Some dettes volvg the partal sum of -bomal coeffcets Bg He Departmet of Mathematcs, Shagha Key Laboratory of PMMP East Cha Normal Uversty 500 Dogchua Road, Shagha 20024, People s Republc of Cha yuhe00@foxmal.com Submtted: Feb 2, 204; Accepted: Jul 2, 204; Publshed: Jul 25, 204 Mathematcs Subect Classfcatos: 05A0; 05A5 Abstract We gve some dettes volvg sums of powers of the partal sum of -bomal coeffcets, whch are -aalogues of Hrschhor s dettes [Dscrete Math. 59 996, 273 278] ad Zhag s detty [Dscrete Math. 96 999, 29 298]. Keywords: bomal coeffcets, -bomal coeffcets, -bomal theorem Itroducto I [2], Cal proved the followg curous detty: 3 2 3 + 2 3 3 0 2 2 2. Hrschhor [5] establshed the followg two dettes o sums of powers of bomal partal sums: 2 + 2, ad 0 0 2 2 2 + 2 2 2 2. 2 the electroc oural of combatorcs 23 204, #P3.7
I [7], Zhag proved the followg alteratg form of 2: 0 2, f 0, 2 2, f s eve ad 0, 3 2 2 /2 /2, f s odd. Several geeralzatos are gve [6, 8, 9]. -dettes: 2 2 2 0 2 Later, Guo et al. [4] gave the followg 2 2, 2 ad 2+ 2 2+ 2 + 2 + 2 + 0 2 2 2 + 2 2 + Here ad what follows, 0 < s the -bomal coeffcet defed by ;, f 0, ; ; 0, otherwse, 2 + where z; z z z s the -shfted factoral for 0. The purpose of ths paper s to study -aalogues of 2 ad establsh a ew -verso of 3. Our ma results may be stated as follows. Theorem. For ay postve teger ad ay o-zero teger m, we have 0 m+ 2 m, m+,, 4 m ad 2 2+2 0 0 ; + ; 2 ; 2 0 2 32 2 + 2 +.. 5 the electroc oural of combatorcs 23 204, #P3.7 2
Theorem 2. For ay o-egatve teger, we have 2+ 2 + 2 2 + 0 0 2 + 22 + 2 ; 4+ ad 2+2 2 + 2 2 2 + 2 0 0 0 2 2+2 2 2 + 2 2 2, 6 22 +3+ 2 ; 4+3. Lettg ad usg L Hôptal s rule ad some famlar dettes, we easly fd that the dettes 4 5 ad 6 7 are -aalogues of 2 ad 3 respectvely. I Sectos 2 ad 3, we wll gve proofs of Theorems. ad.2 respectvely by usg the -bomal theorem ad geeratg fuctos. 2 Proof of Theorem. To gve our proof of Theorem., we eed to establsh a result, whch s a -aalogue of Chag ad Sha s detty see [3]. Lemma 3. For ay postve teger, we have 2 2+2 0 + 0 2 2 32 2 + 2 +. Proof. Accordg to the -bomal theorem see [], we have for all complex umbers z ad wth z < ad <, there holds z, 2 z 8 ad It follows that z; z z ; z + z. z, 0 2 z z, 0 0 2+ z z, 0 the electroc oural of combatorcs 23 204, #P3.7 3 0 7
