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Lerch’s Φ and the Polylogarithm at the Negative Integers

Jose Risomar Sousa

September 17, 2021

Abstract

At the negative integers, there is a simple relation between the Lerch Φ function and

the polylogarithm. The literature has a formula for the polylogarithm at the negative

integers, which utilizes the Stirling numbers of the second kind. Starting from that

formula, we can deduce a simple closed formula for the Lerch Φ function at the negative

integers, where the Stirling numbers are not needed. Leveraging that ﬁnding, we also

produce alternative formulae for the k-th derivatives of the cotangent and cosecant (ditto,

tangent and secant), as simple functions of the negative polylogarithm and Lerch Φ,

respectively, which is evidence of the importance of these functions (they are less exotic

than they seem). Lastly, we present a new formula for the Hurwitz zeta function at the

positive integers using this novelty.

1 Introduction

As seen in reference [4], it’s possible to derive a formula for the Hurwitz zeta function at

the positive integers with results from both [2] and [3]. To greatly summarize the reasoning

presented there, if kis a positive integer greater than one, then:

Z1

0

uk(1 −cos 2π n u) cot πu du ∼ − H(n)

π−k!

π

b(k−1)/2c

X

j=1

(−1)j(2π)−2jζ(2j+ 1)

(k−2j)! ,

which implies the following approximation (with the equality only valid for integer k):

Z1

0

(uk−u)(1−cos 2π n u) cot πu du ∼ − k!

π

b(k−1)/2c

X

j=1

(−1)j(2π)−2jζ(2j+ 1)

(k−2j)! =Z1

0

(uk−u) cot πu du,

which in turn justiﬁes the formula1shown next.

1The formulae that can be derived with this method are not unique and the one shown may be the simplest.

1

For every integer kgreater than one and every non-integer complex b:

ζ(k, b) = 1

2bk+(2πi)k

4 Li−k+1 e−2πib

(k−1)! +e−2πib

k

X

j=1

δ1j+ Li−j+1(e−2πib)

(j−1)!(k−j)! !

−i(2πi)k

2Z1

0

k

X

j=1 δ1j+ Li−j+1(e−2πib)uk−je−2πib u −e−2πib

(j−1)!(k−j)! cot πu du

The discovery of a new, possibly ﬁrst, closed formula for the Lerch transcendent function

at the negative integers was made possible through the analysis of the above formula.

The big breakthrough is really in the following straightforward identity, which only works

for the analytic continuation of the Lerch Φ and the polylogarithm functions at the negative

integers, and makes it possible to obtain the former as a sum of the latter:

k

X

j=0

Li−jebuk−j

j!(k−j)! =eb

k!Φ(eb,−k, u + 1) (1)

In fact, this identity also applies to the sum of Lerches:

k

X

j=0

Φeb,−j, vuk−j

j!(k−j)! =1

k!Φ(eb,−k, u +v)(2)

Last but not least, from two known facts from the literature, namely, a recurrence for the

Bernoulli polynomials and a relation between them and the Hurwitz zeta function:

Bk(u+v) = k!

k

X

j=0

Bj(v)uk−j

j!(k−j)! and Bj(v) = −j ζ(1 −j, v),

one may conclude that,

ζ(−k, u +v) = −uk+1

k+ 1 +k!

k

X

j=0

ζ(−j, v)uk−j

j!(k−j)! ,

and from this recurrence, one can obtain a natural expression relating the Hurwitz and the

Riemann zeta functions, which completes the picture:

ζ(−k, u + 1) = −uk+1

k+ 1 +k!

k

X

j=0

ζ(−j)uk−j

j!(k−j)! (3)

Now, back to the Hurwitz formula, from relation (1) we obtain,

k

X

j=1

Li−j+1 e−2πibuk−j

(j−1)!(k−j)! =e−2πib

(k−1)!Φ(e−2πib,−k+ 1, u + 1),

2

which leads to the below analytic continuation for the Hurwitz zeta, which then holds for

<(k)>1:

ζ(k, b) = 1

2bk+(2πi)k

4(k−1)! e−2πib+e−4πibΦ(e−2πib,−k+ 1,2) + Li−k+1 e−2πib

−i(2πi)k

2(k−1)! Z1

0uk−1e−2πib u −e−2πib

+e−2πib(u+1) Φ(e−2πib,−k+ 1, u + 1) −e−4πibΦ(e−2πib,−k+ 1,2)cot πu du (4)

Since this formula depends on the Lerch’s Φ analytic continuation at the negative integers,

we can rewrite (4), if can we ﬁgure out Lerch’s Φ at the negative integers.

Relation (1) may be the counterpart to the functional equation that relates the Lerch Φ

to the Hurwitz zeta at the positive integers, as demonstrated in a previous paper5. This new

relation therefore presupposes the existence of another relation between the polylogarithm and

the Hurwitz zeta at the negative integers.

