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1 Provided by the author(s) ad Uiversity College Dubli Library i accordace with publisher policies., Please cite the published versio whe available. Title Supercogrueces for sporadic sequeces Authors(s) Sahu, Brudaba; Osbur, Robert; Straub, Armi Publicatio date Publicatio iformatio Proceedigs of the Ediburgh Mathematical Society, 59 (2): Publisher Cambridge Uiversity Press Item record/more iformatio Publisher's versio (DOI) 0.07/S Dowloaded T5:4:00Z The UCD commuity has made this article opely available. Please share how this access beefits you. Your story matters! (@ucd_oa) Some rights reserved. For more iformatio, please see the item record li above.

2 SUPERCONGRUENCES FOR SPORADIC SEQUENCES ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB Dedicated to Frits Beuers o the occasio of his 60th birthday Abstract. We prove two-term supercogrueces for geeralizatios of recetly discovered sporadic sequeces of Cooper. We also discuss recet progress ad future directios cocerig other types of supercogrueces.. Itroductio The term supercogruece first appeared i Beuers wor [4] ad was the subject of the Ph.D. thesis of Coster [3]. It refers to the fact that cogrueces of a certai type are stroger tha those suggested by formal group theory. A motivatig example i [4] ad [3] is the Apéry umbers ( ) 2 ( ) + 2 () A() = which ot oly satisfy [7] (2) A(mp) A(m) (mod p 3 ), but the two-term supercogruece [3] (3) A(mp r ) A(mp r ) (mod p 3r ) for primes p 5 ad itegers m, r. I 985, Beuers related these umbers to the p-th Fourier coefficiet a(p) of η 4 (2z)η 4 (4z), where η(z) = q /24 = ( q ) is the Dedeid eta fuctio ad q = e 2πiz, with z i the upper half-plae. He proved that [5] ( ) p (4) A a(p) (mod p) 2 ad the cojectured that (4) holds modulo p 2. I [2], Ahlgre ad Oo proved this modular supercogruece usig Gaussia hypergeometric series [2]. The techiques i [2] have bee the basis for several recet results of this type (see [6], [25], [27], [29] [32], [36]). Other types of supercogrueces are Date: Jue 7, Mathematics Subject Classificatio. Primary: A07, B83.

3 2 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB also of cosiderable iterest. Ramauja-type supercogrueces are p-adic versios of formulas of Ramauja which relate biomial sums to special values of the gamma fuctio (or /π a, a ). For example, va Hamme [39] cojectured that for Ramauja s formula ( ) /2 5 (4 + ) = we have the p-adic aalogue (5) (p )/2 ( ) /2 5 { p (4 + ) 2 Γ(3/4) 4, Γ p(3/4) 4 (mod p 3 ), if p (mod 4), 0 (mod p 3 ), if p 3 (mod 4), where Γ p ( ) is the p-adic gamma fuctio. For a proof of (5), see [28]. For recet progress i this directio, see [], [6], [22], [23], [26], [33] or [4]. Fially, Ati Swierto-Dyer supercogrueces have bee recetly studied i [4], [5], [24] ad [37]. I this paper, we cosider the sequeces of umbers give by ( ) 2 ( )( ) + 2 (6) s 7 () = as well as (7) [/3] ( s 8 () = ( ) )( 2 )( 2( ) ) [( 2 3 ) + ( 2 3 )], with s 8 (0) =. These sporadic sequeces were recetly discovered by Cooper [2] while performig a umerical search for sequeces which appear as coefficiets of series for /π ad of series expasios i t of modular forms where t is a modular fuctio. Here, the subscripts 7 ad 8 are used i (6) ad (7) as the associated modular fuctio is of level 7 ad 8, respectively (see Theorem 3. i [2]). I [2], Cooper searched for parameters (a, b, c, d) such that the recurrece relatio (8) ( + ) 3 s( + ) = (2 + )(a 2 + a + b)s() (c 2 + d)s( ), with iitial coditios s( ) = 0, s(0) =, produces oly iteger values s() for all 0. The tuple (7, 5,, 0) correspods to the Apéry umbers (), while the tuples (3, 4, 27, 3) ad (4, 6, 92, 2) correspod to the sequeces s 7 () ad s 8 (), respectively. See [3] for the case d = 0. This search was motivated by Beuers [6] ad Zagier s [40] wor o sequeces t() defied by (9) ( + ) 2 t( + ) = (a 2 + a + b)t() c 2 t( ), with iitial coditios t( ) = 0, t(0) =, such that t() Z for all 0. Zagier s search yielded six sequeces that are ot either termiatig,

