SIMULTANEOUS GENERATION OF KOECHER AND ALMKVIST GRANVILLE'S APERY LIKE FORMULAE

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SIMULTANEOUS GENERATION OF KOECHER AND ALMKVIST-GRANVILLE'S APERY-LIKE FORMULAE T. RIVOAL Abstract. We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the familly (?(2r + 4s + 3))r,s≥0: it unifies two identities, proved by Koecher in 1980 and Almkvist & Granville in 1999, for the generating functions of (?(2r+3))r≥0 and (?(4s+3))s≥0 respectively. As a consequence, we obtain that, for any integer j ≥ 0, there exist at least [j/2] + 1 Apery-like formulae for ?(2j + 3). 1. Introduction In proving that ?(3) = ∑∞k=1 1/k3 is irrational, Apery [2] noted that ?(3) = 52 ∞∑ k=1 (?1)k+1(2k k )k3 . (1.1) Although the series on the right hand side converges much faster than the defining series for ?(3), formula (1.1) is not essential in Apery's proof since truncations of this series are not diophantine approximations to ?(3). On the other hand, it is very likely that (1.1) was a source of inspiration for Apery1 and many authors have looked for similar identities, in the hope that they might give some idea of how to prove the irrationality of ?(2s + 1) =∑∞ k=1 1/k2s+1

  • proof since

  • given any integer

  • apery

  • apery-like formulae

  • k4 ?

  • k2 ?

  • since


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SIMULTANEOUS GENERATION OF KOECHER AND ´ ALMKVIST-GRANVILLE’S APERY-LIKE FORMULAE
T. RIVOAL
Abstract.We prove a very general identity, conjectured by Henri Cohen, involving the generating function of the familly (ζ(2r+ 4s+ 3))r,s0unifies two identities, proved: it by Koecher in 1980 and Almkvist & Granville in 1999, for the generating functions of (ζ(2r+ 3))r0and (ζ(4s+ 3))s0respectively. As a consequence, we obtain that, for any integerj0, there exist at least [j/Ap+1ry´e2]umalferol-kiferoζ(2j+ 3).
1.Introduction P 3 In proving thatζ(3) = 1/kthatoted[2]n´ery,lpAoianrrtaiis k=1 k+1 X 5 (1) ζ(3) =¡ ¢.(1.1) 2k 3 2k k=1k Although the series on the right hand side converges much faster than the defining series forζihtfosnoitacnurtresaieerssnipAitlassneonetinceoofsspr´ery(3of,)lumr.1(asi)1 not diophantine approximations toζ(3). On the other hand, it is very likely that (1.1) was 1 asourceofinspirationforApe´ryandmanyauthorshavelookedforsimilaridentities,in the hope that they might give some idea of how to prove the irrationality ofζ(2s+ 1) = P 2s+1 1/kfor any integers2: see This problem is farfor example [4, 6, 8, 10, 13]. k=1 2 frombeingsolved,butmanybeautifulAp´ery-likeformulaehavebeenproved.Infact,two apparently unrelated families of such formulae forζ(2s+ 3) andζ(4shave emerged,+ 3) both of which are more easily explained with the help of the generating functions ∞ ∞ ∞ ∞ X X X X 1n 2s4s ζ(2s+ 3)a= andζ(4s+ 3)b=. 2 2 4 4 n(na)nb s=0n=1s=0n=1
(The series on the left hand sides converge only for|a|<1 and|b|<1, whereas the right hand sides converge on much larger domains.) Koecher [8] (and independently
2000Mathematics Subject Classification.Primary 11M06; Secondary 05A15, 11J72. Key words and phrases.anemRi-lik´eryies,eserfanuznte,npAtcoinctingfuratigeneno.s 1 See[5,12]foradetailedexplanationofApe´rysoriginalmethod. 2 We now know that infinitely many of the valuesζ(2s+ 1) (s1) areQ–linearly independent [3, 11] and that at least one amongstζ(5), ζ(7), ζ(9), ζ(11) is irrational [14]. 1