Simultaneous Generation of Koecher and Almkvist-Granville s Apéry-Like Formulae

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1 Simultaeous Geeratio of Koecher ad Almvist-Graville s Apéry-Lie Formulae T. Rivoal CONTENTS. Itroductio. First Step 3. Secod Step 4. Third Step 5. The Fial Step Acowledgmets Refereces We prove a very geeral idetity cojectured by Heri Cohe ivolvig the geeratig fuctio of the family (ζ(r+4s+3 rs 0 : it uifies two idetities proved by Koecher i 980 ad Almvist ad Graville i 999 for the geeratig fuctios of (ζ(r+3 r 0 ad (ζ(4s+3 s 0 respectively. As a cosequece we obtai that for ay iteger j 0 there exists at least [j/] + Apéry-lie formulae for ζ(j +3.. INTRODUCTION I provig that ζ(3 /3 is irratioal Apéry [Apéry 79] oted that ζ(3 5 ( + 3. ( Although the series o the right-had side coverges much faster tha the defiig series for ζ(3 Formula ( is ot essetial i Apéry s proof sice trucatios of this series are ot diophatie approximatios to ζ(3. O the other had it is very liely that ( was a source of ispiratio for Apéry ad may authors have looed for similar idetities i the hope that they might give some idea of how to prove the irratioality of ζ(s + /s+ for ay iteger s ; see for example [Borwei ad Bradley 97 Cohe 8 Koecher 80 Leshchier 8 va der Poorte 80]. This problem is far from beig solved but may beautiful Apéry-lie formulae have bee proved. I fact two apparetly urelated families of such formulae for ζ(s+3 ad ζ(4s+3 have emerged both of which are more easily explaied 000 AMS Subject Classificatio: Primary M06; Secodary 05A5 J7 Keywords: Riema zeta fuctio Apéry-lie series geeratig fuctios See [Cohe 78 va der Poorte 79] for a detailed explaatio of Apéry s origial method. We ow ow that ifiitely may of the values ζ(s + (s are Q-liearly idepedet [Ball ad Rivoal 0 Rivoal 00] ad that at least oe amogst ζ(5ζ(7ζ(9ζ( is irratioal [Zudili 04]. c A K Peters Ltd /004$ 0.50 per page Experimetal Mathematics 3:4 page 503

2 504 Experimetal Mathematics Vol. 3 (004 No. 4 with the help of the geeratig fuctios ad ζ(s +3a s ( a s0 s0 ζ(4s +3b 4s 4 b 4. (The series o the left-had sides of the equal sigs coverge oly for a < ad b < whereas the right-had sides coverge o much larger domais. Koecher [Koecher 80] (ad idepedetly Leshchier [Leshchier 8] i a expaded form proved that ( a ( a a ( a ( for ay complex umber a such that a < ad more recetly Almvist ad Graville [Almvist ad Graville 99] proved aother idetity first cojectured by Borwei ad Bradley [Borwei ad Bradley 97]: 4 b 4 ( b 4 ( 4 +4b 4 4 b 4 ( 3 for ay complex umber b such that b <. For a b 0 these idetities reduce to ( but otherwise produce differet idetities for the values of the zeta fuctio at odd itegers. For example Borwei ad Bradley ote that ( implies ζ(7 + 5 while ( 3 implies ζ(7 5 ( + ( + ( 7 >j 5 j ( + 3 j i >j>i ( >j ( + 3 j 4. The purpose of this article is to prove the followig very geeral geeratig fuctio idetity which was cojectured by H. Cohe o the basis of computatios i Pari. Theorem.. Let a ad b be complex umbers such that a + b 4 <. The 4 a b 4 ( + 5 a 4 a b 4 ( ( a +4b 4 4 a b 4. ( 4 We remar that Idetity ( 4 uifies ( (case b 0 ad ( 3 (case a 0; cosequetly it should yield ew Apéry-lie formulae. This is ideed true sice 4 a b 4 r0 s0 ( r + s r ζ(r+4s+3 a r b 4s ad sice the umber of represetatios of a iteger j 0asj r +s with itegers r s 0is[j/] +. Hece ( 4 produces [j/] + differet idetities for ζ(j+3 for ay iteger j 0 obtaied by differetiatig the right-had side of ( 4 r respectively s times with respect to a respectively b 4 with j r +s ad the by lettig a b 0. For 0 j oe of r ad s is 0 ad we oly obtai idetities resultig from ( or ( 3. This is also the case for j 3(r s (3 0. The first apparetly ew idetity is for j 3(r s ( : ζ( ( + +5 ( + ( 9 >j 5 j 4 >j >j>i ( + 3 j >j ( + 3 j 4 i 5 4 ( + 7 j >j>i ( + 3 j i 4. To prove Theorem. we will use Borwei ad Bradley s method i which the proof of ( 4 was reduced i several steps to the proof of a fiite combiatorial idetity (the last step i [Borwei ad Bradley 97] is due to Wechag Chu which was fially proved by Almvist ad Graville. I our case we will show that Theorem. follows from the idetity a ( +(j a ( +(j + a j ( j ( + j a a (