4 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 3 polyomial, hypergeometric or Legedria. These six sequeces were called sporadic. Iterestigly, Cooper cojectured the followig cogrueces (see Cojecture 5. i [2]) which are remiiscet of (2). Cojecture.. For ay prime p 3, (0) s 7 (mp) s 7 (m) (mod p 3 ). Liewise, for ay prime p, () s 8 (mp) s 8 (m) (mod p 2 ). The purpose of this paper is to exhibit that (0) ad () are special cases of geeral two-term supercogrueces. For itegers A, B, C, let ( ) A ( ) + B ( ) 2 C (2) S(; A, B, C) =. Note that this family of sequeces icludes the Apéry umbers as well as the sequece s 7 (). Our mai results are the followig supercogrueces, the first of which, i particular, geeralizes the supercogruece (3) for the Apéry umbers. Theorem.2. Let A 2 ad B, C 0 be itegers. For ay itegers m, r ad primes p 5, we have (3) S(mp r ; A, B, C) S(mp r ; A, B, C) (mod p 3r ). Theorem.3. For ay itegers m, r ad ay primes p, we have (4) s 8 (mp r ) s 8 (mp r ) (mod p 2r ). Note that by taig (A, B, C) = (2,, ) ad r = i Theorem.2, we prove (0) of Cojecture. for primes p 5. Moreover, Theorem.2 shows a geeral supercogruece for the sporadic sequece ( ) 2 ( ) 2 2, which is case (ɛ) i [3] (see also Table 2 i Sectio 4). O the other had, we remar that Theorem.3 is cosiderably simpler tha Theorem.2 because it suffices to cosider each summad of the sum (7), defiig s 8 (), idividually. I both cases, our proof of the cogrueces of Cojecture. relies o the presece of the biomial sums (6) ad (7) which were discovered by Zudili. Fially, we should metio that Cooper [2] cojectures cogrueces similar to (0) for p = 2, as well as a stroger versio of () for p = 2, 3. These cojectures as well as (0) for p = 3 remai ope. Based o umerical evidece, we actually cojecture that (5) s 7 (m2 r ) s 7 (m2 r ) (mod 2 3r+2 )

5 4 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB for m 4 ad (6) s 7 (m3 r ) s 7 (m3 r ) (mod 3 3r ) for ay positive iteger m, as well as (7) s 8 (m2 r ) s 8 (m2 r ) (mod 2 2r+3 ) for m 2 ad (8) s 8 (m3 r ) s 8 (m3 r ) (mod 3 3r ) for m 3 (for r = the cogruece (8) empirically holds modulo 3 3 ). Slightly weaer cogrueces appear to hold i the cases whe m is ot large eough. We expect that these cojectures, which aturally geeralize the oes from [2], ca be established usig the techiques we use i the case p 5 whe coupled with a careful ad liely very techical aalysis of the id preseted at the ed of our proof of Theorem.3. The remaider of the paper is orgaized as follows. Sectio 2 is devoted to the proofs of Theorems.2 ad.3. We the idicate i Sectio 3 that these proofs readily geeralize to other sequeces of iterest. I Sectio 4, we coclude with remars cocerig future directios. I particular, we discuss both prove ad cojectural two-term supercogrueces for all ow sporadic sequeces. 2. Proof of Theorems.2 ad.3 Throughout this sectio, followig [4], we let deote the sum over idices ot divisible by p. We first recall the followig versio of Jacobsthal s biomial cogruece [7]. For a proof, whe a, b 0, we refer to [8], [20], while the extesio to egative itegers is discussed i [38]. Similar cogrueces hold [8, 38] i the cases p = 2 ad p = 3. Lemma 2.. For primes p 5, itegers a, b ad itegers r, s, ( p r ) ( a p r ) a (9) p s / b p s (mod p r+s+mi(r,s) ). b We will also mae use, i the case = 2, of the followig simple cogrueces. Lemma 2.2. Let p be a prime ad a iteger such that 0 modulo p. The, for all itegers r 0, (20) p r = 0 (mod p r ). If, additioally, is eve, the, for primes p 5, (2) (p r )/2 = 0 (mod pr ).

