Journal of Number Theory 133 (2013) 3995–4009

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Journal of Number Theory

www.elsevier.com/locate/jnt

Rational series for multiple zeta and log gamma functions

Paul Thomas YoungDepartment of Mathematics, College of Charleston, Charleston, SC 29424, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 December 2012Accepted 20 May 2013Available online xxxxCommunicated by David Goss

Keywords:Barnes zeta functionsHurwitz zeta functionMultiple zeta functionsMultiple gamma functionsBernoulli polynomialsDirichlet L-functionsPolygamma functions

Text. We give series expansions for the Barnes multiple zetafunctions in terms of rational functions whose numerators arecomplex-order Bernoulli polynomials, and whose denomina-tors are linear. We also derive corresponding rational expan-sions for Dirichlet L-functions and multiple log gamma func-tions in terms of higher order Bernoulli polynomials. Theseexpansions naturally express many of the well-known prop-erties of these functions. As corollaries many special valuesof these transcendental functions are expressed as series ofhigher order Bernoulli numbers.

Video. For a video summary of this paper, please click hereor visit http://youtu.be/2i5PQiueW_8.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

Let ζr(s, a) denote the Barnes multiple zeta function [3,16] of order r defined by

ζr(s, a) =∞∑

t1=0· · ·

∞∑tr=0

(a + t1 + · · · + tr)−s (1.1)

for �(s) > r and �(a) > 0, and continued meromorphically to s ∈ C with simple polesat s = 1, 2, . . . , r. Note that ζ1(s, a) is the Hurwitz zeta function, and ζ0(s, a) = a−s byconvention. In Theorem 1 below we prove the series expansion

E-mail address: paul@math.cofc.edu.

0022-314X/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jnt.2013.05.016

3996 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

Γ (s)ζr(s, a) =∞∑

n=0

(−1)nB(n+s)n (a)

n!(n + s− r) , (1.2)

where B(z)n (x) is the n-th Bernoulli polynomial of order z, defined by

(t

et − 1

)z

ext =∞∑

n=0B(z)

n (x) tn

n! . (1.3)

Each term in the convergent series (1.2) is a polynomial in a and a rational functionof s and of the order r. We observe in Section 5 below that, for nonnegative integers k,multiplying both sides by s + k and taking the limit as s → −k yields the well-knownvalues

ζr(−k, a) = (−1)rk!(r + k)!B

(r)r+k(a) (1.4)

[16, Eq. (3.10)] of these zeta functions at the negative integers; therefore the n-th term inthe series is basically the “singular part” of Γ (s)ζr(s, a) at s = r−n. The series (1.2) maytherefore be viewed as a polynomial Mittag-Leffler type decomposition of Γ (s)ζr(s, a)by singular parts at each pole.

The multiple log gamma function Ψr(a) of order r is defined [3,16] for �(a) > 0 by

Ψr(a) = ∂

∂sζr(s, a)

∣∣∣∣s=0

. (1.5)

We observe that Ψ0(a) = − log a following the convention ζ0(s, a) = a−s. The name isderived by analogy to the case r = 1 where we have

Ψ1(a) = log(Γ (a)√

2π

)(1.6)

[16, Eq. (3.27)]; its derivative ψ(a) = Ψ ′1(a) = Γ ′(a)/Γ (a) is called the digamma function

and its higher derivatives ψ(m)(a) = Ψ(m+1)1 (a) are called polygamma functions. In

Corollary 4 below we derive the polynomial expansions

Ψr(a) =∑n�0n �=r

(−1)nB(n)n (a)

n!(n− r) + (−1)rB(r)r (a)

r! γ + Pr−1(a), (1.7)

where γ is the Euler–Mascheroni constant and Pr−1(a) is an explicitly given polynomialof degree r − 1, and similar expressions for the derivatives Ψ

(m)r (a). Again each term in

the series (1.7) is a polynomial in a and a rational function of the order r.By Lemma 2 below, the rate of convergence of the series (1.2), (1.7) is comparable to

that of the series ζ(a + 1 − ε) =∑

n n−(a+1−ε) for any ε > 0; the rate of convergence is

driven by �(a) up to logarithmic factors dependent on �(s). Our expansion of ζr(s, a) is