ad z; 2 z; 2 2 0 2 2 z + z. Therefore, for ay o-egetve teger wth, the coeffcet of z z; z s 2, 0 the coeffcet of z z ; z s + [ ] 0 2 + + ad the coeffcet of z z; 2 z; 2 s 2 2. Usg the fact 0 z; z z ; z z; 2, z; 2 euatg the coeffcets of z ad after some smplfcatos, we obta Lemma 2.. Proof of Theorem.. We frst prove 4. 0 m+ 2 0 2 m [ 2+m ] m+ [ 2 0 ] m 0 0 m, m+, m, where the last step, we have used 8. We ext show 5. By 8, we have 2+2 2 ;, ad tag m 4, we obta 0 2, +,. the electroc oural of combatorcs 23 204, #P3.7 4
Hece, by Lemma 2., we get 2 2+2 0 0 2 2 ; 0 + 2 ; 2 0 2 2+2 0 + ; + ; 2, 3 Proof of Theorem.2 2+2 0 2 2 32 2 + 2 +. I order to prove the Theorem.2, we eed the followg result, whch gves a -aalogue of alteratg sums of Chag ad Sha s detty. Lemma 4. For ay o-egatve teger, we have 2 2+ 2 + 2 2 + 0 + 2 2 + 0 2 + 2 2 2. Proof. By 8, we fd that ad z; + z 0 z; z z 2 ; 2 z 2 2 z z, 0 2 z z, 0 0 0 2 2 z 2 z 2. 2 Therefore, for ay o-egetve teger wth, the coeffcet of z z; +z s 2, 0 0 the electroc oural of combatorcs 23 204, #P3.7 5
the coeffcet of z z; z s + ad the coeffcet of z z 2 ; 2 z 2 where [2 ] s defed by Usg the fact /2 0 [ [2 ] 2 s ] 2 2 {, f 2, 2 [2 ], 0, otherwse. z; z z; + z z2 ; 2 z, 2 euatg the coeffcets of z ad after some smplfcatos, we obta Lemma 3.. Proof of Theorem.2. We frst prove 6. By 8, we have ad 2+ 0 0 2 2 + 2 0 2 0 2 +,, 9 2 2 + 2 2 2 + 2 ; 2+. Replacg by 2 + 9, we obta 2+ 2 + 2 ; 2. 0 the electroc oural of combatorcs 23 204, #P3.7 6
Hece, by Lemma 3., we get 2+ 2 + 2 2 + 2 + 2 0 0 2+ 2+ 2 + 2 2 + 22 + 2 ; 2+ 0 + 2 2+ 2 + 22 + 2 ; 4+ 2 2 + 0 + 2 + 22 + 2 ; 4+ 2. 0 We ext show 7. By 8, we have 2 0 2 2 2 2 2 2 2 2 ; 2, ad replacg by 2 9, we obta 2 2 2 ; 2. 0 Hece, by the fact 2 2 2 2 2 0 + 2 2 whch follows easly from the substtuto 2, we have 2 2 2 2 2 0 2 2 0 22 2 ; 4. 2 0 2 22 2 ; 2 2 + 0 2 2 2 2 + 2 2 + 2 Acowledgemet I would le to tha the referee for hs/her helpful commets. the electroc oural of combatorcs 23 204, #P3.7 7
Refereces [] G.E. Adrews, The Theory of Parttos, Cambrdge Uversty Press, Cambrdge, 998. [2] N.J. Cal, A curous bomal detty, Dscrete Math. 3 994, 335 337. [3] G.-Z. Chag, Z. Sha, Problems 83-3: A bomal summato, SIAM Revew, 983, 25: 97. [4] V.J.W. Guo, Y.-J. L, Y. Lu, C. Zhag, A -aalogue of Zhag s bomal coeffcet dettes, Dscrete Math. 309 2009, 593 599. [5] M. Hrschhor, Cal s bomal detty, Dscrete Math. 59 996, 273 278. [6] J, Wag, Z. Zhag, O extesos of Cal s bomal dettes, Dscrete Math. 274 2004, 33 342. [7] Z. Zhag, A d of bomal detty, Dscrete Math. 96999, 29 298. [8] Z. Zhag, J. Wag, Geeralzato of a combatoral detty, Utl. Math. 7 2006, 27 224. [9] Z. Zhag, X. Wag, A geeralzato of Cal s detty, Dscrete Math. 308 2008, 3992 3997. the electroc oural of combatorcs 23 204, #P3.7 8