psi_{e^{\pi}}\left(\frac{1-i}{2}\right)-\psi_{e^{\pi}}\left(\frac{1+i}{2}\right)=-i\pi-\frac{i\Gamma^{2}\left(\frac{1}{4}\right)}{4\sqrt{2\pi}}$

holds true.

###### Proof.

This is almost immediately equivalent to Theorem 7, from the definition in
(7). ∎

Many of our results, with a particular regard toward Theorem 7, are of a
similar appearance relative to the formula

$\sum_{n=0}^{\infty}\text{sech}^{2}\left(\frac{2n+1}{2}\pi\right)=\frac{1}{\pi}$

recorded in [7]. As in [7], we record that the above formula has been proved
by many different authors [33, 40, 44, 67].

######  Theorem 8.

The identity

$\frac{1}{2}-\frac{\sqrt{k^{\prime}}\left(\sqrt{1-k}+\sqrt{1+k}\right)K}{2\pi}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}+1}$

holds if the above series converges.

###### Proof.

Setting $w=\frac{1}{4}$ in the above series expansion for $\text{nc}(2wK,k)$,
we find that $\frac{1}{\text{cn}\left(\frac{k}{2},k\right)}$ admits the
following expansion:

$\frac{\sqrt{2}\pi}{\sqrt{1-k^{2}}K}\left(\frac{1}{2}-\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\frac{K^{\prime}}{K}\pi}+1}\right).$

According to the half-$K$ identity shown in (32), we find that

$\frac{\sqrt{1-k}+\sqrt{1+k}}{\sqrt{2}\sqrt{k^{\prime}}}$

admits the same expansion, and this gives us an equivalent version of the
desired result. ∎

######  Example 13.

The symbolic form

$\frac{1}{2}-\frac{\sqrt{1+\sqrt{2}}\Gamma^{2}\left(\frac{1}{4}\right)}{8\pi^{3/2}}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{(2n+1)\pi}+1}$

holds and may be proved using Theorem 8 together with the elliptic singular
value indicated in (25). This is equivalent to the $q$-digamma evaluation

$-4\pi-\sqrt{\frac{1+\sqrt{2}}{\pi}}\Gamma^{2}\left(\frac{1}{4}\right)=\psi_{e^{8\pi}}\left(\frac{1-i}{8}\right)+\psi_{e^{8\pi}}\left(\frac{3-i}{8}\right)-\psi_{e^{8\pi}}\left(\frac{5-i}{8}\right)-\psi_{e^{8\pi}}\left(\frac{7-i}{8}\right).$

Ramanujan’s closed forms for

$\sum_{n=0}^{\infty}(-1)^{n}(2n+1)^{-4t-1}\text{sech}\left(\frac{2n+1}{2}\pi\right)$
(39)

have been considered, as in [10], as especially notable contributions out of
the results given in Ramanujan’s notebooks. This further motivates our
generalizations of or related to Ramanujan’s sech-sums as in (8) and (39).

######  Example 14.

The symbolic form

$\frac{1}{2}-\frac{\sqrt{\frac{1}{2}\left(2\sqrt{2}+\sqrt{4+3\sqrt{2}}\right)}\Gamma^{2}\left(\frac{1}{4}\right)}{8\pi^{3/2}}=\sum_{n=0}^{\infty}\frac{(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}}{e^{2(2n+1)\pi}+1}$

holds and may be proved using Theorem 8 together with the elliptic singular
value indicated in (27). We may rewrite this a $q$-digamma evaluation in much
the same way as before.

Ramanujan’s many evaluations for infinite hyperbolic sums served as a main
source of motivation behind Xu’s work in [63]. In particular, Ramanujan’s
evaluations whereby

$\sum_{n=0}^{\infty}(2n+1)^{2}\text{sech}\left(\frac{2n+1}{2}\pi\right)=\frac{\Gamma^{6}\left(\frac{1}{4}\right)}{16\sqrt{2}\pi^{9/2}}$

and

$\sum_{n=0}^{\infty}(2n+1)^{2}\text{sech}^{2}\left(\frac{2n+1}{2}\pi\right)=\frac{\Gamma^{8}\left(\frac{1}{4}\right)}{192\pi^{6}}$

were highlighted as a main sources of motivation concerning the results
introduced by Xu in [63].