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 3997

influenced by recent expansions of Rubinstein [14,15] for Γ (s) and ζ1(s, a); however, weavoid the beta function factors which appear in the expansion of [15] by means of thetheory of Bernoulli polynomials of the second kind [4,8,9], while maintaining the samerate of convergence. In Section 4 below we give series for many of the special values ofthese transcendental functions in terms of higher order Bernoulli polynomials. Then inSection 5 we illustrate how many of the well-known properties of these functions areevidenced by the series (1.2), (1.7).

2. Notations and preliminaries

The Barnes multiple zeta functions [3,16] are defined for �(a) > 0 and �(s) > r

by (1.1) and extend to meromorphic functions of s ∈ C with simple poles at s = 1, 2,. . . , r; by convention we have ζ0(s, a) = a−s. The values of ζr(s, a) at the negative integersare given in terms of order r Bernoulli polynomials by (1.4). They satisfy a differenceequation

ζr(s, a) − ζr(s, a + 1) = ζr−1(s, a) (2.1)

for positive integers r, and their a-derivatives correspond to an s-shift

∂

∂aζr(s, a) = −sζr(s + 1, a) (2.2)

[16, Eq. (3.11)] for nonnegative integers r. Their s-derivatives at s = 0 give the multiplelog gamma functions Ψr(a) as in (1.5), which are analytic functions of a for �(a) > 0satisfying the difference equation

Ψr(a) − Ψr(a + 1) = Ψr−1(a) (2.3)

for positive integers r [3,16]. When r = 0 we have Ψ0(a) = − log a, and Ψ1(a) is givenby (1.6).

The gamma function Γ (s) is defined for �(s) > 0 by the r = 0, a = 1 case ofintegral (3.1) and extends to a meromorphic function on C with simple poles at thenonpositive integers. Its logarithmic derivative

Γ ′(a)Γ (a) = Ψ ′

1(a) = ψ(a) (2.4)

is called the digamma function, the negative of whose value at a = 1 gives the Euler–Mascheroni constant

ψ(1) = Ψ ′1(1) = Γ ′(1) = −γ = lim

n→∞

(log n−

n∑ 1k

). (2.5)

k=1

3998 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

Differentiating the functional equation Γ (s + 1) = sΓ (s) and letting s → 0 shows that−γ is also the constant term in the Laurent expansion of Γ (s) about s = 0, namely

−γ = lims→0

(Γ (s) − 1

s

). (2.6)

Similarly, differentiating (2.2) with respect to s and letting s → 0 shows that −ψ(a) isthe constant term in the Laurent expansion of ζ1(s, a) about s = 1, namely

−ψ(a) = lims→1

(ζ1(s, a) −

1s− 1

). (2.7)

The order z Bernoulli polynomials B(z)n (x) are defined [12,4] by (1.3); these are poly-

nomials of degree n in x and of degree n in the order z. They satisfy a difference equation

B(z)n (x + 1) −B(z)

n (x) = nB(z−1)n−1 (x) (2.8)

[4, Eq. (1.5)] and derivative identity

∂

∂xB(z)

n (x) = nB(z)n−1(x) (2.9)

[4, Eq. (1.6)]. Their dual companions are the order z Bernoulli polynomials of the secondkind b

(z)n (x), which are defined [4,13] by the generating function(

t

log(1 + t)

)z

(1 + t)x =∞∑

n=0b(z)n (x)tn. (2.10)

These are also polynomials of degree n in x and of degree n in the order z; Carlitz [4]originally used the notation β

(z)n (x) for n!b(z)n (x), but the latter is in better agreement

with the original motivation and notation of Jordan [8], and seems to have become morecommon [9,1,7,13,11] since it avoids confusion with the degenerate Bernoulli numbers ofCarlitz. Several recent papers have also referred to the b

