# Generalized Cosecant Numbers and the Hurwitz Zeta Function

Victor Kowalenko  Department of Mathematics and Statistics, The University of
Melbourne, Victoria 3010, Australia<EMAIL_ADDRESS>
###### Abstract.

This paper presents recent developments concerning the generalized cosecant
numbers $c_{\rho,k}$, which emerge as the coefficients of the power series
expansion for the important fundamental function $z^{\rho}/\sin^{\rho}z$.
These coefficients can be computed for all, including complex, values of
$\rho$ by using the relatively novel graphical method known as the partition
method for a power series expansion or by using intrinsic routines in
Mathematica. In fact, they represent polynomials in $\rho$ of degree $k$,
where $k$ is the power of $z$. In addition, though related to the Bernoulli
numbers, they possess more properties and do not diverge like the former. The
partition method for a power series expansion has the advantage that it yields
the $k$-behaviour of the highest order coefficients. Thus, general formulas
for such coefficients are derived by considering the properties of the highest
part partitions. It is then shown how the generalized cosecant numbers are
related to the specific symmetric polynomials that arise from summing over
quadratic powers of integers. Consequently, integral values of the Hurwitz
zeta function for even powers are expressed for the first time ever in terms
of ratios of the generalized cosecant numbers.

###### Key words and phrases:

coefficient, generalized cosecant number, multiplicity, partition, Pochhammmer
notation, symmetric polynomial, zeta function

###### 2010 Mathematics Subject Classification:

11M06,11M35,11P83,11Y55,11Z05

##  1\. Introduction

The cosecant numbers, $c_{k}$, are defined in Ref. [1] as the rational
coefficients of the power series expansion for cosecant, in particular by

$\displaystyle\csc z=\frac{1}{\sin
z}\equiv\sum_{k=0}^{\infty}c_{k}\,z^{2k-1}\;.$ (1)

Via the discrete graphical method known as partition method for a power series
expansion a general formula for them is derived in terms of all the integer
partitions summing to $k$, which is

$c_{k}=(-1)^{k}\sum_{\begin{subarray}{c}\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{k}\\\
\sum_{i=1}^{k}i\,\lambda_{i}=k\end{subarray}}^{k,\lfloor k/2\rfloor,\lfloor
k/3\rfloor,\dots,1}(-1)^{N_{k}}N_{k}!\prod_{i=1}^{k}\Bigl{(}\frac{1}{(2i+1)!}\Bigr{)}^{\lambda_{i}}\frac{1}{\lambda_{i}!}\,.$
(2)

In this result $\lambda_{i}$ represents the multiplicity or the number of
occurrences of each part $i$ in the partitions, while the sum of the
multiplicities or length of the partition is represented by $N_{k}$, i.e.
$N_{k}=\sum_{i=1}^{k}\lambda_{i}$. For the partitions summing to $k$, the
multiplicity of a part $i$ ranges from zero to $\lfloor k/i\rfloor$, where
$\lfloor x\rfloor$ denotes the floor function or the greatest integer less
than or equal to $x$. Generally, when $k$ is large, most of the multiplicities
for a partition vanish as we shall see shortly. Using (2) one finds that
$c_{0}=1$, $c_{1}=1/6$, $c_{2}=7/360$, $c_{4}=31/3\cdot 7!$ and so on. In
addition, the reader should observe that an equivalence symbol has been
introduced into (1) because the power series expansion on the rhs is divergent
for $|z|\geq\pi$, while the lhs is always defined. For these values the rhs
must be regularized in the manner described in Refs. [4]-[9]. Nevertheless,
since the power series is convergent for the other values of $z$, it is
permissible to replace the equivalence symbol by an equals sign. Then one
obtains the standard power series for $x\csc x$, which is given as No.
1.411(11) in Ref. [3]. Consequently, the cosecant numbers can be expressed in
terms of the Bernoulli numbers as

$\displaystyle c_{k}=\frac{(-1)^{k+1}}{(2k)!}\,(2^{2k}-2)\,B_{2k}\;\;.$ (3)

As discussed in Ref. [1], the cosecant numbers possess far more properties
than their more famous counterparts. They have also the major advantage that
they do not diverge like them. In fact, in most situations such as LABEL:tw0a
and the Euler Maclaurin summation formula, the divergence of the Bernoulli
numbers is tamed by the factor of $(2k)!$ in the denominator. Thus, one is
often required to divide two very large numbers by each other, which is
avoided when using the cosecant numbers. Hence the above equation can be
regarded as awkward or even clumsy. Nevertheless, for the interested reader it
should be mentioned that Ref. [8] shows how the Bernoulli numbers and
polynomials can be evaluated by using the partition method for a power series
expansion.

A better method of expressing the cosecant numbers is in terms of the Riemann
zeta function. By using No. 9.616 in Ref. [3], we find that

$\displaystyle
c_{k}=2\bigl{(}1-2^{1-2k}\bigr{)}\,\frac{\zeta(2k)}{\pi^{2k}}\;,$ (4)

where $\zeta(2k)$ represents the Riemann zeta function. Thus, we observe that
the $c_{k}\approx 2\pi^{-2k}$ or $\pi^{2k}\approx 2/c_{k}$ for $k\gg 1$,
although this approximation gives the misleading impression that the $c_{k}$
are irrational. Consequently, (2) becomes another method of determining even
integer values of the Riemann zeta function.

Ref. [1] not only presents numerous applications of cosecant numbers, but also
demonstrates how they are related to other numbers such as the secant numbers
and, more importantly, how the sets of the resulting numbers can be
generalized or extended by introducing an arbitrary power $\rho$ to their
generating function. Specifically, the generalized cosecant numbers are given
by

$\displaystyle\csc^{\rho}z=\frac{1}{\sin^{\rho}z}\equiv\sum_{k=0}^{\infty}c_{\rho,k}\,z^{2k-\rho}\;,$
(5)

where

$c_{\rho,k}=(-1)^{k}\sum_{\begin{subarray}{c}\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{k}\\\
\sum_{i=1}^{k}i\,\lambda_{i}=k\end{subarray}}^{k,\lfloor k/2\rfloor,\lfloor
k/3\rfloor,\dots,1}(-1)^{N_{k}}(\rho)_{N_{k}}\prod_{i=1}^{k}\Bigl{(}\frac{1}{(2i+1)!}\Bigr{)}^{\lambda_{i}}\frac{1}{\lambda_{i}!}\,.$
(6)

In (6) $(\rho)_{N_{k}}$ denotes the Pochhammer notation for
$\Gamma(\rho+N_{k})/\Gamma(\rho)$, where $\Gamma(x)$ represents the gamma
function.

##  2\. Computation

To calculate the generalized cosecant numbers via (6), we need to determine
the specific contribution made by each integer partition that sums to $k$. For
example, if we wish to evaluate $c_{\rho,6}$, then we require all the
contributions made from the partitions summing to 6, which appear in the first
column of Table 1. Each part in a partition is assigned a specific value,
which depends on the function being studied. In the case of
$x^{\rho}/\sin^{\rho}x$, the part $i$ is assigned a value of
$(-1)^{i+1}/(2i+1)!$. In addition, since each part occurs $\lambda_{i}$ times
in a partition, we need to multiply $\lambda_{i}$ values or calculate
$(-1)^{(i+1)\lambda_{i}}/((2i+1)!)^{\lambda_{i}}$. The second column in Table
1 displays the multiplicities of the parts in all the partitions summing to 6,
while the third column presents the length $N_{k}$ for each partition. Thus,
we see that most of the multiplicities vanish as stated earlier.

