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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

k22k

k, ζ(3) = 5

2

∞

X

k=1

(−1)k−1

k32k

k, ζ(4) = 36

17

∞

X

k=1

1

k42k

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

k32k

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

k22k

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)

n32n

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

n22n

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)

2n32n

n3n

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)22n

n3n

n.

Similarly, comparing coeﬃcients of a2gives

ζ(5) = 3

16

∞

X

n=1

(4n−1)(16n3−8n2+ 4n−1)

(−1)n−1n5(2n−1)42n

n3n

n+1

4

∞

X

n=1

(−1)n(56n2−32n+ 5)

n3(2n−1)22n

n3n

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

nn(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

n32n

n3

and

ζ(4) =

∞

X

n=1

69n−32

4n52n

n3−

∞

X

n=1

21n−8

n32n

n3

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

n32n

n5

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)

n52n

n5.

Comparing coeﬃcients of a2leads to

ζ(5) =

∞

X

n=1

(−1)n(31n2−20n+ 4)

n72n

n5

+

∞

X

n=1

(−1)n(205n2−160n+ 32)

n52n

n5 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.

References

[1] G. Almkvist, A. Granville, Borwein and Bradley’s Ap´ery-like formulae for ζ(4n+ 3),Experiment.

Math., 8(1999), 197-203.

[2] T. Amdeberhan, Faster and faster convergent series for ζ(3),Electron. J. Combinatorics 3(1)

(1996), #R13.

[3] T. Amdeberhan, D. Zeilberger, Hypergeometric series acceleration via the WZ method, Electron. J.

Combinatorics 4(2) (1997), #R3.

[4] D. H. Bailey, J. M. Borwein, D. M. Bradley, Experimental determination of Ap´ery-like identities for

zeta(2n+ 2),Experiment. Math. 15 (2006), no. 3, 281-289.

[5] D. M. Bradley, More Ap´ery-like formulae: On representing values of the Riemann

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.

[9] T. Rivoal, Simultaneous generation of Koecher and Almkvist-Grainville’s Ap´ery-like formulae, Ex-

periment. Math., 13 (2004), 503-508.

[10] D. Zeilberger, Closed form (pun intended!), Contemporary Math. 143 (1993), 579-607.

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.

E-mail address:hessamik@ipm.ir, hessamit@ipm.ir, hessamit@gmail.com