(z)n (x) as “Cauchy numbers” or

“Cauchy polynomials” (cf. e.g. [10,18]). When z = 1 or x = 0 that part of the notationis often suppressed, so that B(z)

n denotes B(z)n (0), bn(x) denotes b

(1)n (x), and Bn denotes

B(1)n (0). The b

(z)n (x) satisfy a difference equation

b(z)n (x + 1) − b(z)n (x) = b(z)n−1(x) (2.11)

[4, Eq. (2.4)] and derivative identity

∂

∂xb(z)n (x) = b

(z−1)n−1 (x) (2.12)

[4, Eq. (2.3)]. The polynomials B(z)n (x) and b

(z)n (x) may be used interchangeably; we

initially derive our series for Γ (s)ζr(s, a) using a change of variable to expand an

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 3999

“exponential” integral (3.1) in terms of the “logarithmic” sequence {b(z)n (x)}, and thenconvert back to {B(z)

n (x)} by means of Carlitz’s identities

n!b(z)n (x) = B(n−z+1)n (x + 1), B(z)

n (x) = n!b(n−z+1)n (x− 1) (2.13)

[4, Eqs. (2.11), (2.12)] in order to emphasize classical relationships such as (1.4) betweenζr(s, a) and B

(r)n (a).

3. Demonstration of theorems

In this section we prove the series expansions (1.2) and (1.7) for the multiple zeta andlog gamma functions.

Theorem 1. If �(a) > 0 and s ∈ C \ {r, r − 1, . . . , 1, 0,−1,−2, . . .} then

Γ (s)ζr(s, a) =∞∑

n=0

(−1)nB(n+s)n (a)

n!(n + s− r)

for all nonnegative integers r.

Proof. For �(s) > r and �(a) > 0 we begin with the Mellin transform integral formula

Γ (s)ζr(s, a) =∞∫0

tse−at

(1 − e−t)rdt

t(3.1)

[16, Eq. (3.2)], and make the change of variables u = 1− e−t, following [14]. From (2.10)we then obtain

Γ (s)ζr(s, a) =1∫

0

(− log(1 − u)

)s−1u−r(1 − u)a−1 du

=1∫

0

(log(1 − u)

−u

)s−1(1 − u)a−1us−r−1 du

=1∫

0

∞∑n=0

(−1)nb(1−s)n (a− 1)un+s−r−1 du

=∞∑

n=0(−1)nb(1−s)

n (a− 1)1∫

0

un+s−r−1 du

=∞∑ (−1)nb(1−s)

n (a− 1)n + s− r

, (3.2)

n=0

4000 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

once we show that this series converges. We will then deduce the equation of the theoremfrom Carlitz’s identity (2.13).

The convergence of the series

∞∑n=0

(−1)nb(1−s)n (a− 1)

n + s− r(3.3)

was proved by Rubinstein [14] in the case a = r = 1, via the bound

∣∣b(1−s)n (0)

∣∣ � cs(1 + log(n + 1))|s|+1

n + 1 , (3.4)

where cs is a constant depending only on s; that paper used the notation αk(s) to denote(−1)kb(1−s)

k (0). Lemma 2 below demonstrates the convergence of this series for �(s) > r

and �(a) > 0, except for the cases where a is a positive integer, by comparison withthe series

∑n n

−(a+1−ε) for any ε > 0. In the case of positive integer a we observe fromCarlitz’s identity

b(1−s)n (a) =

n∑r=0

(a)nn! b

(1−s)n−r (0) (3.5)

[4, Eq. (2.6)], where (a)n = a(a − 1) · · · (a − n + 1), that b(1−s)n (a − 1) is a sum of a

terms of the form b(1−s)k (0), with coefficients bounded independent of n; combined with

Rubinstein’s bound (3.4), this shows that the series (3.3) converges when a ∈ Z+, at a

rate comparable to∑

n n−(2−ε) for any ε > 0.

