Liu et al. Advaces i Differece Euatios 05 05:54 DOI 0.86/s366-05-0568-6 R E S E A R C H Ope Access A geeralizatio of Morley s cogruece Jiaxi Liu,HaoPa ad Yog Zhag 3* * Correspodece: yogzhag98@63.com 3 Departmet of Mathematics ad Physics, Najig Istitute of Techology, Najig, 67, People s Republic of Chia Full list of author iformatio is available at the ed of the article Abstract We establish a explicit formula for -aalog of Morley s cogruece. MSC: Primary B68; secodary B65; B83 Keywords: Morley s cogruece;-aalog; biomial sums Itroductio For arbitrary positive iteger,let, which is the -aalog of a iteger sice lim /. Also,for, m Z, defie the -biomial coefficiets by m m m m whe m 0, ad if m <0weset m 0.Itiseasytochecthat m m m m Some combiatorial ad arithmetical properties of the biomial sums. 0 a ad 0 a have bee ivestigated by several authors e.g., Cali, Cusic, McItosh 3, Perlstadt 4. Ideed, we ow cf. 5, euatios 3.8 ad 6.6 that 0. 05 Liu et al. This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese http://creativecommos.org/liceses/by/4.0/, which permits urestricted use, distributio, ad reproductio i ay medium, provided you give appropriate credit to the origial authors ad the source, provide a li to the Creative Commos licese, ad idicate if chages were made.
Liuet al. Advaces i Differece Euatios 05 05:54 Page of 7 ad 0 3 3.. However,byusigasymptoticmethods,deBruij6 has showed that o closed form exists for the sum 0 a whe a 4. Wilf proved i a persoal commuicatio with Cali; see that the sum a 0 hasoclosedformprovidedthat3 a 9. As a -aalog of., we have 0..3 Ideed, from the well-ow -biomial theorem cf. Corollary 0.. of 7 where x x;, 0 x x x, if, x;, if 0, it follows that x ; x; x; 4 m0 x m 0 0 m x m, 0 x whece.3 is derived by comparig the coefficiets of x i the euatio above. As early as 895, with the help of De Moivre s theorem, Morley 8provedthat p p 4 p mod p 3..4 p / I 9, Pa gave a -aalog of Morley s cogruece ad showed that where 4 / e πij/ j j, ; 4 mod 3,.5
Liuet al. Advaces i Differece Euatios 05 05:54 Page 3 of 7 is the th cyclotomic polyomial. I this sectio, we shall establish a geeralizatio of Morley s cogruece.4 proved by Cai ad Graville 0, Theorem 6: p 0 a p a ap mod p 3.6 for ay prime p 5 ad positive iteger a.wealsoshallobtaiageeralizatioof.5i view of.3. Theorem. Let be a positive odd iteger. The a a 0 a ; a aa mod 3..7 4 Furthermore, we have a /4 a / 0 a ; a aa mod 3..8 4 Remar Clearly.6isthespecialcaseof.7 i the limitig case >for p. Some lemmas I this sectio, the followig lemmas will be used i the proof of Theorem.. Lemma. Proof mod 3.. j j j0 j j mod 3. Lemma. Let be a positive odd iteger. The / j / j j / j 4 Q, mod,. where the -Fermat uotiet is defied by Q m, m ; m /;.
Liuet al. Advaces i Differece Euatios 05 05:54 Page 4 of 7 Lemma.3 Let be a positive odd iteger. The / Q, Q, j Q, mod..3 8 Whe is a odd prime, the above two lemmas have bee proved i 9, euatio.7 ad 9, Theorem., respectively. Of course, clearly the same discussios are also valid for geeral odd. 3 ProofsofTheorem. I this sectio, we shall prove.7ad.8. Proof By the properties of the -biomial coefficiets, we ow that j j j mod 3. j i j Thus a a a a a j Notig that j we have a a i<j a i<j i j a a j a j j a a i j, j a j i<j i<j j i<j i<j a i j j mod 3. a a i j j i j j j a mod 3. 3. j j
Liuet al. Advaces i Differece Euatios 05 05:54 Page 5 of 7 Thus lettig a i3., we get i<j j whece by.3ad3. i<j j ; i j Q, i j j Recallig. ad.3, the we obtai a a a 0 a Q, a j0 a j mod 3, 3. Q, 6 mod. Q, a a j j Q, j a ; a aa mod 3. 4 Let us tur to.8. Similarly a 0 a a /4 a / 0 a a a a a a a j a mod 3. Recallig., the we have a a a / mod 3, therefore a a a a a a j j j a a a a j j j j j mod 3. j
Liuet al. Advaces i Differece Euatios 05 05:54 Page 6 of 7 Sice ad we have a a a a mod 3 a a / a / mod, a a a a / a a a Notig that a mod 3. j j j j j j j j j j j j j j j j j j 4 mod, we have a 0 a / a a /4 a / aa 4 a mod 3. 8 0 a 4 a I view of.7, this cocludes the proof of.8. Competig iterests The authors declare that they have o competig iterests. Authors cotributios All authors cotributed eually to the writig of this paper. All authors read ad approved the fial mauscript. Author details Departmet of Teachig Affairs, Najig Istitute of Techology, Najig, 67, People s Republic of Chia. Departmet of Mathematics, Najig Uiversity, Najig, 0093, People s Republic of Chia. 3 Departmet of Mathematics ad Physics, Najig Istitute of Techology, Najig, 67, People s Republic of Chia. Acowledgemets The authors tha the aoymous referee for his her valuable commets ad suggestios. The first ad the third authors are supported by the foudatio of Jiagsu Educatioal Committee No. 4KJB0008 ad Natioal Natural Sciece Foudatio of Chia Grat No. 4030. The secod author is also supported by NNSF Grat No. 785. Received: 5 April 05 Accepted: 8 July 05
Liuet al. Advaces i Differece Euatios 05 05:54 Page 7 of 7 Refereces. Cali, NJ: Factors of sums of powers of biomial coefficiets. Acta Arith. 86, 7-6 998. Cusic, TW: Recurreces for sums of powers of biomial coefficiets. J. Comb. Theory, Ser. A 5, 77-83 989 3. McItosh, RJ: Recurreces for alteratig sums of powers of biomial coefficiets. J. Comb. Theory, Ser. A 63, 3-33 993 4. Perlstadt, MA: Some recurreces for sums of powers of biomial coefficiets. J. Number Theory 7, 304-309 987 5. Gould, HW: Combiatorial Idetities: A Stadardized Set of Tables Listig 500 Biomial Coefficiet Summatios. Hery W. Gould, Morgatow 97 6. de Bruij, NG: Asymptotic Methods i Aalysis. Dover, New Yor 98 7. Adrews, GE, Asey, R, Roy, R: Special Fuctios. Cambridge Uiversity Press, Cambridge 999 8. Morley, F: Note o the cogruece 4!/!,where is a prime. A. Math. 9, 68-70 895 9. Pa,H:A -aalogue of Lehmer s cogrueces. Acta Arith. 8,303-38 007 0. Cai, T-X, Graville, A: O the residues of biomial coefficiets ad their products modulo prime powers. Acta Math. Si. Egl. Ser. 8, 77-88 00