6 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 5 Proof. Sice 0 modulo p, we fid a iteger λ, ot divisible by p, such that λ modulo p. The, pr λ = = p r = (λ) p r = (mod p r ), sice the secod ad third sum ru over the same residues modulo p r. As λ is ot divisible by p, the cogruece (20) follows. Cogruece (2) follows sice the sum i (2), modulo p r, is exactly half of the sum i (20) if is eve. Lemma 2.3. For itegers, ad A, B, C 0, defie ( ) A ( ) + B ( ) 2 C B(, ) = B(, ; A, B, C) =. The, for primes p 5 ad itegers A 2, r, s ad 0 such that p, (22) B(p r, p s ) B(p r, p s ) (mod p 3r ). Proof. By Jacobsthal s cogruece (9), we have ( ) ( ) p r p r p s / p s (mod p r+s+mi(r,s) ) as well as (p r + p s ) ( p r + p s ) p r / p r (mod p r+2 mi(r,s) ) ad ( ) ( ) 2p s 2p s p r / p r (mod p r+s+mi(r,s) ). Thus, if s r the cogruece (22) follows immediately upo applyig Jacobsthal s cogruece to each biomial coefficiet. O the other had, suppose s r. The the same approach yields (23) B(p r, p s ) = λb(p r, p s ) with λ modulo p r+2s. Moreover, sice p, we have ( ) ( ) p r p r p s p s 0 (mod p r s ). As A 2, it follows that p 2(r s) divides B(p r, p s ). Sice r+2s+2(r s) = 3r, cogruece (22) follows from (23). I the followig, [x] deotes the largest iteger m such that m x. Lemma 2.4. For primes p, itegers m ad itegers 0, r, ( mp r ) ( mp (24) ( ) r ) ( ) [/p] (mod p r ). [/p]

7 6 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB Proof. Followig [4, Lemma 2], we split the defiig product of the biomial coefficiet, accordig to whether the idex is divisible by p or ot, to obtai ( mp r ) = = = j= j= p j mp r j j mp r j j [/p] λ= ( mp r ) [/p] The claim follows upo reducig modulo p r. j= p j mp r λ λ mp r j. j Lemma 2.5. For itegers,, j ad A, B, C 0, defie ( ) A ( ) + B ( ) j C (25) C(,, j) = C(,, j; A, B, C) =. The, for primes p ad itegers,, j 0, r, C(p r,, j) ( ) (+[/p])a C(p r, [/p], [j/p]) (mod p r ). Proof. Note that (26) ( ) + ( ) = ( ). Usig Lemma 2.4, we fid that ( p ( ) r ) ( p ( ) [/p] r ) [/p] or, equivaletly, ( p r ) ( + p r ) + [/p] (27) [/p] (mod p r ). I particular, ( p r + (j p r ) ( ) p r + [j/p] p r ) j p r [j/p] p r which is equivalet to (28) ( ) ( ) j [j/p] p r p r (mod p r ). The proof thus follows upo combiig (24), (27), (28). (mod p r ) (mod p r ),