By means of Rubinstein’s bound (3.4) and the estimate of [6,18] (Lemma 2 below)we have demonstrated the uniform absolute convergence of the series (3.3) on compactsubsets of �(a) > 0 and �(s) > r, showing that the rearrangement of integration andsummation in Eq. (3.2) is valid for �(a) > 0 and �(s) > r. However, for fixed a with�(a) > 0 the series (3.3) converges absolutely and uniformly for s in compact subsetsof C \ {r, r− 1, . . . , 1, 0,−1,−2, . . .}, and therefore this series provides the meromorphiccontinuation of Γ (s)ζr(s, a) to all s ∈ C. This completes the proof of the theorem. �

The convergence of the series (3.3) for �(a) > 0 and s ∈ C may be established by thefollowing asymptotic estimate [6,18], together with the observation that logn = o(nε)for every ε > 0.

Lemma 2. For s ∈ C and a ∈ C \ Z+ we have the asymptotic estimate

∣∣b(1−s)n (a− 1)

∣∣ ∼ 1|(logn)snaΓ (1 − a)|

as n → ∞.

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 4001

Proof. This is Lemma 3.2 of [18]; see also Lemma 1 of [6]. �The r = 1 case of Theorem 1 immediately gives a series representation for Dirichlet

L-functions. For nontrivial Dirichlet characters χ of conductor f , the series below con-verges somewhat slowly, at a rate comparable to that of ζ(1 + (1/f)) =

∑n n

−(1+(1/f))

up to a logarithmic factor depending on s.

Corollary 3. If χ is a nontrivial Dirichlet character of conductor f then for all s ∈ C wehave

L(s, χ) = 1Γ (s)fs

∞∑n=1

(−1)n

n!(n + s− 1)

f∑a=1

χ(a)B(n+s)n (a/f).

Proof. For nontrivial Dirichlet characters χ of conductor f we have the formula

L(s, χ) :=∞∑

m=1χ(m)m−s = f−s

f∑a=1

χ(a)ζ1(s, a/f) (3.6)

for �(s) > 0 [17, p. 30]. The corollary then follows immediately from the r = 1 case ofTheorem 1; the n = 0 term in the series vanishes since

∑fa=1 χ(a) = 0 for nontrivial

characters of conductor f , leaving an everywhere-convergent expansion for the entirefunction L(s, χ). �

We now use Theorem 1 to derive the series expansion (1.7) for the multiple log gammafunctions and their derivatives.

Corollary 4. For �(a) > 0 the function Ψr(a) satisfies

Ψr(a) =∑n�0n �=r

(−1)nB(n)n (a)

n!(n− r) + (−1)r

r!

[γB(r)

r (a) + ∂

∂sB(r+s)

r (a)]s=0

.

For integers m with 1 � m � r its m-th derivative Ψ(m)r (a) satisfies

Ψ (m)r (a) =

∑n�0

n �=r−m

(−1)m+nB(m+n)n (a)

n!(n + m− r) + (−1)r

(r −m)!

[γB

(r)r−m(a) + ∂

∂sB

(r+s−m)r−m (a)

]s=m

,

and for m � r + 1 we have

Ψ (m)r (a) =

∞∑n=0

(−1)m+nB(m+n)n (a)

n!(n + m− r) .

4002 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

Proof. We begin the series expansion

Γ (s)ζr(s, a) =∞∑

n=0

(−1)nB(n+s)n (a)

n!(n + s− r) (3.7)

of Theorem 1. Using the Laurent expansion (2.6) we may express the left side of (3.7) inthe form

Γ (s)ζr(s, a) = ζr(s, a)s

− γζr(s, a) +∞∑

n=1ans

n (3.8)

near s = 0, where f(s) =∑∞

n=1 ansn is analytic and vanishes at s = 0. We now subtract

the n = r term of the sum on the right side of (3.7) from both sides of that equation.On the left side we obtain

ζr(s, a)s

− (−1)rB(r+s)r (a)

r!s − γζr(s, a) + f(s). (3.9)

Using (1.4) we rewrite this expression as

ζr(s, a) − ζr(0, a)s

+ (−1)r

r!

(B

(r)r (a) −B

(r+s)r (a)

s

)− γζr(s, a) + f(s). (3.10)

By (1.5) and (1.4), the limit as s → 0 of this expression is

Ψr(a) −(−1)r

r!