By applying term-by-term differentiation with respect to the elliptic modulus
in Theorem 8, we can show that the expression

$\displaystyle\big{(}K^{2}\sqrt{k^{\prime}}\big{(}\big{(}k\big{(}\sqrt{1-k}-\sqrt{1+k}\big{)}+2\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}\big{)}K-2\big{(}\sqrt{1-k}+\sqrt{1+k}\big{)}E\big{)}\big{)}\big{/}$
$\displaystyle\big{(}\pi^{2}(K^{\prime}(E-K)+E^{\prime}K)\big{)}$

equals

$\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)\text{{sech}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

if the above series converges. By direct analogy with the material in Section
2.3, we can mimic our above derivation so as to evaluate

$\sum_{n=0}^{\infty}(-1)^{\left\lfloor\frac{n}{2}\right\rfloor}(2n+1)^{s}\text{{sech}}^{2}\left(\frac{2n+1}{2}\frac{K^{\prime}}{K}\pi\right)$

for odd $s$.

##  4 Powers of Jacobi elliptic functions

Using (12) together with the technique in Section 1.2, we can prove that

$-\frac{K^{2}\left(2EKk^{\prime}+E^{2}+K^{2}\left(k^{2}(k^{\prime}+1)-2k^{\prime}-1\right)\right)}{4\pi^{3}(K^{\prime}(E-K)+E^{\prime}K)}$

equals

$\sum_{n=1}^{\infty}(-1)^{n}n^{2}\text{{csch}}^{2}\left(\frac{2n\pi
K^{\prime}}{K}\right)$

if the above series converges. For example, this gives us that

$-\frac{1}{32\pi^{2}}-\frac{\left(1+\sqrt{2}\right)\Gamma^{4}\left(\frac{1}{4}\right)}{128\pi^{4}}+\frac{\left(1+\sqrt{2}\right)\Gamma^{8}\left(\frac{1}{4}\right)}{2048\pi^{6}}=\sum_{n=1}^{\infty}(-1)^{n}n^{2}\text{csch}^{2}(2n\pi).$

We leave it to a separate project to generalize this result and to investigate
the use of our method together with Fourier series expansions apart from (12)
that involve powers of Jacobi elliptic functions.

##  5 Further considerations

The computation of the inverse functions among (2)–(4) often turns out to be
difficult, even with state-of-the-art Computer Algebra Systems [6]. For
example, there is no direct way of using built-in Mathematica commands such as
JacobiSN, JacobiCN, or JacobiDN to compute the “actual” Jacobi elliptic
functions defined via (2)–(4) relative to the Mathematica definition for the
incomplete elliptic $F$-integral

$\int_{0}^{\phi}\frac{dt}{\sqrt{1-k\sin^{2}t}}$

with arguments $\phi$ and $k$, which is in contrast to (1). This kind of
practical computational problem motivates the development of new and efficient
ways of expressing and applying the Jacobi elliptic functions in the form of
series expansions involved in this article.

Ramanujan’s identity in (9) was applied by Berndt in [8] to prove the
following remarkable closed form:

$\int_{-\infty}^{\infty}\frac{dx}{\cos\sqrt{x}+\cosh\sqrt{x}}=\frac{\pi}{4}\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{\Gamma^{2}\left(\frac{3}{4}\right)}.$

How can we obtain similar results using our evaluations related to Ramanujan’s
formula in (9), as in Section 3? We leave this for a separate project. Also,
we leave it to a separate project to apply our technique using identities such
as the third-$K$ formula, as opposed to the half-$K$ formula we have applied.

### Acknowledgements

The author wants to express his sincere thanks to Alexey Kuznetsov for many
useful discussions.

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John M. Campbell

Department of Mathematics

Toronto Metropolitan University

<EMAIL_ADDRESS>