8 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 7 Proof of Theorem.2. Adaptig the origial approaches of [4] ad [7], we split the biomial sum (2) as S(mp r ; A, B, C) = s 0 G s (mp r ), where G s () = B(, p s ). It follows from Lemma 2.3 that, for s, It therefore remais to show that G s (mp r ) G s (mp r ) (mod p 3r ). (29) G 0 (mp r ) = B(mp r, ) 0 (mod p 3r ). Note that (30) ( mp r ) = mpr ( mp r ) is divisible by p r if p. Hece, if A 3 the (29) is obviously true ad (3) follows. I the remaider, we cosider the case A = 2. With (30) substituted ito (29), we fid that we eed to show that (3) 2 ( mp r ) 2 ( mp r + ) B ( 2 mp r ) C 0 (mod p r ). If p the [( )/p] = [/p] so that, by Lemma 2.4, ( mp r ) 2 ( mp r ) 2 (32) (mod p r ). [/p] By (32) ad Lemma 2.5 with A = 0, the left-had side of (3) is cogruet modulo p r to ( mp r ) 2 ( mp r ) B ( ) + [/p] [2/p] C (33) 2 [/p] [/p] mp r. Usig the otatio of (25) ad (33), cogruece (3) is equivalet to (34) 2 C(mpr, [/p], [2/p]) 0 (mod p r ). I order to establish (34), we ow show that (35) 2 C(mpr,, 2) 2 C(mpr s, [/p s ], [2/p s ]) (mod p r )

9 8 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB for s = 0,,..., r. The case s = 0 is trivial, while the case s = follows from Lemma 2.5. If we ow let { : p s } := p s [/p s ], the remaider of divided by p s, the observe that { [2/p s ] = 2[/p s, if { : p ] + s } > p s /2, 0, otherwise. Hece, (36) 2 C(mpr s, [/p s ], [2/p s ]) = = + [/p s ]= 2 C(mpr s, [/p s ], [2/p s ]) C(mp r s,, 2) C(mp r s,, 2 + ) [/p s ]= {:p s }<p s /2 2 [/p s ]= {:p s }>p s /2 2. It follows from (2) of Lemma 2.2 that each of the ier sums i the last expressio of (36) is divisible by p s. Suppose that s < r. Thus, by (36) ad Lemma 2.5, we ow have C(mpr s, [/p s ], [2/p s ]) C(mp r s, [/p], [2/p]) [/p s ]= {:p s }<p s /2 C(mp r s, [/p], [(2 + )/p]) [/p s ]= {:p s }<p s /2 [/p s ]= {:p s }>p s /2 2 [/p s ]= {:p s }>p s /2 2 C(mpr s, [[/p s ]/p], [[2/p s ]/p]) 2 (mod p r ) 2 C(mpr s, [[/p s ]/p], [[2/p s ]/p]) (mod p r ) 2 C(mpr s, [/p s+ ], [2/p s+ ]) (mod p r ). Hece (35) follows by iductio o s. Moreover, the case s = r i (36) shows that, for s = r ad hece all s = 0,,..., r, the sums i (35) are divisible by p r. I particular, the case s = proves (34) ad thus (3).

10 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 9 Proof of Theorem.3. As i [35], we use the idetity ( )( ) ( a )( c a c ) a b b c = d)( b d ( c d d a, b) to obtai that ( )( ) 2 2( ) = ( 2 )( 2 ( ) 2 ) = 2 ( ) (2)!(2( ))! =!!( )! where S(m, ) are the super Catala umbers S(m, ) = (2m)!(2)! m!!(m + )!. ( ) S(, ), We refer to [9] ad the refereces therei for the history ad properties of these umbers. Here, we oly eed that S(, ) is a iteger; i fact, it is a eve iteger if. Deote the summad of (7) by D(, ), that is so that ( 2 [( 2 3 D(, ) = ( ) ) S(, ) s 8 (mp r ) = s 0 D(mp r, p s ). ) + ( 2 3 )], I aalogy with Lemma 2.3, we claim that, for primes p 5 ad itegers r, s, (37) D(mp r, p s ) D(mp r, p s ) (mod p 3r ). A direct applicatio of Lemma 2. shows that ( 2mp r 3p s ) ( 2mp r 3p s ) mp r / mp r (mod p r+2 mi(r,s) ) as well as S(mp r p s, p s )/S(mp r p s, p s ) (mod p r+2 mi(r,s) ). O the other had, for all itegers a ad itegers b 0, ( ) ( ) a + b a (38) = ( ) b, b b so that ( ) 2 3 ( ) 3 = ( ). Hece, Lemma 2. implies that ( 2mp r 3p s ) ( 2mp r 3p s ) mp r / mp r (mod p r+2 mi(r,s) ).

