Generating Function Identities for $\zeta(2n+2), \zeta(2n+3)$ via the WZ Method

Abstract

Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and
Bailey-Borwein-Bradley's identities for generating functions of the sequences
$\{\zeta(2n+2)\}_{n\ge 0}, \{\zeta(2n+3)\}_{n\ge 0}.$ By the same method we
give several new representations for these generating functions yielding faster
convergent series for values of the Riemann zeta function.

Full text

arXiv:0801.1591v2 [math.NT] 17 Jan 2008
GENERATING FUNCTION IDENTITIES FOR ζ(2n+ 2), ζ (2n+ 3) VIA
THE WZ METHOD
KH. HESSAMI PILEHROOD1AND T. HESSAMI PILEHROOD2
Abstract. Using the WZ method we present simpler proofs of Koecher’s, Leshchiner’s
and Bailey-Borwein-Bradley’s identities for generating functions of the sequences {ζ(2n+
2)}n≥0,{ζ(2n+ 3)}n≥0.By the same method we give several new representations for
these generating functions yielding faster convergent series for values of the Riemann
zeta function.
1. Introduction
The Riemann zeta function is defined by the series
ζ(s) =
∞
X
n=1
1
ns,for Re(s)>1.
Ap´ery’s irrationality proof of ζ(3) and series acceleration formulae for the first values of
the Riemann zeta function going back to Markov’s work [8]
ζ(2) = 3
∞
X
k=1
1
k22k
k, ζ(3) = 5
2
∞
X
k=1
(−1)k−1
k32k
k, ζ(4) = 36
17
∞
X
k=1
1
k42k
k
stimulated intensive search of similar formulas for other values ζ(n), n ≥5.Many Ap´ery-
like formulae have been proved with the help of generating function identities (see [6, 1,
5, 9, 4]). M. Koecher [6] (and independently Leshchiner [7]) proved that
(1)
∞
X
k=0
ζ(2k+ 3)a2k=
∞
X
n=1
1
n(n2−a2)=1
2
∞
X
k=1
(−1)k−1
k32k
k
5k2−a2
k2−a2
k−1
Y
m=1 1−a2
m2,
for any a∈C,with |a|<1.For even zeta values, Leshchiner [7] (in an expanded form)
showed that (see [4, (31)])
(2)
∞
X
k=0 1−1
2k+1 ζ(2k+ 2)a2k=
∞
X
n=1
(−1)n−1
n2−a2=1
2
∞
X
k=1
1
k22k
k
3k2+a2
k2−a2
k−1
Y
m=1 1−a2
m2,
1991 Mathematics Subject Classification. 11M06, 05A10, 05A15, 05A19.
Key words and phrases. Riemann zeta function, Ap´ery-like series, generating function, convergence
acceleration, Wilf-Zeilberger method, WZ pair.
1This research was in part supported by a grant from IPM (No. 86110025).
2This research was in part supported by a grant from IPM (No. 86110020).
1

2 KH. HESSAMI PILEHROOD1AND T. HESSAMI PILEHROOD2
for any complex a, with |a|<1.Recently, D. Bailey, J. Borwein and D. Bradley [4] proved
another formula
(3)
∞
X
k=0
ζ(2k+ 2)a2k=
∞
X
n=1
1
n2−a2= 3
∞
X
k=1
1
2k
k(k2−a2)
k−1
Y
m=1 m2−4a2
m2−a2,
for any a∈C,|a|<1.
In this paper, we present simpler proofs of identities (1)–(3) using the WZ method.
By the same method we give some new representations for the generating functions (1),
(3) yielding faster convergent series for values of the Riemann zeta function.
We recall [10] that a discrete function A(n, k) is called hypergeometric or closed form
(CF) if the quotients
A(n+ 1, k)
A(n, k)and A(n, k + 1)
A(n, k)
are both rational functions of nand k. A pair of CF functions F(n, k) and G(n, k) is
called a WZ pair if
(4) F(n+ 1, k)−F(n, k) = G(n, k + 1) −G(n, k).
We need the following summation formulas.
Proposition 1. ([3, Formula 2]) For any WZ pair (F, G)
∞
X
k=0
F(0, k)−lim
n→∞
n
X
k=0
F(n, k) =
∞
X
n=0
G(n, 0) −lim
k→∞
k
X
n=0
G(n, k),
whenever both sides converge.
Proposition 2. ([3, Formula 3]) For any WZ pair (F, G)we have
∞
X
n=0
G(n, 0) =
∞
X
n=0
(F(n+ 1, n) + G(n, n)) −lim
n→∞
n−1
X
k=0
F(n, k),
whenever both sides converge.
As usual, (λ)νis the Pochhammer symbol (or the shifted factorial) defined by
(λ)ν=Γ(λ+ν)
Γ(λ)=(1, ν = 0;
λ(λ+ 1) ...(λ+ν−1), ν ∈N.
2. Proof of Koecher’s identity
Consider
F(n, k) = (−1)nk!(1 + a)n(1 −a)n
(2n+k+ 1)!((n+k+ 1)2−a2).
Then we have
F(n+ 1, k)−F(n, k) = G(n, k + 1) −G(n, k)
with
G(n, k) = (−1)nk!(1 + a)n(1 −a)n(5(n+ 1)2−a2+k2+ 4k(n+ 1))
(2n+k+ 2)!((n+k+ 1)2−a2)(2n+ 2) ,

