Some identities involving the partial sum of q-binomial coefficients

Transcription

1 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 , ] ad Zhag s detty [Dscrete Math , ]. Keywords: bomal coeffcets, -bomal coeffcets, -bomal theorem Itroducto I [2], Cal proved the followg curous detty: Hrschhor [5] establshed the followg two dettes o sums of powers of bomal partal sums: 2 + 2, ad the electroc oural of combatorcs , #P3.7

2 I [7], Zhag proved the followg alteratg form of 2: 0 2, f 0, 2 2, f s eve ad 0, /2 /2, f s odd. Several geeralzatos are gve [6, 8, 9]. -dettes: Later, Guo et al. [4] gave the followg 2 2, 2 ad 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 ; the electroc oural of combatorcs , #P3.7 2

3 Theorem 2. For ay o-egatve teger, we have ; 4+ ad , ; 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 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, z z, 0 the electroc oural of combatorcs , #P

4 ad z; 2 z; 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 + [ ] 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 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 ;, ad tag m 4, we obta 0 2, +,. the electroc oural of combatorcs , #P3.7 4

5 Hece, by Lemma 2., we get ; ; ; + ; 2, 3 Proof of Theorem 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 Proof. By 8, we fd that ad z; + z 0 z; z z 2 ; 2 z 2 2 z z, 0 2 z z, 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 , #P3.7 5

6 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+. Replacg by 2 + 9, we obta ; 2. 0 the electroc oural of combatorcs , #P3.7 6

7 Hece, by Lemma 3., we get ; ; ; We ext show 7. By 8, we have ; 2, ad replacg by 2 9, we obta ; 2. 0 Hece, by the fact whch follows easly from the substtuto 2, we have ; ; Acowledgemet I would le to tha the referee for hs/her helpful commets. the electroc oural of combatorcs , #P3.7 7

8 Refereces [] G.E. Adrews, The Theory of Parttos, Cambrdge Uversty Press, Cambrdge, 998. [2] N.J. Cal, A curous bomal detty, Dscrete Math , [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 , [5] M. Hrschhor, Cal s bomal detty, Dscrete Math , [6] J, Wag, Z. Zhag, O extesos of Cal s bomal dettes, Dscrete Math , [7] Z. Zhag, A d of bomal detty, Dscrete Math , [8] Z. Zhag, J. Wag, Geeralzato of a combatoral detty, Utl. Math , [9] Z. Zhag, X. Wag, A geeralzato of Cal s detty, Dscrete Math , the electroc oural of combatorcs , #P3.7 8