11 0 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB Proceedig as i the proof of Lemma 2.3, we therefore obtai (37). The cogruece (4) the follows if we ca prove that, for itegers such that p, D(mp r, ) 0 (mod p 2r ). This is a immediate cosequece of the fact that D(, ) is divisible by ( ) 2. Fially, let us briefly idicate how to obtai the correspodig cogrueces i the case that p = 2 or p = 3. For p = 3, cogruece (9) of Lemma 2. oly holds modulo p r+s+mi(r,s). The same argumets as above the show that cogruece (37) holds modulo p 3r, ad hece modulo p 2r, for p = 3. The case p = 2 requires some more attetio. I that case, the couterpart of cogruece (9) is ( 2 r a 2 s b ) / ( 2 r ) a 2 s ε (mod 2 r+s+mi(r,s) 2 ) b with ε =, if 2 r a 0, 2 s b modulo 2, ad ε = otherwise. Hece, applyig the same argumets as for p > 2 shows that D(2 r m, 2 s ) = λd(2 r m, 2 s ) with λ ± modulo 2 r+2 mi(r,s) 2 ad, hece, λ modulo 2. Moreover, both sides are divisible by 2 2 max(r s,0)+ because ( ) 2 r m 2 s is divisible by 2 r s, if r s, ad the super Catala umbers S(, ) are eve whe. I the cases r = or s =, this suffices to coclude that (37) holds modulo p 2r for p = 2. O the other had, if r 2 ad s 2, the goig through the above computatios reveals that λ modulo 2 r+2 mi(r,s) 2, ad hece modulo 2 r+mi(r,s). Together with the divisibility of D(2 r m, 2 s ) by 2 r s, if r s, we agai fid that (37) holds modulo p 2r for p = Commets o direct geeralizatios The approaches of the proof of Theorems.2 ad.3, which are based o [7] ad [4], geeralize easily to other sequeces. Example 3.. For istace, cosider the sequece ( ) 3 (( ) ( )) (39) Z() = ( ) +, 3 3 which is case (η) i [3] (see also Table 2 i Sectio 4). We claim that the proof of Theorem.3 aturally exteds to show that, for primes p 5, (40) Z(mp r ) Z(mp r ) (mod p 3r ). Similar to the proof of Theorem.3, write Z(mp r ) = s 0 A(mp r, p s ),

12 where SUPERCONGRUENCES FOR SPORADIC SEQUENCES ( ) 3 (( ) ( )) A(, ) = ( ) As i the proof of Theorem.3, we fid that for primes p 5, A(mp r, p s ) A(mp r, p s ) (mod p 3r ). O the other had, the presece of ( ) 3 shows that, for p, A(mp r, ) 0 (mod p 3r ). Combiig these two cogrueces, we coclude that the supercogruece (40) ideed holds. Example 3.2. The proof of Theorem.3 directly geeralizes to supercogrueces for the followig family of sequeces which icludes s 8 () as the case (A, B, C, D, E) = (,,,, ). For oegative itegers A, B, C, D, E, defie T (; A, B, C, D, E) to be the sequece [/3] ( ) A ( ) 2 B ( ) [ 2( ) C (2 ) 3 D ( ) ] 2 3 E ( ) +. If A, B ad C, the T (mp r ; A, B, C, D, E) T (mp r ; A, B, C, D, E) (mod p 2r ) for all primes p. More geerally, we have, for istace, that if A 2, B ad C, the T (mp r ; A, B, C, D, E) T (mp r ; A, B, C, D, E) (mod p 3r ) for all primes p 5. Example 3.3. I [8] ad [9], the sequece ( )( ) 2 u() = ( ) is studied. I particular, it is proved that u(mp) u(m) modulo p 3 for all primes p 5. Followig the approach of Theorem.2, we obtai that, more geerally, u(mp r ) u(mp r ) (mod p 3r ) for all primes p 5. Let a, b be oegative itegers. It is show i [8] that the more geeral sequeces ( ) a ( ) 2 b u ε a,b () = ( ) ε satisfy, for primes p 5, the cogruece u ε a,b (p) uε a,b () (mod p3 )