2 Stirling numbers of the second kind

The existing formula for the polylogarithm Li−j+1(z), available in the literature, makes use

of the Stirling numbers of the second kind (the S(j, q) in the curly brackets):

Li−j+1(z) =

j

X

q=1

(q−1)!j

q z

1−zq

(5)

Let’s see how to obtain the Lerch Φ from it. Making z=e−2πib, we have:

z

1−z=−1 + icot π b

2

Therefore, back to equation (1):

k

X

j=1

Li−j+1 e−2πibu−j

(j−1)!(k−j)! =

k

X

j=1

j

X

q=1

(q−1)!j

q−1 + icot π b

2q

⇒

k

X

j=1

j

X

q=1

(q−1)!u−j

(j−1)!(k−j)!j

q−1 + icot π b

2q

⇒

k

X

q=1

(q−1)! −1 + icot π b

2qk

X

j=q

u−j

(j−1)!(k−j)!j

q(6)

3

Now we need to study the second sum from equation (6) and see if it’s possible to rewrite

it. An expression for a similar sum exists in the literature,

k

X

j=q

1

j!(k−j)!j

q=1

k!k+ 1

q+ 1,

but this new one is much more complicated.

3 Binomial formula for Stirling numbers

Let’s try and rewrite the below ﬁnite sum,

k

X

j=q

u−j

(j−1)!(k−j)!j

q(7)

First oﬀ, the literature provides us with the below relation:

k

X

j=qj

qxj

j!=(ex−1)q

q!,

therefore,

∞

X

k=q

k

X

j=qj

qxj

j!

yk−j

(k−j)! =ey(ex−1)q

q!,

and diﬀerentiating with respect to x,

∞

X

k=q

k

X

j=qj

qxj−1

(j−1)!

yk−j

(k−j)! =eyex(ex−1)q−1

(q−1)!

Now, making x=u z and y=z, we have:

∞

X

k=q

zk−1

k

X

j=qj

quj−1

(j−1)!(k−j)! =e(u+1)z(eu z −1)q−1

(q−1)! =

q−1

X

j=0

(−1)q−1−je(u+1+j u)z

j!(q−1−j)! ,

and hence diﬀerentiating k−1 times with respect to z, we conclude that,

k

X

j=q

uj

(j−1)!(k−j)!j

q=u

(k−1)!

q

X

j=1

(−1)q−j(j u + 1)k−1

(j−1)!(q−j)! ,(8)

To obtain a more proper version of this formula, we integrate as below,

Zu

0

k

X

j=q

xj−1

(j−1)!(k−j)!j

qdx =1

(k−1)! Zu

0

q

X

j=1

(−1)q−j(j x + 1)k−1

(j−1)!(q−j)! dx,

4

which gives us the neat expression below, which holds for every non-negative integer qand

every u:

k

X

j=q

uj

j!(k−j)!j

q=1

k!

q

X

j=0

(−1)q−j(j u + 1)k

j!(q−j)! (9)

Though it’s not going to be used here, another pattern similar to the binomial theorem

emerges in the factorial power of the sum of two numbers, xand y:

(x+y)(k)=k!

k

X

j=0

x(j)y(k−j)

j!(k−j)!, where x(j)=x!

(x−j)!

4 Lerch’s Φat the negative integers

Now, if we combine equations (1), (6) and (8), we obtain:

Φ(e−2πib,−k+ 1, u + 1) = e2πib

k

X

q=1

(q−1)! 1 + icot π b

2qq

X

j=1

(−1)j(j+u)k−1

(j−1)!(q−j)! (10)

For integer bthe formula is not deﬁned as the cotangent is inﬁnity, so we can’t extract the

Hurwitz zeta at the negative integers from it. But from the relation,

Φ(e−2πib,−k+ 1,1) = e2πibLi−k+1 e−2πib,

we can derive the polylogarithm, which however is just a rewrite of equation (5) with an

expression for S(k, q) known from the literature, which nonetheless conﬁrms equation (10):

Li−k+1 e−2πib=

k

X

q=1

(q−1)! 1 + icot π b

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)! (11)

Looking at (10) and (11) now, it might look simple to go from the latter straight to the

former without having to ﬁgure (8), but that’s misleading.

Finally, we can turn (10) into another form, which holds for every non-negative integer k:

Φ(z, −k, u) = −1

z−1

k

X

q=0

q!z

z−1qq

X

j=0

(−1)j(j+u)k

j!(q−j)! (12)

It’s interesting to note how much simpler the analytic continuation of the Lerch Φ at the

negative integers is than the proper function at the positive integers. And also how strikingly

similar it is to the power series for the Lerch Φ at the positive integers available in the literature:

Φ(z, k, u) = −1

z−1

∞

X

q=0

q!z

z−1qq

X

j=0

(−1)j(j+u)−k

j!(q−j)!