[∂

∂sB(r+s)

r (a)]s=0

− γζr(0, a)

= Ψr(a) −(−1)r

r!

[γB(r)

r (a) + ∂

∂sB(r+s)

r (a)]s=0

(3.11)

and the limit as s → 0 of the right side of (3.7), with the n = r term removed, is

∑n�0n �=r

(−1)nB(n)n (a)

n!(n− r) , (3.12)

from which the first statement follows. The remaining statements follow by differentiatingthis series for Ψr(a) using (2.9). �Remark. The s-derivative term in the above series for Ψ

(m)r for 0 � m � r may be

evaluated as[∂

∂sB

(r+s−m)r−m (a)

]=

r−m∑ (r −m

k

)(−1)k+1Bk

kB

(r)r−m−k(a) (3.13)

s=m k=1

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 4003

where Bk = B(1)k (0) is the ordinary Bernoulli number; this may be obtained by differen-

tiating both sides of Ericksen’s exponential representation

∞∑n=0

B(z)n (x) t

n

n! = exp(xt + z

∞∑n=1

(−1)n+1Bntn

n · n!

)(3.14)

[5, Eq. (19)] with respect to z and equating coefficients of tn/n!, yielding

∂

∂zB(z)

n (x) =n∑

k=1

(n

k

)(−1)k+1Bk

kB

(z)n−k(x), (3.15)

as desired.

4. Special cases

In this section we give selected cases of special values of transcendental functions whichmay be expressed as series of higher order Bernoulli numbers by means of Theorem 1,Corollary 3, and Corollary 4. Special cases of Theorem 1 with r = 0 include

Corollary 5. For �(a) > 0 and any positive integer m we have

∞∑n=0

(−1)nB(n+m)n (a)

n!(n + m) = (m− 1)!am

and for �(a) > 0 we have

∞∑n=0

(−1)nB(n+(1/2))n (a)

n!(2n + 1) = 12

√π

a.

Equivalently we may write

∞∑n=0

(−1)nb(1/2)n (a− 1)(2n + 1) = 1

2

√π

a.

Proof. The first statement is obtained by taking r = 0 and s = m ∈ Z+ in Theorem 1;

the second statement is obtained by taking r = 0 and s = 1/2, using Γ (1/2) =√π.

The last statement is a restatement of the second by means of (2.13). (We remarkthat we are unaware of any other known identity involving fractional order Bernoullipolynomials.) �

Special cases of Theorem 1 with r = 1 include

4004 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

Corollary 6.

∞∑n=0

(−1)nB(n+2)n (1)

n!(n + 1) = π2

6 and∞∑

n=0

(−1)nB(n+4)n (1)

n!(n + 3) = π4

15 .

More generally for every positive integer k we have

∞∑n=0

(−1)nB(n+2k)n (1)

n!(n + 2k − 1) = (−1)k+1(2π)2kB2k

4k

where B2k = B(1)2k (0) is the ordinary (order 1) Bernoulli number.

Proof. These equations all follow from the r = 1, a = 1 cases of Theorem 1 by means ofEuler’s formula

ζ(2k) = (−1)k+1(2π)2kB2k

2(2k)! (4.1)

for the values of ζ(s) = ζ1(s, 1) at the positive even integers. �Here we give a special case arising from Corollary 3 for the nontrivial Dirichlet char-

acter of conductor 4; the first series below, consisting only of positive terms, convergesslowly to the value π. Many similar series may be derived from Corollary 3 and theidentities in [2].

Corollary 7.

∞∑n=1

(−1)n

n · n!(B(n+1)

n (1/4) −B(n+1)n (3/4)

)= π

and∞∑

n=1

(−1)n

n!(n + 2)(B(n+3)

n (1/4) −B(n+3)n (3/4)

)= 4π3.

