BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES

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1 #A37 INTEGERS (20) BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES Derot McCarthy Departet of Matheatics, Texas A&M Uiversity, Texas Received: /3/, Accepted: 5/3/, Published: 5/27/ Abstract We establish two bioial coefficiet geeralized haroic su idetities usig the partial fractio decopositio ethod These idetities are a ey igrediet i the proofs of uerous supercogrueces I particular, i other wors of the author, they are used to establish odulo p ( > ) cogrueces betwee trucated geeralized hypergeoetric series, ad a fuctio which exteds Greee s hypergeoetric fuctio over fiite fields to the p-adic settig A specializatio of oe of these cogrueces is used to prove a outstadig cojecture of Rodriguez-Villegas which relates a trucated geeralized hypergeoetric series to the p-th Fourier coefficiet of a particular odular for Itroductio ad Stateet of Results For o-egative itegers i ad, we defie the geeralized haroic su, H (i), by H (i) : ad H (i) 0 : 0 I [3] Chu proves the followig bioial coefficiet-geeralized haroic su idetity usig the partial fractio decopositio ethod If is a positive iteger, the j j i H() 4H() 0 () This idetity had previously bee established usig the WZ ethod [] ad was used by Ahlgre ad Oo i provig the Apéry uber supercogruece [2] This wor was supported by the UCD Ad Astra Research Scholarship progra

2 INTEGERS: (20) 2 I [4], [5] the author establishes various supercogrueces betwee trucated geeralized hypergeoetric series, ad a fuctio which exteds Greee s hypergeoetric fuctio over fiite fields to the p-adic settig Specifically, let p be a odd prie ad let Z For i, let i d i Q Z p such that 0 < i d i < Let Γ p ( ) deote Morita s p-adic gaa fuctio The defie G d, 2 d 2,, d p : p p 2 j0 ( ) j Γ j Γ p i p p i d i j p Γ p i d i ( p) i d j i p Note that whe p (od d i ) this fuctio recovers Greee s hypergeoetric fuctio over fiite fields For a coplex uber a ad a o-egative iteger let (a) deote the risig factorial defied by (a) 0 : ad (a) : a(a )(a 2) (a ) for > 0 The, for coplex ubers a i, b j ad z, with oe of the b j beig egative itegers or zero, we defie the trucated geeralized hypergeoetric series a, a 2, a 3,, a p (a ) pf q z : (a 2 ) (a 3 ) (a p ) z b, b 2,, b q (b ) (b 2 ) (b q )! A exaple of oe the supercogruece results fro [5] is the followig theore Theore (Theore 26 i [5] ) Let r, d Z such that 2 r d 2 ad gcd(r, d) Let p be a odd prie such that p ± (od d) or p ±r (od d) with r 2 ± (od d) If s(p) : Γ r d r d p d Γp d Γp d Γp d, the 4G d, r d, r d, d p 4 F 3 d, 0 r d, r d, d,, p s(p)p (od p 3 ) A specializatio of this cogruece is used to prove a outstadig supercogruece cojecture of Rodriguez-Villegas, which relates a trucated geeralized hypergeoetric series to the p-th Fourier coefficiet of a particular odular for [4],[6] Siilar results to Theore exist for 4 G with other paraeters, ad also 2 G ad 3G The ai results of the curret paper, Theores 2 ad 3 below, are two bioial coefficiet geeralized haroic su idetities which factor heavily ito the proofs of all the 4 G cogrueces Taig particular values for,, l, c ad c 2 i these idetities allows the vaishig of certai ters i the proofs Note that lettig i Theore 2 recovers ()

3 INTEGERS: (20) 3 Theore 2 Let, be positive itegers with The 0 H() H() H() 4H() ( ) ( ) Theore 3 Let l,, be positive itegers with l > l 2 ad c, c 2 Q soe costats The 0 c H() l H() H() H() 4H() c 2 c 2 H (2) H(2) l H() l c H (2) H(2) l ( ) c H() l c 2 l H() 0 The reaider of this paper is spet provig Theores 2 ad 3 2 Proofs We first develop two algebraic idetities of which the bioial coefficiet haroic su idetities are liitig cases Theore 4 Let x be a ideteriate ad let, positive itegers with The 0 (x ) 2 ( ) x H() H() H() 4H() x x( x) ( x) (x) (x) (2)

4 INTEGERS: (20) 4 Proof Usig partial fractio decopositio we ca write f(x) : x( x) ( x) (x) (x) A x B (x ) 2 C x D x for soe A, B, C ad D Q We ow isolate these coefficiets by taig various liits of f(x) as follows A li xf(x) li ( x) ( x) ( x) ( x) For, B li x (x )2 f(x) li x x( x) ( x) (x) 2 (x ) (x ) ( ) ( ) ( ) 2 () () ( ) ( ) ( ) 2! 2 ( )!( )!, ad, usig L Hôspital s rule, (x ) 2 f(x) B C li x x d li (x ) 2 f(x) x dx d x( x) li ( x) x dx (x) 2 (x ) (x )

5 INTEGERS: (20) 5 li x ( x) ( x) (x) 2 (x ) (x ) x ( x s) ( x s) (x s) (x s) 2 (x s) ( ) ( ) ( ) 2 () () s0 ( s) ( s) (s) (s) 2 ( s) s0 H() H() H() 4H() Siilarly, for, D li (x )f(x) x li x x( x) ( x) (x) (x) (x ) ( ) ( ) ( ) ( ) () ( ) Theore 5 Let x be a ideteriate ad let l,, be positive itegers with l > l 2 ad c, c 2 Q soe costats The 0 x c 2 H() l c H() l x x H() H() H() 4H() c H (2) H(2) l c 2 H (2) H(2) l

6 INTEGERS: (20) 6 ( ) x c H() l c 2 H() l x( x) ( x) (x) (x) c sl ( x s) c 2 sl ( x s) (3) Proof Usig partial fractio decopositio we ca write f(x) : x( x) ( x) (x) (x) A x c sl B (x ) 2 C x ( x s) c 2 D x sl ( x s) for soe A, B, C ad D Q As i the proof of Theore 4, we isolate the coefficiets A, B, C ad D by taig various liits of f(x) For brevity, we first let ad The we have A li xf(x) c li T (r) a : c sl (a s) r c 2 sl (a s) r U (r) : c H (r) H(r) l c 2 H (r) H(r) l sl c sl ( x) ( x) ( x) ( x) (s x) c 2 li s c 2 c l sl s c 2 H () l sl ( x) ( x) ( x) ( x) (s x)

7 INTEGERS: (20) 7 For, B li x (x )2 f(x) li x x( x) ( x) (x) 2 (x ) (x ) x ( ) ( ) ( ) 2 () () ad d C li (x ) 2 f(x) x dx d x( x) li ( x) x dx (x) 2 (x ) (x ) U () x ( x) li ( x) x (x) 2 (x ) (x ) x T (2) x x x x ( x s) ( x s) (x s) (x s) 2 (x s) ( ) ( ) ( ) 2 () () T (2) ( s) ( s) (s) (s) 2 ( s) U (2) H() H() H() 4H() U () s0 s0

8 INTEGERS: (20) 8 For, x( x) D li (x )f(x) li ( x) x x x (x) (x) (x ) ( ) ( ) ( ) ( ) () ( ) U () Proofs of Theores 2 ad 3 Multiply both sides of (2) ad (3) respectively by x ad tae the liit as x Refereces [] S Ahlgre, S B Ehad, K Oo, D Zeilberger, A bioial coefficiet idetity associated to a cojecture of Beuers, Electro J Cobi 5 (998), Research Paper 0, [2] S Ahlgre, K Oo, A Gaussia hypergeoetric series evaluatio ad Apéry uber cogrueces, J Reie Agew Math 58 (2000), [3] W Chu, A bioial coefficiet idetity associated with Beuers cojecture o Apéry ubers, Electro J Cobi (2004), o, Note 5, 3 [4] D McCarthy, p-adic hypergeoetric series ad supercogrueces, PhD thesis, Uiversity College Dubli, 200 [5] D McCarthy, Extedig Gaussia hypergeoetric series to the p-adic settig, subitted [6] D McCarthy, O a supercogruece cojecture of Rodriguez-Villegas, Proc Aer Math Soc, accepted for publicatio