GENERATING FUNCTION IDENTITIES FOR ζ(2n+ 2), ζ(2n+ 3) VIA THE WZ METHOD 3
i.e., (F, G) is a WZ pair and by Proposition 1, we get
∞
X
k=0
F(0, k) =
∞
X
n=0
G(n, 0),
or
∞
X
k=1
1
k(k2−a2)=
∞
X
n=0
(−1)n(1 + a)n(1 −a)n(5(n+ 1)2−a2)
(2n+ 2)!(2n+ 2)((n+ 1)2−a2)
=1
2
∞
X
n=1
(−1)n−1(5n2−a2)
n32n
n(n2−a2)
n−1
Y
m=1 1−a2
m2.
3. Proof of Leshchiner’s identity
Consider
F(n, k) = (−1)kk!(1 + a)n(1 −a)n(n+k+ 1)
(2n+k+ 1)!((n+k+ 1)2−a2).
Then we have
F(n+ 1, k)−F(n, k) = G(n, k + 1) −G(n, k)
with
G(n, k) = (−1)kk!(1 + a)n(1 −a)n(3(n+ 1)2+a2+k2+ 4k(n+ 1))
2(2n+k+ 2)!((n+k+ 1)2−a2)
and by Proposition 1, we get
∞
X
k=0
F(0, k) =
∞
X
n=0
G(n, 0),
or
∞
X
k=1
(−1)k−1
k(k2−a2)=1
2
∞
X
n=0
(1 + a)n(1 −a)n(3(n+ 1)2+a2)
(2n+ 2)!((n+ 1)2−a2)
=1
2
∞
X
n=1
3n2+a2
n22n
n(n2−a2)
n−1
Y
m=1 1−a2
m2.
4. Proof of the Bailey-Borwein-Bradley identity
Consider
F(n, k) = n!2(1 + a)k(1 −a)k(1 + 2a)n(1 −2a)n
(2n)!(1 + a)n+k+1(1 −a)n+k+1
.
Then we have
F(n+ 1, k)−F(n, k) = G(n, k + 1) −G(n, k)
with
G(n, k) = (1 + a)k(1 −a)k(1 + 2a)n(1 −2a)nn!(n+ 1)!(3n+ 3 + 2k)
(1 + a)n+k+1(1 −a)n+k+1(2n+ 2)! ,