5

5 Derivatives of trigonometric functions

In his paper On the Hurwitz function for rational arguments 1, Victor Adamchik provides

the ﬁrst ever formula for the intricate patterns of the k-th derivatives of the cotangent. It

looks like this:

dk(cot a x)

dxk= (2 ia)k(−i+ cot a x)

k

X

q=1

q!k

q−1 + icot a x

2q

It’s possible to express this formula as a simple function of the polylogarithm. First, we

rewrite it as:

dk(cot a x)

dxk= (2 ia)k(−i+ cot a x)

k

X

q=1

q!1−icot a x

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)! (13)

Secondly, we note how similar it looks to the polylogarithm from (11):

Li−k+1 e2ia x=

k

X

q=1

(q−1)! 1−icot a x

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)!

If the above polylog is diﬀerentiated once with respect to xand transformed,

Li−ke2ia x=1

1−e2ia x

k

X

q=1

q!1−icot a x

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)!,(14)

an alternative expression is obtained for the polylogarithm of order k, which, however, is not

exactly equal to form (11). That stems from a property of polylogs, that when diﬀerentiated

they yield the next order polylog. Finally, comparing the two expressions, (13) and (14), we

conclude that:

dk(cot a x)

dxk=−iδ0k−2i(2 ia)kLi−ke2ia x, where δ0k= 1 iﬀ k= 0 (15)

To obtain the cosecant, we can resort to a simple logic:

cos a x +isin a x

sin a x =eia x

sin a x =i+ cot a x ⇒1

sin a x =e−ia x (i+ cot a x)

Then the Leibniz rule for the derivative of a product of two functions leads to:

dk

dxk1

sin a x=−2ie−ia x

k

X

q=0

k! (−ia)k−q

q!(k−q)! (2 ia)qLi−qe2ia x

Lastly, formula (1) allows us to rewrite the above expression as:

dk

dxk1

sin a x=−2i(2 ia)keia x Φe2ia x ,−k, 1

2,(16)

which holds for every non-negative integer k.

6

5.1 Tangent and secant

To be able to obtain the tangent and secant, ﬁrst we need to produce a formula for the

cotangent and cosecant of a translated arc. Adamchik’s formula1becomes:

dk(cot (a x +b))

dxk= (2 ia)k(−i+ cot (a x +b))

k

X

q=1

q!1−icot (a x +b)

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)!

The polylog formula then changes to:

Li−ke2i(a x+b)=1

1−e2i(a x+b)

k

X

q=1

q!1−icot (a x +b)

2qq

X

j=1

(−1)jjk−1

(j−1)!(q−j)!,

and then the ﬁnal formula is not diﬀerent from the simple case:

dk(cot (a x +b))

dxk=−iδ0k−2i(2 ia)kLi−ke2i(a x+b), where δ0k= 1 iﬀ k= 0

Similarly, the cosecant for a translated arc is:

dk

dxk1

sin (a x +b)=−2i(2 ia)kei(a x+b)Φe2i(a x+b),−k, 1

2

Finally, to obtain the tangent and secant, we just need to set bto π/2. And since the

formulae for the translated arc are not very diﬀerent from b= 0, for the tangent we have:

dk(tan (a x +b))

dxk=iδ0k+ 2 i(2 ia)kLi−k−e2i(a x+b), where δ0k= 1 iﬀ k= 0, (17)

and for the secant:

dk

dxk1

cos (a x +b)= 2 (2 ia)kei(a x+b)Φ−e2i(a x+b),−k, 1

2(18)

It’s surprising that these derivatives can be expressed by means of negative Lerch and

polylogs. For example, the negative polylog is known to yield the derivatives of a simple

exponential function at a point, but not the derivative itself:

dk

dxkx

ea x+b−1x=0

=−kδ1k+ Li−k+1 ebak−1

7

6 Hurwitz zeta at the positive integers

Now that fomula (10) is known, the Hurwitz zeta formula at the positive integers greater

than one can be rewritten with only elementary functions references:

ζ(k, b) = 1

2bk+(2πi)k

4(k−1)!

e−2πib+

k

X

q=1

(q−1)! 1 + icot π b

2qq

X

j=1

(−1)je−2πib(j+ 1)k−1+jk−1

(j−1)!(q−j)!

−i(2πi)k

2(k−1)! Z1

0 uk−1e−2πib u −e−2πib+

k

X

q=1

(q−1)! 1 + icot π b

2qq

X

j=1

(−1)je−2πib u(j+u)k−1−e−2πib(j+ 1)k−1

(j−1)!(q−j)! !cot πu du

The pattern of the above formula becomes clearer and looks better if we include the outside

terms into the summation symbol, assuming the convention that (−1)!/(−1)! = 1:

ζ(k, b) = 1

2bk+(2πi)k

4(k−1)!

k

X

q=0

(q−1)! 1 + icot π b

2qq

X

j=0

(−1)je−2πib(j+ 1)k−1+jk−1

(j−1)!(q−j)!

−i(2πi)k

2(k−1)! Z1

0

k

X

q=0

(q−1)! 1 + icot π b

2qq

X

j=0

(−1)je−2πib u(j+u)k−1−e−2πib(j+ 1)k−1

(j−1)!(q−j)! cot πu du

One of the advantages of this new formula is the fact it allows one to get rid of its non-real

parts more easily, though the resulting formula is inevitably more complicated.

6.1 When the parameter bis a half-integer

The below result stems from e−2πib=−1 and cot π b = 0 when bis a half-integer:

ζ(k, b) = 1

2bk−(2πi)k

4(k−1)!

1 +

k

X

q=1

(q−1)! 1

2qq

X

j=1

(−1)j(j+ 1)k−1−jk−1

(j−1)!(q−j)!

−i(2πi)k

2(k−1)! Z1

0

uk−1e−2πib u +1+

k

X

q=1

(q−1)! 1

2qq

X

j=1

(−1)je−2πib u(j+u)k−1+ (j+ 1)k−1

(j−1)!(q−j)!

cot πu du

7 A new formula for the Hurwitz zeta

In [4], we had created a generating function for the Hurwitz zeta function, f(x). When b

is not a half-integer, the expression is:

f(x) =

∞

X

k=2

xkζ(k, b) = −x2

2b(x−b)−1

2 sin πb

πx sin πx

sin π(x−b)

−πx Z1

0sin 2πu(x−b)

sin 2π(x−b)−sin 2πbu

sin 2πb cot πu du (19)

8

The k-th derivative of f(x) yields the Hurwitz zeta function of order k:

ζ(k, b) = f(k)(0)

k!

And now that we know how to diﬀerentiate the cosecant successively, it’s possible to pro-

duce an explicit formula from f(x), again through the Leibniz rule. However, to make this

process simpler, we resort to two artiﬁces. First, to get rid of the extra xfactor in the integral,

we divide f(x) by xand take the (k−1)-th derivative instead of the k-th. Second, to avoid the

complications of diﬀerentiating the sine, we replace it with an equivalent sum of exponential

functions.

The ﬁrst and second parts of (19) are straightforward, they coincide with the terms outside

of the integral from (4), that is:

1

k!

dk

dxk−1

2 sin πb

πx sin πx

sin π(x−b)x=0

=(2πi)k

4(k−1)! e−2πib+e−4πibΦ(e−2πib,−k+ 1,2) + Li−k+1 e−2πib

The same isn’t true for the third part of (19), since f(x) was created using a diﬀerent

process than (4) (see [4] for details). The integrals evaluate to the same number, but the

integrands are not the same.

After all is put together, the ﬁnal formula holds for any positive integer kgreater than one,

and any 2bthat’s not an integer:

ζ(k, b) = 1

2bk+(2πi)k

4(k−1)! e−2πib+e−4πibΦ(e−2πib,−k+ 1,2) + Li−k+1 e−2πib+

−i(2πi)ke−2πib

4Z1

0

k

X

j=1

2juk−je−2πib u −(−1)k−je2πib u

(j−1)!(k−j)! Φe−4πib,−j+ 1,1

2cot πu du (20)

Finally, by using the relation (2), an analytic continuation can be produced:

ζ(k, b) = 1

2bk+(2πi)k

4(k−1)! e−2πib+e−4πibΦ(e−2πib,−k+ 1,2) + Li−k+1 e−2πib+

−i(4πi)ke−2πib

4(k−1)! Z1

0e−2πib uΦe−4πib,−k+ 1,u+ 1

2−e2πib uΦe−4πib,−k+ 1,−u+ 1

2cot πu du

9

References

[1] Adamchik, Victor S. On the Hurwitz function for rational arguments, Applied Mathematics

and Computation, Volume 187, Issue 1, 2007, Pages 3–12.

[2] Risomar Sousa, Jose Generalized Harmonic Numbers, eprint arXiv:1810.07877, 2018.

[3] Risomar Sousa, Jose Generalized Harmonic Progression Part II, eprint arXiv:1902.01008,

2019.

[4] Risomar Sousa, Jose The Hurwitz Zeta Function at the Positive Integers, eprint

arXiv:1902.06885, 2019.

[5] Risomar Sousa, Jose Lerch’s Φand the Polylogarithm at the Positive Integers, eprint

arXiv:2006.08406, 2020.

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