Partition | $\lambda_{1}$ | $\lambda_{2}$ | $\lambda_{3}$ | $\lambda_{4}$ | $\lambda_{5}$ | $\lambda_{6}$ | $N_{k}$
---|---|---|---|---|---|---|---
$\\{6\\}$ |  |  |  |  |  | 1 | 1
$\\{5,1\\}$ | 1 |  |  |  | 1 |  | 2
$\\{4,2\\}$ |  | 1 |  | 1 |  |  | 2
$\\{4,1,1\\}$ | 2 |  |  | 1 |  |  | 3
$\\{3,3\\}$ |  |  | 2 |  |  |  | 2
$\\{3,2,1\\}$ | 1 | 1 | 1 |  |  |  | 3
$\\{3,1,1,1\\}$ | 3 |  | 1 |  |  |  | 4
$\\{2,2,2\\}$ |  | 3 |  |  |  |  | 3
$\\{2,2,1,1\\}$ | 2 | 2 |  |  |  |  | 4
$\\{2,1,1,1,1\\}$ | 4 | 1 |  |  |  |  | 5
$\\{1,1,1,1,1,1\\}$ | 6 |  |  |  |  |  | 6
Table 1. Multiplicities of the partitions summing to 6

Associated with each partition is a multinomial factor that is determined by
taking the factorial of $N_{k}$ and dividing by the factorials of all the
multiplicities. E.g., for the partition $\\{2,1,1,1,1\\}$ in Table 1, we have
$\lambda_{1}=4$ and $\lambda_{2}=1$, while the other multiplicities vanish.
Hence, the multinomial factor becomes $5!/(4!\,1!)=5$. When the function is
accompanied by an arbitrary power, say $\rho$, a further modification must be
made. Each partition is then multiplied by the Pochhammer factor of
$\Gamma(N_{k}+\rho)/\Gamma(\rho)$ divided by $N_{k}!$. That is, for each
partition we must include the extra factor of $(\rho)_{N_{k}}/N_{k}!$. For
$\rho=1$, this simply yields unity and thus, the multinomial factor remains
unaffected. Consequently, (6) reduces to (2) for $\rho=1$. For $\rho=2$ we
obtain the cosecant-squared numbers as given in Theorem 5 of Ref. [1], which
are given by

$\displaystyle c_{2,k}$
$\displaystyle=(-1)^{k}\sum_{\begin{subarray}{c}\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{k}\\\
\sum_{i=1}^{k}i\,\lambda_{i}=k\end{subarray}}^{k,\lfloor k/2\rfloor,\lfloor
k/3\rfloor,\dots,1}(-1)^{N_{k}}(2)_{N_{k}}\prod_{i=1}^{k}\Bigl{(}\frac{1}{(2i+1)!}\Bigr{)}^{\lambda_{i}}\frac{1}{\lambda_{i}!}$
$\displaystyle=\;\;\frac{(2k-1)}{\bigl{(}1-2^{1-2k}\bigr{)}}\;c_{k}\;\;.$ (7)

In the above result one can replace $(2)_{N_{k}}$ by $(N_{k}+1)(1)_{N_{k}}$.
Hence we obtain (2) again except $N_{k}!$ is now multiplied by $N_{k}+1$. On
the other hand, if $\rho=-1$, then the coefficients are simply equal to the
power series expansion for $\sin z$ divided by $z$. Therefore, we arrive at

$\sum_{\begin{subarray}{c}\lambda_{1},\lambda_{2},\lambda_{3},\dots,\lambda_{k}\\\
\sum_{i=1}^{k}i\,\lambda_{i}=k\end{subarray}}^{k,\lfloor k/2\rfloor,\lfloor
k/3\rfloor,\dots,1}(-1)^{N_{k}}(\rho)_{N_{k}}\prod_{i=1}^{k}\Bigl{(}\frac{1}{(2i+1)!}\Bigr{)}^{\lambda_{i}}\frac{1}{\lambda_{i}!}=\frac{1}{(2k+1)!}\,.$
(8)

Hence, we have an expression for the reciprocal of $(2k+1)!$ in terms of a sum
over partitions summing to $k$.

If we examine (6) more closely, then we see that the product deals with
calculating the contribution made by each partition based on the values of the
multiplicities, while the sum refers to all partitions summing to $k$. Hence
the sum covers the range of values for each multiplicity. For example,
$\lambda_{1}$ attains a maximum value of $k$, which corresponds to the
partition with $k$ ones, while $\lambda_{2}$ attains a maximum value of
$[k/2]$, which corresponds to the partition with $[k/2]$ twos in it. For odd
values of $k$, the partition with $[k/2]$ twos also possesses a one, i.e.
$\lambda_{1}=1$. Thus, it can be seen that the maximum value of $\lambda_{i}$
is always $[k/i]$, which becomes the upper limit for each multiplicity in both
(2) and (6). Moreover, each partition in the sums must satisfy the constraint,
$\sum_{i=1}^{k}i\lambda_{i}=k$.

As an example, let us calculate $c_{\rho,6}$. According to Table 1 there are
eleven partitions summing to 6. Therefore, we need to determine eleven
contributions in the sum over the partitions. By applying the steps mentioned
above to (6), we find that the contributions from the partitions in the same
order as the table are

$\displaystyle c_{\rho,6}$
$\displaystyle=-(\rho)_{1}\;\frac{1}{13!}+\frac{(\rho)_{2}}{2!}\;\frac{2!}{1!\cdot
1!}\;\frac{1}{3!\cdot 11!}+\frac{(\rho)_{2}}{2!}\;\frac{2!}{1!\cdot
1!}\;\frac{1}{5!\cdot 9!}-\frac{(\rho)_{3}}{3!}\;\frac{3!}{1!\cdot
2!}\;\frac{1}{3!^{2}\cdot 9!}$
$\displaystyle+\;\;\frac{(\rho)_{2}}{2!}\;\frac{2!}{2!}\;\frac{1}{(7!)^{2}}-\frac{(\rho)_{3}}{3!}\;\frac{3!}{1!\cdot
1!\cdot 1!}\;\frac{1}{3!\cdot 5!\cdot
7!}+\frac{(\rho)_{4}}{4!}\,\frac{4!}{1!\cdot 3!}\;\frac{1}{(3!)^{3}\cdot
7!}-\frac{(\rho)_{3}}{3!}\;\frac{3!}{3!}\;\frac{1}{(5!)^{3}}$
$\displaystyle+\;\;\frac{(\rho)_{4}}{4!}\,\frac{4!}{2!\cdot
2!}\;\frac{1}{(3!)^{2}\cdot(5!)^{2}}-\frac{(\rho)_{5}}{5!}\,\frac{5!}{4!\cdot
1!}\;\frac{1}{(3!)^{4}\cdot
5!}+\frac{(\rho)_{6}}{6!}\,\frac{6!}{6!}\;\frac{1}{(3!)^{6}}\;\;.$ (9)

The interesting property of the above result is that when the length $N_{k}$
appearing in the Pochhammer terms is even, the contribution from the partition
is positive while if it is odd, then the contribution is negative. This
behaviour applies to all even values of $k$. On the other hand, if $k$ is odd,
then the contributions with an odd number of parts are positive, while those
from an even number of parts are negative. Furthermore, by introducing (9)
into Mathematica [10] and wrapping it entirely around the combination of the
Simplify and Expand routines, one obtains

$\displaystyle c_{\rho,6}$
$\displaystyle=\frac{1}{5884534656000}\,\Bigl{(}1061376\rho+3327584\rho^{2}+4252248\rho^{3}+2862860\rho^{4}$
$\displaystyle+\;\;1051050\rho^{5}+175175\rho^{6}\Bigr{)}\;.$ (10)

Since the denominator equals $2/(9\cdot 15!)$, we arrive at the $k=6$ result
in Table 2.