4 KH. HESSAMI PILEHROOD1AND T. HESSAMI PILEHROOD2
and (F, G) is a WZ pair. Then
∞
X
k=0
F(0, k) =
∞
X
n=0
G(n, 0),
and therefore,
∞
X
k=1
1
(k2−a2)= 3
∞
X
n=0
(1 + 2a)n(1 −2a)n(n+ 1)!2
(1 + a)n+1(1 −a)n+1 (2n+ 2)!
= 3
∞
X
n=1
1
2n
n(n2−a2)
n−1
Y
m=1 m2−4a2
m2−a2,
as required. 
5. New generating function identities for ζ(2n+ 2) and ζ(2n+ 3)
Theorem 1. Let abe a complex number not equal to a non-zero integer. Then
(5)
∞
X
k=1
1
k(k2−a2)=
∞
X
n=1
a4−a2(32n2−10n+ 1) + 2n2(56n2−32n+ 5)
2n32n
n3n
n((2n−1)2−a2)(4n2−a2)
n−1
Y
m=1 a2
m2−1.
Expanding both sides of (5) in powers of a2and comparing coefficients of a2ngives
Ap´ery-like series for ζ(2n+3) for every non-negative integer nconvergent at the geometric
rate with ratio 1/27.In particular, comparing constant terms recovers Amdeberhan’s
formula [2] for ζ(3)
ζ(3) = 1
4
∞
X
n=1
(−1)n−156n2−32n+ 5
n3(2n−1)22n
n3n
n.
Similarly, comparing coefficients of a2gives
ζ(5) = 3
16
∞
X
n=1
(4n−1)(16n3−8n2+ 4n−1)
(−1)n−1n5(2n−1)42n
n3n
n+1
4
∞
X
n=1
(−1)n(56n2−32n+ 5)
n3(2n−1)22n
n3n
n
n−1
X
k=1
1
k2.
Proof. Consider
F(n, k) = (−1)nn!(2n)!k!(1 + a)k(1 −a)k(1 + a)n(1 −a)n(1 + a)2n(1 −a)2n
(3n)!(2n+k+ 1)!(1 + a)2n+k+1(1 −a)2n+k+1
.
Then application of the WZ algorithm produces the WZ mate
G(n, k) = (−1)nk!n!(2n)!(1 + a)k(1 −a)k(1 + a)n(1 −a)n(1 + a)2n(1 −a)2n
6(3n+ 2)!(2n+k+ 2)!(1 + a)2n+k+2(1 −a)2n+k+2
q(n, k)
satisfying (4), with
q(n, k) = 2(2n+ 1)(a4−a2(32n2+ 54n+ 23) + 2(n+ 1)2(56n2+ 80n+ 29))
+k4(9n+ 6) + k3(90n2+ 132n+ 48) + k2(348n3+ 792n2−15a2n+ 594n+ 147
−9a2) + k(624n4+ 1932n3+ 2214n2−84a2n2−117a2n+ 1113n+ 207 −39a2).

GENERATING FUNCTION IDENTITIES FOR ζ(2n+ 2), ζ(2n+ 3) VIA THE WZ METHOD 5
By Proposition 1, we have
∞
X
k=0
F(0, k) =
∞
X
n=0
G(n, 0)
or equivalently,
∞
X
k=1
1
k(k2−a2)=
∞
X
n=0
(−1)nn!(1 −a)n(1 + a)n(a4−a2(32n2+ 54n+ 23) + 2(n+ 1)2(56n2+ 80n+ 29))
2(3n+ 3)!((2n+ 1)2−a2)((2n+ 2)2−a2),
and the theorem follows. 
Theorem 2. Let abe a complex number not equal to a non-zero integer. Then
(6)
∞
X
k=1
1
k2−a2=
∞
X
n=1
n2(21n−8) −a2(9n−2)
2n
nn(n2−a2)(4n2−a2)
n−1
Y
k=1 k2−4a2
(k+n)2−a2.
Formula (6) generates Ap´ery-like series for ζ(2n+ 2) for every non-negative integer n
convergent at the geometric rate with ratio 1/64.In particular, it follows that
(7) ζ(2) =
∞
X
n=1
21n−8
n32n
n3
and
ζ(4) =
∞
X
n=1
69n−32
4n52n
n3−
∞
X
n=1
21n−8
n32n
n3
n−1
X
k=1 4
k2−1
(k+n)2.
Another proof of formula (7) can be found in [10, §12].
Proof. Consider
F(n, k) = n!2(1 + 2a)n(1 −2a)n(1 + a)n+k(1 −a)n+k
(2n)!(1 + a)2n+k+1(1 −a)2n+k+1
.
Application of the WZ algorithm produces the WZ mate
G(n, k) = n!2(1 + a)n+k(1 −a)n+k(1 + 2a)n(1 −2a)n
2(2n+ 1)!(1 + a)2n+k+2(1 −a)2n+k+2
q(n, k)
satisfying (4), with
q(n, k) = (n+ 1)2(21n+ 13) −a2(9n+ 7) + 2k3+k2(13n+ 11) + k(28n2+ 48n+ 20 −2a2).
By Proposition 1,
∞
X
k=0
F(0, k) =
∞
X
n=0
G(n, 0),
which implies (6). 
Theorem 3. Let abe a complex number not equal to a non-zero integer. Then
∞
X
k=1
1
k(k2−a2)=1
4
∞
X
n=0
(1 + a)2
n(1 −a)2
n((n+ 1)2(30n+ 19) −a2(12n+ 7))
(1 + a)2n+2(1 −a)2n+2 (n+ 1)(2n+ 1) .

6 KH. HESSAMI PILEHROOD1AND T. HESSAMI PILEHROOD2
Proof. Consider
F(n, k) = (1 + a)k(1 −a)k(1 + a)2
n(1 −a)2
n
(1 + a)2n+k+1(1 −a)2n+k+1 (n+k+ 1).
Then application of the WZ algorithm produces the WZ companion
G(n, k) = (1 + a)k(1 −a)k(1 + a)2
n(1 −a)2
nq(n, k)
4(1 + a)2n+k+2(1 −a)2n+k+2 (n+k+ 1)(n+ 1)(2n+ 1),
with
q(n, k) = (n+ 1)3(30n+ 19) −a2(n+ 1)(12n+ 7) + 2k3(n+ 1) + 2k2(7n2+ 13n+ 6)
+k(34n3+ 93n2+ 84n−4a2n+ 25 −3a2).
Now by Proposition 1, the theorem follows. 
Theorem 4. Let abe a complex number not equal to a non-zero integer. Then
(8)
∞
X
k=1
1
k(k2−a2)= 2
∞
X
n=1
(−1)n−1
n32n
n5
p(n, a)
(n2−a2)(4n2−a2)
n−1
Y
m=1 (1 −a2/m2)2
1−a2/(n+m)2,
where
p(n, a) = a4−a2(62n2−40n+ 8) + n2(205n2−160n+ 32).
Formula (8) generates Ap´ery-like series for ζ(2n+ 3), n ≥0,convergent at the geometric
rate with ratio 2−10.In particular, if a= 0 we get the formula of Amdeberhan and
Zeilberger [3]
ζ(3) = 1
2
∞
X
n=1
(−1)n−1(205n2−160n+ 32)
n52n
n5.
Comparing coefficients of a2leads to
ζ(5) =
∞
X
n=1
(−1)n(31n2−20n+ 4)
n72n
n5
+
∞
X
n=1
(−1)n(205n2−160n+ 32)
n52n
n5 n−1
X
m=1
1
m2−
n
X
m=0
1
2(m+n)2!.
Proof. Consider
F(n, k) = (−1)k(1 + a)k(1 −a)k(1 + a)2
n(1 −a)2
n(2n−k−1)!k!n!2
2(n+k+ 1)!2(2n)!(1 + a)2n(1 −a)2n
.
Then
G(n, k) = (−1)k(1 + a)k(1 −a)k(1 + a)2
n(1 −a)2
n(2n−k)!k!n!2q(n, k)
4(2n+ 1)!(n+k+ 1)!2(1 + a)2n+2(1 −a)2n+2
,
with
q(n, k) = (n+ 1)3(30n+ 19) −a2(n+ 1)(12n+ 7) + k(21n3+ 55n2+ 47n+ 13 −3a2n−a2),

GENERATING FUNCTION IDENTITIES FOR ζ(2n+ 2), ζ(2n+ 3) VIA THE WZ METHOD 7
is a WZ mate such that
∞
X
n=0
G(n, 0) =
∞
X
n=0
(1 + a)2
n(1 −a)2
n((n+ 1)2(30n+ 19) −a2(12n+ 7))
4(n+ 1)(2n+ 1)(1 + a)2n+2(1 −a)2n+2
=
∞
X
k=1
1
k(k2−a2),
by Theorem 3. Now by Proposition 2, the theorem follows. 
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(1996), #R13.
[3] T. Amdeberhan, D. Zeilberger, Hypergeometric series acceleration via the WZ method, Electron. J.
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zeta function by infinite series damped by central binomial coefficients, August 1, 2002.
http://www.math.umaine.edu/faculty/bradley/papers/bivar5.pdf
[6] M. Koecher, Letter (German), Math. Intelligencer, 2(1979/1980), no. 2, 62-64.
[7] D. Leshchiner, Some new identities for ζ(k),J. Number Theory, 13 (1981), 355-362.
[8] A. A. Markoff, M´emoir´e sur la transformation de s´eries peu convergentes en s´eries tres convergentes,
M´em. de l’Acad. Imp. Sci. de St. P´etersbourg, t. XXXVII, No.9 (1890), 18pp.
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Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
Current address : Mathemetics Department, Faculty of Science, Shahrekord University, Shahrekord,
P.O. Box 115, Iran.
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