3 Rivoal: Simultaeous Geeratio of Koecher ad Almvist Graville s Apéry-Lie Formulae 505 for ay iteger which we will the prove as corollary of the followig result. Theorem.. Let g(x C[X] be of degree at most. For ay iteger ad ay complex umbers a ad t witha {± ±...±} we have that ( 4 ( a ( ( t( a +g(j 0 j< or <j<+ 0 j< or <j<+ g(j 0. ( 5 For the special case a 0 we obtai the ey idetity proved i [Almvist ad Graville 99]. 3. SECOND STEP We defie t ( to be the summad of the series i ( 3 ad δ to be a (for ay fixed brach of the logarithm. We observe that t ( ca be exteded to a meromorphic fuctio of the complex variable : t ( ( e iπ Γ( ± iδ(5 a Γ( ± iδγ( ± iδ Γ( + Γ( + Γ( ± ± iδ Γ( +± Γ( +± iδ (3 where Γ(x ± y ± z is defied to be Γ(x + y + z Γ(x + y zγ(x y + zγ(x y z etc. We ote that as a result of the factor Γ( + i the deomiator of (3 we have t ( 0for.... Furthemore simple computatios give that t (0 a /( a ad for {...}. FIRST STEP We trasform the right-had side of ( 4 by a partial fractio decompositio with respect to b 4 : ( a +4b 4 4 a b 4 4 a b 4 C (a 4 a b 4 ( where C (a ( +(j a ( +(j + a. j (j (j + a ( Isertig ( i the right-had side of ( 4 ad ivertig the summatios we see that it will be eough to show that (ad i fact this is equivalet 4 a b 4 4 a b 4 ( + 5 a C (a. Clearly it is eough to show that for ay iteger ad ay complex a with a < ( + 5 a C (a. ( 3 From ow o ad uless otherwise specified we assume that a <. t ( 3 ( a a ( 5 a ( +( a ( +( + a ( j (j + a ( +(j a ( +(j + a. (3 We are ow ready to prove our secod step. Propositio 3.. For ay give Equatio ( 3 is equivalet to the followig fiite combiatorial idetity: ( 5 a ( +( a ( +( + a ( j (j + a ( +(j a ( +(j + a a. (3 3 Remar 3.. Give ay iteger if (3 3 is true for a < it is true for ay complex umber a such that a ca ot be writte a + m with a iteger m {0 ±...±}. Proof of Propositio 3.: We will prove below that + t ( 0. (3 4

4 506 Experimetal Mathematics Vol. 3 (004 No. 4 Equatio (3 4 ca be writte t ( t (0 t ( t ( a a t ( ad t ( is clearly equivalet to t ( 3 a which give (3 is exactly (3 3. We ow prove (3 4 ad for that we closely follow Borwei ad Bradley whose method is based o Gosper s hypergeometric summatio algorithm (see [Graham et al. 94 pages 5 7] for details. We ote that t ( + t ( 5( + a 5 a + ( ± ± iδ ( +± ( +± iδ p ( +q ( p (r ( + is a ratioal fuctio of with q ( ( ± iδ r ( ( ( + ad p ( (5 a ( j( + j ± iδ. Sice q ad r do ot have roots differig by itegers 3 Gosper s algorithm esures that there exists a polyomial s of degree at most deg(p deg(q r 3 3 such that p ( s ( +q ( r (s (. We ow defie T ( r (s (t ( p ( which satisfies T ( + T ( t (. Sice t ( is fiite ad p ( 0r ( we have T ( 0. Hece for ay T ( j t (. Sice deg(r s deg(p we have T ( O(t ( as + which implies that T ( teds to 0 as +. It follows that (3 4 holds. Propositio 4.. Equatio (3 3 for every iteger is equivalet to the followig idetity for every iteger : a ( +(j a ( +(j + a j ( j ( + j a a (. (4 Remar 4.. The simplificatio (4 below shows that give ay iteger if (4 is true for a < it is true for ay complex umber a such that a {±...±}. Furthermore it ca also be writte as C (a a ( + a where C (a is defied i (. ( Proof of Propositio 4.: We use Krattethaler s iversio formula [Krattehaler 96]: f( where r a d + b c d ϕ(c /d ; ψ ( c /d ; + g( iff g( r ϕ(x; (a j + xb j j0 ψ( c /d ; ϕ(c /d ; + f( ψ(x; (c j + xd j ad j0 ψ m (x; (c j + xd j. j0 j m Applied to (3 3 it yields the result with the choices r a j (j a b j 4c j j 4 a j d j ( f( ( (5 a 4. THIRD STEP Here we geeralise the last reductio step of [Borwei ad Bradley 97] (due to Wechag Chu. 3 Sice a < ad iδ ca t be a iteger. ad g( a 4 4 4a +(a 4 a.

5 Rivoal: Simultaeous Geeratio of Koecher ad Almvist Graville s Apéry-Lie Formulae 507 Usig the same tric as Almvist ad Graville it is easy to write (4 i a more coveiet form that we will prove below: for ay ( 4 ( a 5. THE FINAL STEP 0 j< or <j<+ (! a ( + j a (. (4 Note that (4 is simply Theorem. with g(x X ad t : ideed the first product i the left-had side of ( 5 correspods exactly to the left-had side of (4 ad (sice oly the th summad is ozero ( 4 ( a a <j< j 0 j< or <j<+ g(j 4 (! a! (! ( a. Hece Theorem. follows from Theorem.. Proof of Theorem.: So far we have bee very lucy i that every step of [Borwei ad Bradley 97] geeralises without problems to this more geeral settig. But here the geeral Theorem i [Almvist ad Graville 99] is apparetly ot strog eough to prove (4. Fortuately we ca adapt the method used there for our purpose. For ay we defie the polyomial of degree F (X (X g(j. 0 j< or <j<+ Propositio i [Almvist ad Graville 99] establishes the existece of polyomials Q r (X of degree d r r such that F (X F (0 Q r ( a X r. (5 r0 The importat poit for us is the fact that sice F (X F (0 vaishes at X 0 the the sum i (5 termiates at. (I fact Q r (X c r (X + a with the polyomials c r give i [Almvist ad Graville 99]. We write Q r (X d r i0 q ri X i. Equatio ( 5 ca be expressed as ( ( 4 ( a (F ( t( a F (0 ( d r ( t r q ri r0 i0 ( 4 ( a ( a i+ r. (5 Sice i 0adr we have 4 a ( a i+ r P ( where P (X 4X(X a +i r is a polyomial of degree i + r d r + r such that P (0 0. Lemma i [Almvist ad Graville 99] which reads ( ( l 0 (5 3 for ay l the gives that ( 4 ( a ( a i+ r ( ( P ( 0. This proves that the left-had side of (5 is 0 for all t ad the proof of Theorem. is complete. We coclude this sectio with the followig remar. Almvist ad Graville proved (5 3 by expressig its left-had side as the lth Taylor coefficiet of the fuctio e z (e z. Aother proof is as follows: defie S(z z l /z(z (z for ay itegers l 0 ad 0. The by the residue theorem for ay closed direct cotour Γ eclosig the poles of S wehave Res (S S(z dz Res (S iπ Γ 0 ( l (!( +!. 0 If we assume that l the Res (S 0adif l the (5 3 follows after multiplicatio by (!/.

6 508 Experimetal Mathematics Vol. 3 (004 No. 4 ACKNOWLEDGMENTS The mai result (Theorem. of this paper was cojectured by Heri Cohe ad it was his ecouragemet that prompted me to wor o this problem. I warmly tha him for this. REFERENCES [Almvist ad Graville 99] G. Almvist ad A. Graville. Borwei ad Bradley s Apéry-Lie Formulae for ζ( Exp. Math. 8: ( [Apéry 79] R. Apéry. Irratioalité de ζ( et ζ(3. Astérisque 6 ( [Ball ad Rivoal 0] K. Ball ad T. Rivoal. Irratioalité d ue ifiité de valeurs de la foctio zêta aux etiers impairs. Ivet.Math. 46: ( [Borwei ad Bradley 97] J. Borwei ad D. Bradley. Empirically Determied Apéry-Lie Formulae. Exp. Math. 6:3 ( [Cohe 78] H. Cohe. Démostratio de l irratioalité de ζ(3 (d après Apéry. I Sém. de Théorie des Nombres de Greoble. Uiversité Joseph Fourier Greoble 978. [Cohe 8] H. Cohe. Gééralisatio d ue costructio de R. Apéry. Bull. Soc. Math. Frace 09 ( [Graham et al. 94] R. L. Graham D. E. Kuth ad O. Patashi. Cocrete Mathematics: A Foudatio for Computer Sciece d editio. Readig MA: Addiso- Wesley 994. [Koecher 80] M. Koecher. Letter to the Editor. Math. Itelligecer ( [Krattehaler 96] C. Krattethaler. A New Matrix Iverse. Proc.Amer.Math.Soc4: ( [Leshchier 8] D. H. Leshchier. Some New Idetities for ζ(. J. Number Theory 3:3 ( [Rivoal 00] T. Rivoal. La foctio Zêta de Riema pred ue ifiité de valeurs irratioelles aux etiers impairs. C. R. Acad. Sci. Paris Série I Math. 33:4 ( [va der Poorte 79] A. va der Poorte. A Proof that Euler Missed... Apéry s Proof of the Irratioality of ζ(3. A Iformal Report. Math. Itelligecer :4 (978/ [va der Poorte 80] A. va der Poorte. Some Woderful Formulas... A Itroductio to Polylogarithms. Quee s Papers i Pure ad Applied Mathematics (Proc. 979 Quee s Number Theory Coferece ( [Zudili 04] W. Zudili. Arithmetic of Liear Forms Ivolvig Odd Zeta Values. To appear i J. Théor. Nombres Bordeaux (004. Available at T. Rivoal Laboratoire de Mathématiques Nicolas Oresme CNRS UMR 639 Uiversité de Cae BP Cae cedex Frace (rivoal@math.uicae.fr Received Jue 5 004; accepted August