13 2 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB uless (ε, a, b) = (0, 0, ) or (0,, 0). Agai, we ca geeralize this cogruece by usig the approach of Theorem.2 to show that, for primes p 5, provided that a + b 2. u ε a,b (mpr ) u a,b (mp r ) (mod p 3r ) 4. Cocludig remars There are several directios for future wor. First, we have umerically checed that each of the six sporadic examples of Zagier (labelled A, B, C, D, E, F i [40]) ad the six sporadic examples i [3] (labelled (δ), (η), (α), (ɛ), (ζ), (γ)) satisfies a two-term supercogruece. Precisely, cases A ad D are modulo p 3r while cases B, C, E, F are modulo p 2r. All six cases from [3] are modulo p 3r. Cases A ad D have bee prove by Coster [3] ad cases C ad E were settled by the first two authors i [34] ad [35]. Cases (α) ad (γ) have bee prove i [35] ad [3], respectively. As metioed i the itroductio, the proof of Theorem.2 implies case (ɛ). O the other had, case (η) is prove i Example 3. of Sectio 3. For a discussio cocerig coectios betwee the six sporadic examples i [3] ad [40], see Theorem 4. i [3] or Theorem 3.5 ad Tables ad 2 i [0]. The iformatio o supercogrueces for Apéry-lie umbers is summarized i Tables ad 2. The value for idicates that the sequece A() (at least cojecturally i cases B, F, (δ) ad (ζ)) satisfies the supercogruece A(mp r ) A(mp r ) (mod p r ) for primes p 5. I the cases where this cogruece has bee prove, a referece is idicated i the fial colum. It would be of iterest to prove cases B, F, (δ) ad (ζ) ad, more geerally, provide a framewor for all two-term supercogrueces. Secodly, for each of the 5 sporadic cases i Tables ad 2, oe could as if there exists a modular form f(z) whose p-th Fourier coefficiet satisfies a modular supercogruece. This is true for cases D [] ad (γ) [2]. Fially, it also appears that all ow Ramauja-type series for /π a, a, have a p-adic aalogue which satisfies a Ramauja-type supercogruece. Differet techiques have bee employed as there is curretly o geeral explaatio for this occurrece. For example, the first author ad McCarthy [28] utilized Gaussia hypergeometric series ad Whipple s trasformatio to prove (5). Zudili [4] proved several Ramauja-type supercogrueces usig the Wilf Zeilberger method while Log [26] used a combiatio of combiatorial idetities, p-adic aalysis ad trasformatios ad strage evaluatios of ordiary hypergeometric series due to Whipple, Gessel ad Gosper. Acowledgemets. The first author would lie to tha Frits Beuers ad Matthijs Coster for their cotiued iterest. The secod author is partially fuded by SERB grat SR/FTP/MS-053/202. The third author

14 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 3 (a, b, c) [40] [3] A() (7, 2, 8) A (a) ) 3 3 [3] (, 3, ) D (b) (0, 3, 9) C (c) (2, 4, 32) E (d) (9, 3, 27) B (f) (7, 6, 72) F (g) ( ( ) 2 ( ) + ( ) 2 ( ) 2 ( )( )( ) 2 2( ) ( ) (3)! ( ) ! 3 ( )( ) 3 ( ) 8 2 l,l 3 [3] 2 [34] 2 [35] Table. Sporadic sequeces from [40] for (9) (a, b, c, d) [3], [2] A() (7, 3, 8, 0) (δ) ( )( ) + (3)! ( ) ! 3 (, 5, 25, 0) (η) defied i (39) 3 (40) (0, 4, 64, 0) (α) (2, 4, 6, 0) (ɛ) (9, 3, 27, 0) (ζ) (7, 5,, 0) (γ),l (6, 2, 64, 4) s 0 (3, 4, 27, 3) s 7 ( ) 2 ( 2 ( ) 2 ( 2 ( ) 2 ( l )( 2( ) ) 3 [35] ) 2 3 (3) )( l ( ) 2 ( + )( ) + l ) 2 3 [3] ( ) 4 3 [3] ( ) 2 ( + )( ) (3) (4, 6, 92, 2) s 8 defied i (7) 2 (4) Table 2. Sporadic sequeces from [3] ad [2] for (8) would lie to tha the Max-Plac-Istitute for Mathematics i Bo, where he was i residece whe this wor was completed, for providig