Proof. If χ denotes the nontrivial Dirichlet character of conductor 4, then from Corol-lary 3 we have

L(s, χ) = 1Γ (s)4s

∞∑n=1

(−1)n

n!(n + s− 1)(B(n+s)

n (1/4) −B(n+s)n (3/4)

)(4.2)

for any s ∈ C. The first result above follows by evaluating this series at s = 1 and notingthat

L(1, χ) =∞∑ (−1)m−1

2m− 1 = arctan 1 = π

4 . (4.3)

m=1

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 4005

The second follows by evaluating (4.2) at s = 3 and comparing with the value L(3, χ) =π3/32 [2, Eq. (5.11)]. �Remark. For purposes of computation at positive integer values of s, it is easy to accel-erate the convergence of series such as (4.2) by summing separately over the first severalperiods of the character. For any nonnegative integer N , writing

L(s, χ) =∞∑

m=1χ(m)m−s =

Nf∑a=1

χ(a)a−s +∞∑

m=Nf+1χ(m)m−s (4.4)

we have

L(s, χ) =Nf∑a=1

χ(a)a−s + f−s

f∑a=1

χ(a)ζ1(s,N + a

f

), (4.5)

which leads to a modified Corollary 3, namely,

L(s, χ) =Nf∑a=1

χ(a)a−s + 1Γ (s)fs

∞∑n=1

(−1)n

n!(n + s− 1)

f∑a=1

χ(a)B(n+s)n (N + a/f). (4.6)

The advantage is that the series in (4.6) then converges like∑

n n−(N+1+(1/f)) rather

than like∑

n n−(1+(1/f)). In the case where χ is the nontrivial character of conductor 4

and s = 1, 3, for example, (4.6) with N = 1 reads

4L(1, χ) = 83 +

∞∑n=1

(−1)n

n · n!(B(n+1)

n (5/4) −B(n+1)n (7/4)

)= π (4.7)

and

128L(3, χ) = 128 · 2627 +

∞∑n=1

(−1)n

n!(n + 2)(B(n+3)

n (5/4) −B(n+3)n (7/4)

)= 4π3, (4.8)

which are equivalent to the series of the above corollary but converge much more rapidly.

Among the special cases of Corollary 4 with r = 0 we have the following, whichgeneralize the a = 1 cases which were given by Jordan [9, pp. 280, 277].

Corollary 8. For �(a) > 0 we have

γ + log a =∞∑ (−1)n+1B

(n)n (a)

n · n! =∞∑ (−1)n+1bn(a− 1)

n

n=1 n=1

4006 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

and

log(

1 + 1a

)=

∞∑n=0

(−1)nB(n)n (a)

(n + 1)! =∞∑

n=0

(−1)nbn(a− 1)n + 1

where bn(x) = b(1)n (x) is the ordinary (order 1) Bernoulli polynomial of the second kind.

Proof. The first statement is immediate from Corollary 4 with r = 0. For the second,use Corollary 4 to obtain a series for Ψ0(a) − Ψ0(a + 1) and use (2.8). �

Special cases of Corollary 4 with r = 1 include

Corollary 9. We have a convergent series of positive terms

∞∑n=2

(−1)nB(n)n (1)

n!(n− 1) =∞∑

n=2

(−1)nbnn− 1 = 1

2(γ + 1 − log(2π)

).

Furthermore, for �(a) > 0 the digamma function ψ(a) satisfies

γ + ψ(a) = γ + Γ ′(a)Γ (a) =

∞∑n=1

(−1)n+1B(n+1)n (a)

n · n! ,

and the polygamma function ψ(m)(a) satisfies

ψ(m)(a) =∞∑

n=0

(−1)n+m+1B(n+m+1)n (a)

n!(n + m)

for all positive integers m.

Proof. Using (3.13), the r = 1, m = 0 case of Corollary 4 reads

Ψ1(a) = log(Γ (a)√

2π

)=

∑n�0n �=1

(−1)nB(n)n (a)

n!(n− 1) −(a− 1

2

)γ + 1

2

=∞∑

n=2

(−1)nB(n)n (a)

n!(n− 1) −(a− 1

2

)γ − 1

2

=∞∑

n=2

(−1)nbn(a− 1)n− 1 −

(a− 1

2

)γ − 1

2 , (4.9)

of which the first statement is the a = 1 case. Since ψ(m) = Ψ(m+1)1 , the remaining

statements restate the r = 1, m � 1 cases of Corollary 4; the ψ = Ψ ′1 case follows from

the corollary by observing that B(1)0 (a) = 1. �

P.T. Young / Journal of Number Theory 133 (2013) 3995–4009 4007

We remark that all these identities may be expressed in terms of b(z)n (x) by means

of (2.13); in the above identities we have given the b(z)n (x) forms in the cases when the

order z is positive. For the purpose of computing the series, it may be preferable toexpress these series in their b(z)n (x) form so that the order z is constant; for example, thelast equation of Corollary 9 is equivalent to

ψ(m)(a) =∞∑

n=0

(−1)n+m+1B(n+m+1)n (a)

n!(n + m) =∞∑

n=0

(−1)n+m+1b(−m)n (a− 1)

(n + m) . (4.10)

5. Properties of multiple zeta and log gamma functions

In this final section we illustrate briefly how several well-known properties of thesefunctions are naturally expressed by the series in Theorem 1 and Corollary 4. The caser = 0, a = 1 of Theorem 1 reads

Γ (s) =∞∑

n=0

(−1)nB(n+s)n (1)

n!(n + s) . (5.1)

For nonnegative integers k, multiplying both sides by s + k and taking the limit ass → −k yields

lims→−k

(s + k)Γ (s) = (−1)k

k! B(0)k (1),

but since it is clear from (1.3) that B(0)k (1) = 1 for all k, we recover the well-known

residues

Ress=−k

Γ (s) = (−1)k

k! (5.2)

of the gamma function at its poles at the nonpositive integers. Then returning to ther > 0 cases of Theorem 1, multiplying both sides by s+k and taking the limit as s → −k

yields

ζr(−k, a) · Ress=−k

Γ (s) =(−1)r+kB

(r)r+k(a)

(r + k)! , (5.3)

which gives the values (1.4) of ζr(s, a) at the nonpositive integers by comparisonwith (5.2). Similarly, when r > 0 and k ∈ {1, 2, . . . , r}, multiplying both sides of Theo-rem 1 by s− k and taking the limit as s → k yields the residues

Res ζr(s, a) =(−1)r−kB

(r)r−k(a) (5.4)

s=k (k − 1)!(r − k)!

4008 P.T. Young / Journal of Number Theory 133 (2013) 3995–4009

[16, Eq. (3.9)] of ζr(s, a) at each of its r poles at s = 1, . . . , r. Furthermore, lettings → −k in Corollary 3 gives the well-known values

L(−k, χ) = −fk

k + 1

f∑a=1

χ(a)Bk+1(a/f) = −Bk+1,χ

k + 1 (5.5)

[17, p. 31] at the nonpositive integers of Dirichlet L-function for a nontrivial character χof conductor f in terms of the generalized Bernoulli numbers Bn,χ, which may be defined[17, p. 30] by

f∑a=1

χ(a)teat

eft − 1 =∞∑

n=0Bn,χ

tn

n! . (5.6)

The fundamental difference equations (2.1) and (2.3) for ζr(s, a) and Ψr(a) are readilyseen to be equivalent via Theorem 1 and Corollary 4 to the difference equation (2.8) forthe order z Bernoulli polynomials. The derivative-shift property (2.2) of ζr(s, a) is alsomanifest in Theorem 1 by means of the derivative relation (2.9) of the order z Bernoullipolynomials and the translation functional equation Γ (s + 1) = sΓ (s).

The functions ζr(s, a) are traditionally defined as multiple Dirichlet series (1.1) and assuch require r to be a nonnegative integer; however the series in Theorem 1 makes sensefor any complex r. Therefore the functions ζr(s, a) and Ψr(a) could in fact be defined bymeans of Theorem 1 and (1.5) with arbitrary complex order r. With such a definitionthe difference equations (2.1) and (2.3) for ζr(s, a) and Ψr(a) are still satisfied, as is thederivative-shift property (2.2) of ζr(s, a).

Appendix A. Supplementary material

The online version of this article contains additional supplementary material. Pleasevisit http://dx.doi.org/10.1016/j.jnt.2013.05.016.

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