$k$ | $c_{\rho,k}$
---|---
$0$ | $1$
$1$ | $\frac{1}{3!}\;\rho$
$2$ | $\frac{2}{6!}\;\bigl{(}2\rho+5\rho^{2}\bigr{)}$
$3$ | $\frac{8}{9!}\;\bigl{(}16\rho+42\rho^{2}+35\rho^{3}\bigr{)}$
$4$ | $\frac{2}{3\cdot 10!}\;\bigl{(}144\rho+404\rho^{2}+420\rho^{3}+175\rho^{4}\bigr{)}$
$5$ | $\frac{4}{3\cdot 12!}\;\bigl{(}768\rho+2288\rho^{2}+2684\rho^{3}+1540\rho^{4}+385\rho^{5}\bigr{)}$
$6$ | $\frac{2}{9\cdot 15!}\;\bigl{(}1061376\rho+3327594\rho^{2}+4252248\rho^{3}+2862860\rho^{4}+1051050\rho^{5}+175175\rho^{6}\bigr{)}$
$7$ | $\frac{1}{27\cdot 15!}\;\bigl{(}552960\rho+1810176\rho^{2}+2471456\rho^{3}+1849848\rho^{4}+820820\rho^{5}+210210\rho^{6}$
| $+25025\rho^{7}\bigr{)}$
$8$ | $\frac{2}{45\cdot 18!}\bigl{(}200005632\rho+679395072\rho^{2}+978649472\rho^{3}+792548432\rho^{4}+397517120\rho^{5}$
| $+125925800\rho^{6}+23823800\rho^{7}+2127125\rho^{8}\bigr{)}$
$9$ | $\frac{4}{81\cdot 21!}\bigl{(}129369047040\rho+453757851648\rho^{2}+683526873856\rho^{3}+589153364352\rho^{4}$
|
$+323159810064\rho^{5}+117327450240\rho^{6}+27973905960\rho^{7}+4073869800\rho^{8}+282907625\rho^{9}\bigr{)}$
$10$ | $\frac{2}{6075\cdot 22!}\bigl{(}38930128699392\rho+140441050828800\rho^{2}+219792161825280\rho^{3}+199416835425280\rho^{4}$
|
$+117302530691808\rho^{5}+47005085727600\rho^{6}+12995644662000\rho^{7}+2422012593000\rho^{8}$
| $+280078548750\rho^{9}+15559919375\rho^{10}\bigr{)}$
$11$ | $\frac{8}{243\cdot 25!}\bigl{(}494848416153600\rho+1830317979303936\rho^{2}+2961137042841600\rho^{3}$
|
$+2805729689044480\rho^{4}+1747214980192000\rho^{5}+755817391389984\rho^{6}+232489541684400\rho^{7}$
|
$+50749166067600\rho^{8}+7607466867000\rho^{9}+715756291250\rho^{10}+32534376875\rho^{11}\bigr{)}$
$12$ | $\frac{2}{2835\cdot 27!}\bigl{(}1505662706987827200\rho+5695207005856038912\rho^{2}+9487372599204065280\rho^{3}$
|
$+9332354263294766080\rho^{4}+6096633539052376320\rho^{5}+2806128331871953088\rho^{6}$
|
$+937291839756592320\rho^{7}+229239926321406000\rho^{8}+40598842049766000\rho^{9}$
|
$+5005999501002500\rho^{10}+390802935022500\rho^{11}+14803141478125\rho^{12}\bigr{)}$
$13$ | $\frac{232}{81\cdot 30!}\bigl{(}844922884529848320\rho+3261358271400247296\rho^{2}+5576528334428209152\rho^{3}$
|
$+5668465199488266240\rho^{4}+3858582205451484160\rho^{5}+1870620248833400064\rho^{6}$
|
$+667822651436228288\rho^{7}+178292330746770240\rho^{8}+35600276746834800\rho^{9}$
|
$+5225593531158000\rho^{10}+539680243602500\rho^{11}+35527539547500\rho^{12}+1138703190625\rho^{13}\bigr{)}$
$14$ | $\frac{2}{1215\cdot 30!}\;\bigl{(}138319015041155727360\rho+543855095595477762048\rho^{2}+952027796641042464768\rho^{3}$
|
$+996352286992030556160\rho^{4}+703040965960031795200\rho^{5}+356312537387839432192\rho^{6}$
|
$+134466795172062184832\rho^{7}+38526945410311117760\rho^{8}+8436987713444690400\rho^{9}$
|
$+1404048942958662000\rho^{10}+173777038440005000\rho^{11}+15258232341852500\rho^{12}$
| $+858582205731250\rho^{13}+23587423234375\rho^{14}\bigr{)}$
$15$ | $\frac{1088}{729\cdot 35!}\;\bigl{(}562009739464769840087040\rho+2247511941596311764074496\rho^{2}$
| $+4019108379306905439830016\rho^{3}+4317745925208072594259968\rho^{4}$
| $+3145163776677939429416960\rho^{5}+1656917203539032341530624\rho^{6}$
| $+655643919364420586023424\rho^{7}+199227919419039256217472\rho^{8}$
|
$+46995751664475880185920\rho^{9}+8614026107092938211680\rho^{10}+1214778349162323946000\rho^{11}$
|
$+128587452922193265000\rho^{12}+9720180867524627500\rho^{13}+472946705787806250\rho^{14}$
| $+11260635852090625\rho^{15}\bigr{)}$
Table 2. Generalized cosecant numbers $c_{\rho,k}$ up to $k=15$

Table 2 displays the generalized cosecant numbers up to $k=15$, which have
been obtained by introducing the multiplicities of all partitions summing to
$k$ into the sum in (2). For $k\\!>\\!10$, the partition method for a power
series expansion becomes laborious due to the exponential increase in the
number of partitions. To circumvent this problem, a general computing
methodology has been developed in Refs. [2] and [13], which is based on
representing all the partitions summing to a specific order $k$ by a partition
tree and invoking the bivariate recursive central partition (BRCP) algorithm.
From this computing methodology general expressions for the coefficients of
any power series expansion obtained. For example, the symbolic form in the
case of the partitions summing to 6 is given by
DS[6]:= p[6] q[1] a + p[1] p[5] q[2] a∧(2) 2! + p[1]∧(2) p[4] q[3] a∧(3) 3!/2!
+
p[1]∧(3) p[3] q[4] a∧(4) 4!/3! + p[1]∧(4) p[2] q[5] a∧(5) 5!/4! + p[1]∧(6)
q[6] a∧(6) +
p[1]∧(2) p[2]∧(2) q[4] a∧(4) 4!/(2! 2!) + p[1] p[2] p[3] q[3] a∧(3) 3! +
\+ p[2] p[4] q[2] a∧(2) 2! + p[2]∧(3) q[3] a∧(3) + p[3]∧(2) q[2] a∧(2)
$\quad.$
Such an expression can easily be imported into Mathematica [10]. Then the
coefficients of the inner series p[k] are set equal to the assigned values of
the parts. For the generalized cosecant numbers, this means we set
p[k-]:=(-1)∧(k+1)/(2k+1)! $\;\;,$
while the q[k], which are referred to as the coefficients of the outer series,
are set equal to the coefficients of the binomial series, viz.
q[k-]:=Pochhammer[$\rho$,k]/k! $\;\;.$
In addition, we set the parameter a equal to unity, thereby obtaining
$c_{\rho,6}$. On the other hand, changing p[k] to
p[k-] := (-1)∧(k + 1)/(2 k)! $\;\;,$
will yield an entirely different set numbers known as the generalized secant
numbers $d_{\rho,k}$ [1, 2]. These numbers represent the coefficients in the
power series expansion for $\sec^{\rho}z$. In terms of the partition method
for a power series expansion they are given by

$\displaystyle
d_{\rho,k}=(-1)^{k}\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3},\dots,n_{k}=0\\\
\sum_{i=1}^{k}in_{i}=k\end{subarray}}^{k,[k/2],[k/3],\dots,1}(-1)^{N}\,(\rho)_{N}\prod_{i=1}^{k}\left(\frac{1}{(2i)!}\right)^{\\!n_{i}}\frac{1}{n_{i}!}\;\;.$
(11)

Thus, the symbolic representation for DS[6] is not only general, but also
powerful.

It should also be mentioned that Mathematica [10] via its SeriesCoefficient
routine is able to determine the generalized cosecant numbers very quickly due
to the fact that it has been optimized. For example, to obtain the first
fifteen generalized cosecant numbers, one need only type:
In[1]:= Table[SeriesCoefficient[(z/Sin[z])${}^{\wedge}\rho$, {z, 0, k}], {k,
0, 30, 2}].
The result for $k=7$ generated by this command is:
Out[2]= (8191 $\rho$)/37362124800 + (15019 (-1 + $\rho$) $\rho$)/12454041600 +
(517457 (-2 + $\rho$) (-1 + $\rho$) $\rho$)/301771008000 + ( 169 (-3 + $\rho$)
(-2 + $\rho$) (-1 + $\rho$) $\rho$)/183708000 + (83 (-4 + $\rho$) (-3 +
$\rho$) (-2 +$\rho$) (-1 + $\rho$) $\rho$)/391910400 \+ ( 7 (-5 + $\rho$) (-4
+ $\rho$) (-3 + $\rho$) (-2 + $\rho$) (-1 + $\rho$) $\rho$)/335923200 + ((-6 +
$\rho$) (-5 + $\rho$) (-4 + $\rho$) (-3 + $\rho$) (-2 + $\rho$) (-1 + $\rho$)
$\rho$)/1410877440 .
Although there are far less contributions than by the partition method for a
power series expansion, this form for the generalized cosecant numbers is
still cumbersome because one must, once again, employ the Expand and Simplify
routines in Mathematica to obtain the forms displayed in Table 2. Moreover, by
using this approach one cannot determine the $k$-dependence of the
coefficients of the polynomials, whereas the partition method for a power
series expansion is able to reveal this behaviour for the highest order
coefficients as explained in the next section.

##  3\. Coefficients of the Generalized Cosecant Numbers

Despite the fact that the contributions in (6) alternate in sign according to
whether the length $N_{k}$ for each partition is even or odd, the final forms
for the generalized cosecant numbers only possess positive coefficients.
Moreover, the highest order terms in the $c_{\rho,k}$ is $O(\rho^{k})$. For
example, from (10) we see that $c_{\rho,6}$ is a sixth order polynomial in
$\rho$. The term that is responsible for the $\rho^{6}$ term in (6) emanates
from the partition with six ones in it because it yields $(\rho)_{6}$.
Consequently, we see that the generalized cosecant numbers are polynomials of
degree $k$ with fixed coefficients. We can express them as
$c_{\rho,k}=\sum_{i=1}^{k}C_{k,i}\rho^{i}$. Our aim in this section is to
determine the highest order coefficients as a function of $k$.

Since we know that the highest order term in the generalized cosecant numbers
is determined by the partition with the most number of ones, i.e. $k$ ones, we
evaluate the contribution from this partition in (6), which is
$(\rho)_{k}/(3!)^{k}k!$. Hence the highest order term in $\rho$ is the
coefficient of $\rho^{k}$ in the Pochhammer factor, which is

$\displaystyle C_{k,k}=\frac{1}{(3!)^{k}k!}\;\;.$ (12)

The second highest order term, namely $C_{k,k-1}$, is the sum of the
contributions from two partitions. First, there is the $\rho^{k-1}$ term from
the partition with $k$ ones and second, there is the highest order term from
the partition with $k-2$ ones and one two. The first term represents the
coefficient of $\rho^{k-1}$ in the Pochhammer factor in the previous
calculation, while the second term represents the $\rho^{k-1}$ power in
$-(\rho)_{k-1}/(5!\cdot(3!)^{k-1}(k-2)!)$. Combining the contributions yields

$\displaystyle
C_{k,k-1}=\frac{1}{(3!)^{k}k!}\sum_{i=1}^{k-1}i-\frac{1}{5\cdot(3!)^{k-2}(k-2)!}=\frac{1}{5\cdot(3!)^{k}(k-2)!}\;\;.$
(13)

The next highest order term or $C_{k,k-2}$ is the sum of the contributions
from four partitions, viz. the $\rho^{k-2}$ power from the partition with $k$
ones, the second leading order term from the partition with $k-2$ ones and one
two and the leading order terms in the partitions with $k-3$ ones and one
three and $k-4$ ones and two twos.

To evaluate each of these contributions in general form, we require the
formula that gives the coefficient of $\rho$ to an arbitrary power in each
Pochhammer factor. According to Chapters 24 and 18 of Refs. [11] and [12]
respectively, the Pochhammer polynomials can be expressed as

$\displaystyle(y)_{k}=\frac{\Gamma(y+k)}{\Gamma(y)}=(-1)^{k}\sum_{j=0}^{k}(-1)^{j}s^{(j)}_{k}\,y^{j}\;\;,$
(14)

where the integers $s^{(j)}_{k}$ are known as the (signed) Stirling numbers of
the first kind with $s^{(0)}_{k}\\!=\\!s^{(k)}_{0}\\!=\\!0$ for $k\\!\geq\\!1$
and $s_{0}^{0}\\!=\\!1$. It should be mentioned that the Stirling numbers of
the first kind are often represented as $s(k,j)$. They also satisfy the
following recurrence relation:

$\displaystyle s^{(j)}_{k+1}=s^{(j-1)}_{k}-k\,s^{(j)}_{k}\;\;.$ (15)

In the appendix of Ref. [8], a general formula for these numbers is derived,
which is given as

$\displaystyle
s_{k}^{(k-j)}=(-1)^{j}\sum_{i_{j}=j}^{k-1}i_{j}\sum_{i_{j-1}=j-1}^{i_{j}-1}\sum_{i_{j-2}=j-2}^{i_{j-1}-1}i_{j-2}\cdots\sum_{i_{1}=1}^{i_{2}-1}i_{1}\;\;.$
(16)

By using this result we can calculate general formulas for specific values of
$j$ ranging from 0 to 4. These are

$\displaystyle s^{(k)}_{k}=1\quad,\quad s^{(k-1)}_{k}=-\binom{k}{2}\quad,\quad
s^{(k-2)}_{k}=\frac{(3k-1)}{4}\,\binom{k}{3}\;\;,$ $\displaystyle
s^{(k-3)}_{k}=-\binom{k}{2}\binom{k}{4}\quad,\quad
s^{(k-4)}_{k}=\frac{1}{48}\bigl{(}15k^{3}-30k^{2}+5k+2\bigr{)}\binom{k}{5}\;\;.$
(17)

Note that the denominator of the last result has been corrected here compared
with Ref. [8], where 336 appears instead of 48. These results have been
obtained by introducing the nested sum formula given by (16) as a module in
Mathematica [10]. Thus, in general one obtains a polynomial in $k$ of degree
$2j$ with one of the terms in the polynomials being equal to
$(-1)^{j}\binom{k}{j+1}$. In Chapter 6 of Ref. [2] the partition method for a
power series expansion is used to derive an alternative formula for the
Stirling numbers of the first kind, which will enables general formulas for
$s^{(k-\ell)}_{k}$ to be determined more easily than using (16). Then it is
found that the Stirling numbers of the first kind can be represented as
$s^{(k-\ell)}_{k}=(-1)^{\ell}\binom{k}{\ell+1}r_{\ell}(k)$, where the
$r_{\ell}(k)$ are polynomials of degree $\ell-1$ and are displayed in Table 3.
For odd values of $\ell$ the polynomials possess a common external factor of
$k(k-1)$. As an aside, it should be mentioned that the corresponding results
for the Stirling numbers of the second kind are also derived in Ref. [2].

$\ell$ | $r_{\ell}(k)$
---|---
$1$ | $1$
$2$ | $\frac{1}{4}(3k-1)$
$3$ | $\frac{1}{2}\,k(k-1)$
$4$ | $\frac{1}{48}\bigl{(}15k^{3}-30k^{2}+5k+2\bigr{)}$
$5$ | $\frac{1}{16}\,k(k-1)\bigl{(}3k^{2}-7k-2\bigr{)}$
$6$ | $\frac{1}{576}\bigl{(}63k^{5}-315k^{4}+315k^{3}+91k^{2}-42k-16\bigr{)}$
$7$ | $\frac{1}{144}\,k(k-1)\bigl{(}9k^{4}-54k^{3}+51k^{2}+58k+16\bigr{)}$
$8$ | $\frac{1}{3840}\bigl{(}135k^{7}-1260k^{6}+3150k^{5}-840k^{4}-2345k^{3}-540k^{2}-66262636k$
| $+596367504\bigr{)}$
$9$ | $\frac{1}{768}\,k(k-1)\bigl{(}15k^{7}-180k^{6}+630k^{5}-448k^{4}-665k^{3}+100k^{2}+404k-144\bigr{)}$
$10$ | $\frac{1}{9216}\bigl{(}99k^{9}-1485k^{8}+6930k^{7}-8778k^{6}-8085k^{5}+8195k^{4}+11792k^{3}$
| $+2068k^{2}-2288k-768\bigr{)}$
Table 3. The polynomials $r_{\ell}(k)$ in the Stirling numbers of the first
kind

With the above results $C_{k,k-2}$ reduces to

$\displaystyle C_{k,k-2}$
$\displaystyle=\frac{s^{(k-2)}_{k}}{(3!)^{k}k!}+\frac{s^{(k-2)}_{k-1}}{5!\cdot(3!)^{k-2}(k-2)!}+\frac{s^{(k-2)}_{k-2}}{7!\cdot(3!)^{k-3}(k-3)!}$
$\displaystyle+\;\;\frac{s^{(k-2)}_{k-2}}{2!\cdot(5!)^{2}(3!)^{k-4}(k-4)!}\;\;.$
(18)

Introducing the results in (17), we find that the coefficients reduce to

$\displaystyle C_{k,k-2}=\frac{21k+17}{175\,(3!)^{k+1}\,(k-3)!}\;\;.$ (19)

The expressions for $C_{k,k-\ell}$ become more difficult to evaluate as $\ell$
increases. However, from the above results, we see that a pattern is
developing. First, the power of $3!$ appears to be increasing as the power of
$\rho$ decreases. Next the factorial in the denominator decrements by unity.
That is, $C_{k,k-1}$ goes as $1/(3!)^{k}(k-2)!$, while $C_{k,k-2}$ goes as
$1/(3!)^{k+1}(k-3)!$. Furthermore, the numerator for $C_{k,k-1}$ is constant,
whereas it is linear in $k$ for $C_{k,k-2}$. So we conjecture that the next
coefficient is given by

$\displaystyle C_{k,k-3}=\frac{ak^{2}+bk+c}{(3!)^{k+2}(k-4)!}\;\;.$ (20)

The reason for choosing $1/(k-\ell-1)!$ in the denominator for $C_{k,k-\ell}$
is that the first value of these coefficients only begins when $k=\ell+1$.
Thus, we can put $k=4$ in the above result and equate it to $C_{4,1}$ in Table
2, which equals 144/5443200. Similarly, we put $k=5$ and $k=6$ in (20) and
equate the values respectively to $C_{5,2}$ and $C_{6,3}$ in the same table.
Then we arrive at the following set of equations:

$\displaystyle 16a+4b+c=\frac{216}{175}\;\;,$ (21) $\displaystyle
25a+5b+c=\frac{312}{175}\;\;,$ (22) $\displaystyle
36a+6b+c=\frac{2124}{875}\;\;.$ (23)

The solution to the above set of equations is $a=6/125$, $b=102/875$ and
$c=0$, which means in turn that $C_{k,k-3}$ is given by

$\displaystyle C_{k,k-3}=\frac{k^{2}+17k/7}{125(3!)^{k+1}(k-4)!}\;\;.$ (24)

Putting $k=8$ in this formula yields 73/26453952000, which agrees with the
coefficient of 397517120 in Table 2 for $k=8$ when it is multiplied by the
external factor of $2/45\cdot 18!$. In using this approach it did not matter
whether our conjecture had the correct power of $3!$ in the denominator
initially provided the dependence upon $k$ was correct. However, it is crucial
that the correct factorial appears in the denominator.

In the case of $C_{k,k-4}$ it would be conjectured by

$\displaystyle C_{k,k-4}=\frac{ak^{3}+bk^{2}+ck+d}{(3!)^{k+4}(k-5)!}\;\;.$
(25)

Then a set of four linear equations is required with $k$ ranging from 5 to 8.
Solving the equations using the LinearSolve routine in Mathematica [10] yields

$\displaystyle
C_{k,k-4}=\frac{3\bigl{(}3k^{3}+102k^{2}/7+289k/49-11170/539\bigr{)}}{625\,(3!)^{k+3}\,(k-5)!}\;\;.$
(26)

Finally, putting $k=9$ into the above result yields a value of
229051/733303549440000, which agrees with the coefficient of $\rho^{5}$ for
$k=9$ in Table 2.

Determining general formulas for the coefficients of the lowest order terms in
the generalized cosecant numbers is a much more formidable problem. If we
consider the preceding methods, then to obtain the equivalent of (13) for the
lowest order coefficient, viz. $C_{k,1}$, we need to evaluate the
contributions from all partitions whereas previously we only needed a fixed
number, e.g., four for determining $C_{k,k-2}$. This is clearly not possible
since the number of partitions is not fixed, but increases exponentially. In
addition, the conjectural approach breaks down when $k$ appears in the second
subscript of the coefficients. For example, to determine $C_{k,2}$ via this
approach, $\ell$ would now be equal to $k-2$.

For $|\rho|\gg k$ we may use the first four coefficients derived above as a
means of approximating the generalized cosecant numbers. That is, the
generalized cosecant numbers can be approximated by

$\displaystyle c_{\rho,k}\overset{|\rho|\gg
k}{\approx}\frac{\rho^{k}}{(3!)^{k}k!}+\frac{\rho^{k-1}}{5\cdot(3!)^{k}(k-2)!}+\frac{(21k+17)\rho^{k-2}}{175\cdot(3!)^{k+1}(k-3)!}+\frac{(k^{2}+17k/7)\rho^{k-3}}{125\cdot(3!)^{k+1}(k-4)!}\;.$
(27)

To gain an appreciation of this approximation, let us denote the ratio of (27)
to the corresponding values of $c_{\rho,k}$ in Table 2 by $\beta(\rho,k)$.
Table 4 presents values of $\beta(\rho,k)$ for integer values of $\rho$
ranging from 10 to 1000. They have been given to six decimal places with no
rounding-off. From the table we see that when $\rho$ is close to $k$ or
smaller, which occurs towards the right hand top corner, $\beta(\rho,k)$ is
not close to unity, but for all other values, it is. Therefore, provided
$\rho$ is significantly greater than $k$, (27) represents an accurate
approximation for the generalized cosecant numbers.

$\rho$ | $k=6$ | $k=8$ | $k=10$ | $k=12$ | $k=15$
---|---|---|---|---|---
$10$ | $0.998905$ | $0.985655$ | $0.929830$ | $0.801477$ | $0.501086$
$15$ | $0.999741$ | $0.996144$ | $0.977941$ | $0.925497$ | $0.751944$
$20$ | $0.999910$ | $0.998571$ | $0.991119$ | $0.966957$ | $0.870459$
$30$ | $0.999981$ | $0.999669$ | $0.997755$ | $0.990752$ | $0.956671$
$50$ | $0.999997$ | $0.999951$ | $0.999644$ | $0.998405$ | $0.991292$
$100$ | $0.999999$ | $0.999997$ | $0.999974$ | $0.999676$ | $0.992370$
$1000$ | $0.999999$ | $0.999999$ | $0.999999$ | $0.999999$ | $0.999999$
Table 4. The ratio $\beta(\rho,k)$ for various values of $\rho$ and $k$

There is one interesting feature in the table that needs to be mentioned. If
one examines the ratios when (1) $k=10$ and $\rho=20$ and (2) $k=15$ and
$\rho=30$, then it is readily observed that the ratio is more accurate for the
first case than in the second case despite the fact they are both good
approximations. This means that as $k$ increases, $|\rho/k|$ must also
increase in order to obtain the same value of $\beta(\rho,k)$ for the lower
values of $k$. For example, the value of $\beta(20,10)$ is about 0.991, which
in the case of $k=15$ only is only reached when $\rho$ is about 50. Hence in
the $k=10$ case it only takes twice the value of $k$ to achieve the same value
of $\beta(\rho,k)$ as the $k=15$ case, which requires at least three times the
value of $k$. This has ramifications when the relationship between the
generalized cosecant numbers and the Hurwitz zeta function is discussed next.

##  4\. Connection to Hurwitz zeta function

It is found in Refs. [14] and [15] that the generalized cosecant numbers can
also be expressed as

$c_{2v,i}=2^{2i}\,\frac{\Gamma(2v-2i)}{\Gamma(2v)}\,s(v,i)\;,\quad i<v\;,$
(28)

where $s(v,n)$ represents the $n$th elementary symmetric polynomial obtained
by summing over quadratic powers or squared integers, viz.
$1^{2},2^{2},\ldots,(v-1)^{2}$. That is,

$s(v,n)=\sum_{1\leq i_{1}<i_{2}<\cdots<i_{n}<v-1}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n}}\;\;,$ (29)

where $x_{i_{1}}<x_{i_{2}}<\cdots<x_{i_{n}}$ and each $x_{i_{j}}$ is equal to
at least one value in the set
$\left\\{1,2^{2},3^{2},\dots,(v-1)^{2}\right\\}$. For the three lowest values
of $n$ the symmetric polynomials are

$s(v,0)=1\;,\quad s(v,1)=(v-1)v(2v-1)/6\;,$

and

$s(v,2)=\frac{(5v+1)}{4\cdot 6!}\;(2v-4)_{5}\;,$ (30)

while for the four highest values of $n$ they are given by

$s(v,v-1)=(v-1)!^{2}\;,\quad
s(v,v-2)=(v-1)!^{2}\,\bigl{(}\zeta(2)-\zeta(2,v)\bigr{)}\;,$
$s(v,v-3)=\frac{(v-1)!^{2}}{2}\Bigl{(}(\zeta(2)-\zeta(2,v))^{2}+\zeta(4,v)-\zeta(4)\Bigr{)}\;,$

and

$\displaystyle s(v,v-4)$
$\displaystyle=\frac{(v-1)!^{2}}{6}\Bigl{(}(\zeta(2)-\zeta(2,v))^{3}-3(\zeta(4)-\zeta(4,v))(\zeta(2)-\zeta(2,v))$
$\displaystyle+\;\;2(\zeta(6)-\zeta(6,v))\Bigr{)}\;.$ (31)

The $n=v-\ell$ results have been obtained by beginning with the general form
given by (29). For $s(v,v-4)$ this becomes

$\displaystyle
s(v,v-4)=\frac{1}{6}\prod_{i=1}^{v-1}i^{2}\sum_{\begin{subarray}{c}j_{1},j_{2},j_{3}=1\\\
j_{1}\neq j_{2}\neq
j_{3}\end{subarray}}^{v-1}\frac{1}{j_{1}^{2}j_{2}^{2}j_{3}^{2}}\;.$ (32)

To evaluate this result, the constraint that none of the $j_{i}$ is equal to
another must be removed. This is accomplished by dropping it and eliminating
all the possibilities where at least one of $j_{i}$ is equal to another.
Consequently, we must consider subtracting all the cases where two of the
indices are equal to another and finally when three indices are equal to one
another. The product over $i$ yields $\Gamma(v)^{2}$. Thus, we arrive at

$\displaystyle
s(v,v-4)=\frac{1}{2}\Gamma(v)^{2}\Bigl{(}\sum_{j_{1},j_{2},j_{3}=1}^{v-1}\frac{1}{j_{1}^{2}j_{2}^{2}j_{3}^{2}}-3\sum_{j_{1},j_{2}=1}^{v-1}\frac{1}{j_{1}^{4}j_{2}^{2}}+2\sum_{j_{1}=1}^{v-1}\frac{1}{j_{1}^{6}}\Bigr{)}\;\;,$
(33)

which yields the result given by (31). In a similar manner one finds that

$\displaystyle s(v,v-5)$
$\displaystyle=\frac{1}{24}\;\Gamma(v)^{2}\Bigl{(}\bigl{(}\zeta(2)-\zeta(2,v)\bigr{)}^{4}-6\bigl{(}\zeta(4)-\zeta(4,v)\bigr{)}\bigl{(}\zeta(2)-\zeta(2,v)\bigr{)}^{2}$
$\displaystyle+\;8\bigl{(}\zeta(6)-\zeta(6,v)\bigr{)}\bigl{(}\zeta(2)-\zeta(2,v)\bigr{)}+3\bigl{(}\zeta(4)-\zeta(4,v)\bigr{)}^{2}\Bigr{)}$
$\displaystyle-\;6\bigl{(}\zeta(8)-\zeta(8,v)\bigr{)}\Bigr{)}\;.$ (34)

In general, $s(v,v-\ell)$ is given by

$\displaystyle
s(v,v-\ell)=\sum_{\begin{subarray}{c}j_{1},\dots,j_{\ell-1}=1\\\
j_{1}<j_{2},\cdots<j_{\ell-1}\end{subarray}}^{v-1}\prod_{i=1}^{v-1}\prod_{k=1}^{\ell-1}\frac{i^{2}}{j_{k}^{2}}=\frac{\Gamma(v)^{2}}{(\ell-1)!}\sum_{\begin{subarray}{c}j_{1},\dots,j_{\ell-1}=1\\\
j_{1}\neq j_{2}\cdots j_{n-2}\neq
j_{\ell-1}\end{subarray}}^{v-1}\prod_{k=1}^{\ell-1}\frac{1}{j_{k}^{2}}\;.$
(35)

In order to solve the sum on the rhs, one needs to remove the constraint that
each of the $j_{i}$ cannot equal one another. This means that we need to
subtract all the possibilities when at least one of the indices is equal to
one another from the sum where all the indices can equal each other other,
viz. $\sum_{j_{1},\dots,j_{\ell-1}=1}^{v-1}\prod_{k=1}^{\ell-1}1/j_{k}^{2}$.
The number of sums that appear on the rhs becomes $p(\ell-1)$, where $p(k)$ is
the partition function or the number of partitions summing to $k$. E.g., in
(33) we see that there are three sums for $s(v,v-4)$, since $p(3)$ is equal to
three as a result of the partitions, {1,1,1}, {2,1} and {3}. In addition, the
powers of the $j_{i}$ will be twice the parts in the partitions. That is, the
sum $\sum_{j_{1},j_{2}=1}^{v-1}1/(j_{1}^{4}j_{2}^{2})$ corresponds to the
partition {2,1}. Moreover, the sign outside each sum is $(-1)^{N_{k}+1}$,
while the factors preceding each sum is the factorial of the length $N_{k}$
divided by the factorials of the multiplicities of each part and the parts
taken to the power of their multiplicities. For example, in the case of the
partition {2,1}, the sum of the parts is three, while the multiplicities are
both equal to 1. Hence the factor outside the sum becomes $3!/(2\cdot 1!\cdot
1\cdot 1!)=3$. See Ref. [15] for more details.

If we introduce the results for $s(v,v-n)$ into (28), then for $n=1$ we obtain

$\frac{\Gamma(v)}{\Gamma(v+1/2)}=\frac{2}{\sqrt{\pi}}\;c_{2v,v-1}\;\;,$ (36)

which is only valid for $v>1$. Alternatively, (36) can be expressed as
$B(v,1/2)=2\,c_{2v,v-1}$, where $B(x,y)$ represents the beta function.
Furthermore, according to No. 2.5.3.1 in Ref. [16], we have

$\displaystyle\int_{0}^{\pi/2}\left\\{\begin{matrix}\sin x\\\ \cos
x\end{matrix}\right\\}^{2v-1}\;dx=\frac{\sqrt{\pi}}{2}\,\frac{\Gamma(v)}{\Gamma(v+1/2)}=c_{2v,v-1}\;\;,$
(37)

while from Appendix I.1.9 of the same reference, we find that

$\displaystyle\left\\{\begin{matrix}\sin x\\\ \cos
x\end{matrix}\right\\}^{2v-1}=\frac{1}{2^{2v-2}}\sum_{k=0}^{v-1}(\mp
1)^{v-k-1}\binom{2v-1}{k}\left\\{\begin{matrix}\sin(2v-2k-1)x\\\
\cos(2v-2k-1)x\end{matrix}\right\\}.$ (38)

A surprising property of the above analysis is that the $c_{2v,k}$ are
coefficients in the power series expansion of $(x/\sin x)^{2v}$, but in (37)
they are related to positive powers of sine and cosine taken to $2v-1$ when
$k=v-1$. Inserting the second result into the first one yields

$\displaystyle\frac{1}{2^{2v-2}}\sum_{k=0}^{v-1}\frac{(-1)^{v-k-1}}{(2v-2k-1)}\,\binom{2v-1}{k}=c_{2v,v-1}\;.$
(39)

This result can be checked by implementing it in Mathematica as follows:
C2vvminus1[v-] := 2∧(2 - 2 v) Sum[(-1)∧(v - k - 1) Binomial[2 v - 1, k]/(2 v -
2 k - 1), {k, 0, v - 1}]
Putting v equal to 5 in this line of code yields 128/315, while putting
$\rho=10$ in the $k=4$ result of Table 2 gives the same value.

It emerges that the terms at the upper limit of the sum in (39) contribute
more to the value of $c_{2v,v-1}$ than those at the lower limit. Therefore, we
express (39) as

$\displaystyle
c_{2v,v-1}=2^{2-2v}\,\binom{2v-1}{v}\sum_{k=0}^{v-1}\frac{(-1)^{k}}{(2k+1)}\,\prod_{j=1}^{k}\Bigl{(}\frac{1-j/v}{1+j/v}\Bigr{)}\;.$
(40)

Now we expand the terms in the sum and product, thereby obtaining

$\displaystyle c_{2v,v-1}$
$\displaystyle=2^{2-2v}\,\binom{2v-1}{v}\Bigl{[}\Bigl{(}1-\frac{2}{v}+\frac{2}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}-\frac{1}{3}\Bigl{(}1-\frac{2}{v}+\frac{2}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}\Bigl{(}1-\frac{4}{v}$
$\displaystyle+\;\;\frac{8}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}+\frac{1}{5}\Bigl{(}1-\frac{2}{v}+\frac{2}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}\Bigl{(}1-\frac{4}{v}+\frac{8}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}$
$\displaystyle\times\;\;\Bigl{(}1-\frac{6}{v}+\frac{18}{v^{2}}+O\bigl{(}\frac{1}{v^{3}}\bigr{)}\Bigr{)}-\cdots\Bigr{]}\;.$
(41)

In more compact notation the above result can be written as

$\displaystyle
c_{2v,v-1}=2^{2-2v}\,\binom{2v-1}{v}\Bigl{(}\sum_{j=0}^{v-1}\frac{(-1)^{j}}{(2j+1)}-\frac{1}{v}\sum_{j=0}^{v-1}\frac{(-1)^{j}}{(2j+1)}\;(j+1)(j+2)+O\bigl{(}\frac{1}{v^{2}}\bigr{)}\Bigr{)}\;.$
(42)

The first sum is a known result given by No. 4.1.3.4 in Ref. [16], while the
second sum requires decomposition. Then we arrive at

$\displaystyle\sum_{j=0}^{v-1}\frac{(-1^{j}}{2j+1}\;(j+1)(j+2)=\frac{1}{2}\sum_{j=0}^{v-1}(-1)^{j}j+\frac{5}{4}\sum_{j=0}^{v-1}(-1)^{j}+\frac{3}{4}\sum_{j=0}^{v-1}\frac{(-1)^{j}}{2j+1}\;\;.$
(43)

The last sum in (43) is simply another occurrence of the first sum in (42).
Consequently, $c_{2v,v-1}$ becomes

$\displaystyle c_{2v,v-1}$
$\displaystyle=2^{2-2v}\,\binom{2v-1}{v}\Bigl{(}\frac{\pi}{4}+\frac{(-1)^{v-1}}{2}\,\beta(v+1/2)+\frac{(-1)^{v-1}}{2v}\,\lfloor
v/2\rfloor$
$\displaystyle-\;\;\frac{5}{8v}\bigl{(}1-(-1)^{v}\bigr{)}+\frac{3}{4v}\frac{(-1)^{v-1}}{2}\,\beta(v+1/2)+O\bigl{(}\frac{1}{v^{2}}\bigr{)}\Bigr{)}\;,$
(44)

where $\lfloor x\rfloor$ denotes the floor function or the greatest integer
less than or equal to $x$, $\beta(v)=(\psi((v+1)/2)-\psi(v/2))/2$ and
$\psi(x)$ is the digamma function. For large values of $v$, the leading term
given by the $\pi/4$ term on the rhs is a good asymptotic approximation to
$c_{2v,v-1}$.

For $n=2$, (28) yields

$\sum_{k=1}^{v-1}\frac{1}{k^{2}}=\frac{\pi^{2}}{6}-\zeta(2,v)=\frac{2}{3}\frac{c_{2v,v-2}}{c_{2v,v-1}}\;\;,$
(45)

where $\zeta(x,y)$ represents the Hurwitz zeta function and $v>2$. By adopting
the same approach for $n=3$ to $n=6$, we arrive at

$\sum_{k=1}^{v-1}\frac{1}{k^{4}}=\frac{\pi^{4}}{90}-\zeta(4,v)=\frac{4}{9}\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{2}-\frac{4}{15}\frac{c_{2v,v-3}}{c_{2v,v-1}}\;\;,$
(46)
$\sum_{k=1}^{v-1}\frac{1}{k^{6}}=\frac{\pi^{6}}{945}-\zeta(6,v)=\frac{4}{105}\;\frac{c_{2v,v-4}}{c_{2v,v-1}}-\frac{4}{15}\frac{c_{2v,v-3}}{c_{2v,v-1}}\;\frac{c_{2v,v-2}}{c_{2v,v-1}}+\frac{8}{27}\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{3}\;,$
(47) $\displaystyle\sum_{k=1}^{v-1}\frac{1}{k^{8}}$
$\displaystyle=\frac{\pi^{8}}{9450}-\zeta(8,v)=\frac{8}{14175}\left(350\,\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{4}-420\,\frac{c_{2v,v-3}}{c_{2v,v-1}}\;\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{2}\right.$
$\displaystyle\left.+\;63\Bigl{(}\frac{c_{2v,v-3}}{c_{2v,v-1}}\Bigr{)}^{2}+\;60\,\frac{c_{2v,v-4}}{c_{2v,v-1}}\,\frac{c_{2v,v-2}}{c_{2v,v-1}}-5\,\frac{c_{2v,v-5}}{c_{2v,v-1}}\right)\;,$
(48)

and

$\displaystyle\sum_{k=1}^{v-1}\frac{1}{k^{10}}$
$\displaystyle=\frac{\pi^{10}}{93555}-\zeta(10,v)=\frac{4}{93555}\left(3080\,\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{5}-4620\,\frac{c_{2v,v-3}}{c_{2v,v-1}}\;\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{3}\right.$
$\displaystyle+\;1386\Bigl{(}\frac{c_{2v,v-3}}{c_{2v,v-1}}\Bigr{)}^{2}\frac{c_{2v,v-2}}{c_{2v,v-1}}+\;660\,\frac{c_{2v,v-4}}{c_{2v,v-1}}\,\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{2}-198\,\frac{c_{2v,v-4}}{c_{2v,v-1}}\;\frac{c_{2v,v-3}}{c_{2v,v-1}}$
$\displaystyle\left.-\;55\,\frac{c_{2v,v-5}}{c_{2v,v-1}}\;\frac{c_{2v,v-2}}{c_{2v,v-1}}+3\,\frac{c_{2v,v-6}}{c_{2v,v-1}}\right)\;.$
(49)

where $v>2$ for the first result, $v>3$ in the second, etc. In principle, this
process can be continued for higher powers of the sum on the lhs by
determining larger values of $\ell$ in the symmetric polynomials,
$s(v,v-\ell)$. Consequently, we see that integer values of the Hurwitz zeta
function for even powers can now be expressed in terms of ratios of the
generalized cosecant numbers, which is indeed fascinating in view of the
intractability of this famous function. Moreover, in the limit as
$v\to\infty$, we obtain new results for the Riemann zeta function such as

$\zeta(4)=\frac{\pi^{4}}{90}=\lim_{v\to\infty}\Bigl{\\{}\frac{4}{9}\Bigl{(}\frac{c_{2v,v-2}}{c_{2v,v-1}}\Bigr{)}^{2}-\frac{4}{15}\frac{c_{2v,v-3}}{c_{2v,v-1}}\Bigr{\\}}\;\;.$
(50)

Unfortunately, we cannot introduce (27) into the above result because
$|\rho/k|$ is approximately equal to 2 when $k$ is equal to either $v-1$ and
$v-2$, whereas we have observed in Table 4 that as $\rho$ or $2v$ increases,
the ratio $2v/k$ needs to increase dramatically in order to ensure that
$\beta(\rho,k)$ remains close to unity. Otherwise, (27) does not represent a
good approximation for the generalized cosecant numbers. Therefore, we require
asymptotic forms as in (44) for the generalized cosecant numbers of the form,
$c_{2v,v-\ell}$, where $\ell=2,3,\dots$. These have yet to be developed.

Finally, it should be mentioned that the series on the lhs of (45) to (49)
also represent specific values of the generalized harmonic numbers, which are
defined as $H_{n,r}=\sum_{k=1}^{n}1/k^{r}$ [17]. In particular, for the case
of $r=2$ given by (45) the numbers are known as Wolstenholme numbers, which
appear as Sequences A007406, A007408, A11354 and A123751 in the online
encyclopedia of integer sequences [18].

##  5\. Acknowledgement

The author is grateful to Professor Carlos M. da Fonseca, Kuwait University,
without whose support and encouragement this paper would not have proceeded.

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