15 4 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB woderful worig coditios. Fially, we tha the referee for their helpful commets ad suggestios. Refereces [] S. Ahlgre, Gaussia hypergeometric series ad combiatorial cogrueces, Symbolic computatio, umber theory, special fuctios, physics ad combiatorics (Gaiesville, FL, 999), 2, Dev. Math., 4, Kluwer Acad. Publ., Dordrecht, 200. [2] S. Ahlgre, K. Oo, A Gaussia hypergeometric series evaluatio ad Apéry umber cogrueces, J. reie agew. Math. 58 (2000), [3] G. Almvist, D. va Strate ad W. Zudili, Geeralizatios of Clause s formula ad algebraic trasformatios of Calabi-Yau differetial equatios, Proc. Edib. Math. Soc. (2) 54 (20), o. 2, [4] F. Beuers, Some cogrueces for the Apéry umbers, J. Number Theory 2 (985), o. 2, [5] F. Beuers, Aother cogruece for the Apéry umbers, J. Number Theory 25 (987), o. 2, [6] F. Beuers, O B. Dwor s accessory parameter problem, Math. Z. 24 (2002), o. 2, [7] V. Bru, J. Stubba, J. Fjeldstad, L. Tambs, K. Aubert, W. Ljuggre, E. Jacobsthal, O the divisibility of the differece betwee two biomial coefficiets, De te Sadiavise Mathematierogress, Trodheim, 949, pp Joha Grudt Taums Forlag, Oslo, 952. [8] M. Chamberlad, K. Dilcher, Divisibility properties of a class of biomial sums, J. Number Theory 20 (2006), o. 2, [9] M. Chamberlad, K. Dilcher, A biomial sum related to Wolsteholme s theorem, J. Number Theory 29 (2009), o., [0] H. Cha, S. Cooper, Ratioal aalogues of Ramauja s series for /π, Math. Proc. Cambridge Philos. Soc. 53 (202), o. 2, [] S. Chisholm, A. Deies, L. Log, G. Nebe, H. Swisher, p-adic aalogues of Ramauja type formulas for /π, Mathematics (203), 9 3. [2] S. Cooper, Sporadic sequeces, modular forms ad ew series for /π, Ramauja J. 29 (202), o. -3, [3] M. Coster, Supercogrueces, Ph.D. thesis, Uiversiteit Leide, 988. [4] M. Coster, L. Va Hamme, Supercogrueces of Ati ad Swierto-Dyer type for Legedre polyomials, J. Number Theory 38 (99), o. 3, [5] J. Kibelbe, L. Log, K. Moss, B. Sheller ad H. Yua, Supercogrueces ad complex multiplicatio, J. Number Theory, to appear. [6] T. Kilbour, A extesio of the Apéry umber supercogruece, Acta Arith. 23 (2006), [7] I. Gessel, Some cogrueces for Apéry umbers, J. Number Theory 4 (982), o. 3, [8] I. Gessel, Some cogrueces for geeralized Euler umbers, Caad. J. Math. 35 (983), o. 4, [9] I. Gessel. Super ballot umbers, J. of Symbolic Comput. 4 (992), o. 2 3, [20] A. Graville, Arithmetic properties of biomial coefficiets I: Biomial coefficiets modulo prime powers, Orgaic mathematics (Buraby, BC, 995), , CMS Cof. Proc., 20, Amer. Math. Soc., Providece, RI, 997. [2] J. Greee, Hypergeometric fuctios over fiite fields, Tras. Amer. Math. Soc. 30 (987), o., [22] J. Guillera, Mosaic supercogrueces of Ramauja type, Exp. Math. 2 (202), o.,

16 SUPERCONGRUENCES FOR SPORADIC SEQUENCES 5 [23] J. Guillera, W. Zudili, Diverget Ramauja-type supercogrueces, Proc. Amer. Math. Soc. 40 (202), o. 3, [24] W.-C. W. Li, L. Log, Ati ad Swierto-Dyer cogrueces ad ocogruece modular forms, preprit available at [25] P. Loh, R. Rhoades, p-adic ad combiatorial properties of modular form coefficiets, It. J. Number Theory 2 (2006), o. 2, [26] L. Log, Hypergeometric evaluatio idetities ad supercogrueces, Pacific J. Math. 249 (20), o. 2, [27] D. McCarthy, O a supercogruece cojecture of Rodriguez-Villegas, Proc. Amer. Math. Soc. 40 (202), o. 7, [28] D. McCarthy, R. Osbur, A p-adic aalogue of a formula of Ramauja, Arch. Math. (Basel) 9 (2008), o. 6, [29] E. Morteso, Supercogrueces betwee trucated 2F hypergeometric fuctios ad their Gaussia aalogs, Tras. Amer. Math. Soc. 355 (2003), o. 3, [30] E. Morteso, A supercogruece cojecture of Rodriguez-Villegas for a certai trucated hypergeometric fuctio, J. Number Theory 99 (2003), o., [3] E. Morteso, Supercogrueces for trucated +F hypergeometric series with applicatios to certai weight three ewforms, Proc. Amer. Math. Soc. 33 (2005), o. 2, [32] E. Morteso, Modularity of a certai Calabi-Yau threefold ad combiatorial cogrueces, Ramauja J. (2006), o., [33] E. Morteso, A p-adic supercogruece cojecture of va Hamme, Proc. Amer. Math. Soc. 36 (2008), o. 2, [34] R. Osbur, B. Sahu, Supercogrueces for Apéry-lie umbers, Adv. i Appl. Math. 47 (20), o. 3, [35] R. Osbur, B. Sahu, A supercogruece for geeralized Domb umbers, Fuct. Approx. Commet. Math. 48 (203), part, [36] R. Osbur, C. Scheider, Gaussia hypergeometric series ad supercogrueces, Math. Comp. 78 (2009), o. 265, [37] A. Scholl, Modular forms ad de Rham cohomology; Ati-Swierto-Dyer cogrueces, Ivet. Math. 79 (985), o., [38] A. Straub, Multivariate Apéry umbers ad supercogrueces of ratioal fuctios, preprit available at [39] L. Va Hamme, Some cojectures cocerig partial sums of geeralized hypergeometric series, p-adic fuctioal aalysis (Nijmege, 996), , Lecture Notes i Pure ad Appl. Math., 92, Deer, 997. [40] D. Zagier, Itegral solutios of Apéry-lie recurrece equatios, Group ad Symmetries, , CRM Proc. Lecture Notes, 47. Amer. Math. Soc., Providece, RI, [4] W. Zudili, Ramauja-type supercogrueces, J. Number Theory 29 (2009), o. 8, School of Mathematical Scieces, Uiversity College Dubli, Belfield, Dubli 4, Irelad address: robert.osbur@ucd.ie School of Mathematical Scieces, Natioal Istitute of Sciece Educatio ad Research, Bhubaeswar 75005, Idia address: brudaba.sahu@iser.ac.i

17 6 ROBERT OSBURN, BRUNDABAN SAHU AND ARMIN STRAUB Departmet of Mathematics, Uiversity of Illiois at Urbaa-Champaig, Urbaa, IL 680, Uited States Curret address: Max-Plac-Istitut für Mathemati, 53 Bo, Germay address: