Note On Some Identities of New Combinatorial Integers

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1 Apped Mathematcs & Informaton Scences 5(3 (20, An Internatona Journa c 20 NSP Note On Some Identtes of New Combnatora Integers Adem Kııçman, Cenap Öze 2 and Ero Yımaz 3 Department of Mathematcs and Insttute for Mathematca Research Unversty Putra Maaysa, UPM, Serdang, Seangor, Maaysa 2,3 Department of Mathematcs, Abant Izzet Baysa Unversty Gooy Kampusu, Bou 4280, Turey Emas Address: acman@putra.upm.edu.my Receved June 22, 200; Revsed Dec. 2, 200; Accepted March 02, 20 The bnoma numbers ( m n are very mportant n severa appcatons and satsfy severa number of denttes. The purpose of ths paper s to ntroduce a new combnatora nteger ( m,n and obtan some agebrac denttes by means of doube combnatora argument. Further severa arthmetc propertes of ths type of ntegers are proven and some nterestng denttes are aso provded. AMS Subect Cassfcaton: Prmary 05A9; Secondary 3F25. Keywords and phrases: bnoma expresson; combnatora numbers; combnatora denttes Introducton The bnoma coeffcents, denoted by ( n, pay an mportant roe n combnatorcs and these numbers appear as coeffcents n the expanson of the bnoma expresson (x y n. That s, (x y n ( n x y n (. and ths dentty s nown as Bnoma theorem. The bnoma coeffcents are aso nown as combnatons or combnatora numbers. In fact the equaton (. has very cose reaton to the to the dscusson of prme numbers. Further, prmes come up n many dfferent paces n the mathematca terature, and there are a ot of dscussons to dstngush prmes from the compostes. In the terature the we nown and the most amazng propertes of prme numbers, dscovered by Fermat that, f n s prme, then n dvdes a n a for a ntegers a. That s 500

2 Note On Some Identtes of New Combnatora Integers 50 a n a (mod n, for a ntegers a and n. Now f n s a prme number then t was proved that (x y n x y x n y n (mod n for a ntegers x, y and prmes n. Then we state the foowng reated theorem whch was proved n []. Theorem.. Integer n s prme f and ony f (x n x n (mod n n Z[x]. The arthmetca propertes of bnoma coeffcents have aso been studed by many authors, for exampe see [3]. The sequence of mdde bnoma coeffcents (a n ( 2n n nown aso as centra bnoma coeffcent s an partcuar nterest to many peope and have the foowng generatng functon 4x 2x 6x 2 20x 3 70x 4 252x (.2 The mdde bnoma coeffcents aso pay an sgnfcant roe n Erdös Conecture that s wdey nown as square-free ntegers, see [2]. In constructng the propertes and denttes of some speca numbers, bnoma coeffcents are frequenty nvoved. Note that the defnton of the bnoma coeffcents was extended n [3] where n can be a compex number. However there are st many propertes and denttes that one can estabsh by usng the bnoma coeffcents. In the next we ntroduce the combnatora ntegers ( n,m whch s usefu for the cacuatons n cohomoogy. 2 Identtes on Combnatora Integers Now reca the equaton (. and consder to mutpy by (x y then we have (x y n (x y More genera, (x y n (x y m ( n x y n (x y 0 ( n x y n ( n x y n (x y m ( n 0 m ( m x ( y m x y n m ( n ( m ( n x y n. (2. ( m x y mn (2.2 where m, n 0. Now we et P n (x denote the Legendre poynomas of nth order. Then the functon P (x, y P (xy ( 2xy y 2 /2

3 502 A. Kııçman, C. Öze and E. Yımaz s the generatng functon for Legendre poynomas. Then we can easy have P ( 2x, y ( y [ 4xy( y 2 ] /2, where Q n (x P n ( 2x, Q(x, y P ( 2x, y and Now we have two expansons P ( 2x m n( 2x n Q(x, y m n( 2x, y P ( 2x, y. n Q(x, y ( y [ 4xy( y 2] /2, ( 2 x y ( y 2 y n n0 ( n 2 ( 2 x, so that Then so that Q(x, y ( ( 2xy [ 4(x x 2 y 2 ( y 2xy 2] /2, y ( ( n 2 n ( 2x n 2 (x x 2, (2.3 2 n0 Q n (x q n, ( n 2 0 ( n 2 0 ( 2 Q n (x ( 2 x n 2 ( n 2 ( 2 q n, x ( x n ( n 2, ( ( ( n n Q n (x ( 2x n 2 (x x 2. where ( 2 n (by Vandermonde convouton ( 2 n x ( x n see [4]. Now by mang substtutons for x we can obtan severa denttes, for exampe, f we repace x 2 n equaton (2.3 then t foows that Q ( 2, y ( y 2 /2 ( 2n ( n 2 2n y 2n, n n0

4 then we obtan the foowng denttes ( 2n ( n n Note On Some Identtes of New Combnatora Integers ( ( 2n 2 ( 2 2n 2 2 ( 2 2n (. Now by usng the rght hand sde of the equaton (2.2 and symmetrc propertes of the bnoma ( coeffcents respectvey we can ntroduce a new and nterestng nteger sequence n, m that w smpfy the cacuaton and we w ca t twn pars bnoma coeffcents. In partcuar we can represent ( x n ( x m 0 nm 0 ( n, m x. Note that f we consder x and x then t s easy to obtan nm ( nm n, m ( n, m ( 0. 0 In fact, ths type of sequences pay an mportant roe for the computaton n cohomoogy, see [5]. Through out ths study we ca t twn pars bnoma coeffcents as foows: ( m, n ( ( m n ( where n, m 0 ntegers and 0 m n, and of course ( m m!!(m! f m, 0 f m <. ( m, n Note that the generatng functon of the nteger sequence of s the functon ( ( m, n x n ( x m. By defnton of we can easy show that ( m, n ( ( m n ( ( ( n m ( ( n, m f even, ( n, m f odd, ( ( n m (

5 504 A. Kııçman, C. Öze and E. Yımaz ( n, n and hence 0 whenever s odd. The cacuaton of the twn pars bnoma coeffcents aso pay a cruca roe n the cacuaton of generators for deas. Further f we et G be a compact sem-smpe Le group, LG the space of smooth oops on the group G and T s the maxma torus of G, then twn pars of bnoma coeffcents w aso be very usefu durng the determnaton of the ran for each modue whch s graded by ntegra cohomoogy agebra of fnte dmensona fag manfods LG/T and ΩG n the oca coeffcent rng Z[ 2 ] for G A 2, see [5]. Now the symmetry and ant-symmetry property can be gven n the foowng theorem. Theorem 2.. Let n be a non-negatve nteger. For 0,, 2,..., n we have ( ( n,, n ( n n, n Proof. By defnton, for 0,, 2,..., n we have (, n f n even, f n odd. ( ( n ( ( n ( ( 0 0 ( ( n ( n 2 0 n ( ( n ( n2, n n 2 n ( ( n ( f n even n n 2 n ( ( n ( f n odd. n n 2 Snce for < n 2, we have n > so t foows that ( n 0 where 0,,..., n 2. Therefore we have (, n n n Hence we have the desred resut. ( 0 ( ( n n 0 ( ( ( n n f f n even n odd. We note that the twn pars bnoma coeffcents have aso smar propertes to the bnoma

6 coeffcents. In partcuar Note On Some Identtes of New Combnatora Integers 505 n 0 n n 2 n 3 n 4 n 5 ( 0,5 0 ( 0, ( 0,0 0 (,0 ( 0,2 0 (, ( 2,0 ( 0,3 0 (,2 ( 2, 2 ( 3,0 ( 0,4 0 (,3 ( 2,2 2 ( 3, 3 ( 4,0 0 (,4 ( 2,3 2 ( 3,2 3 ( 4, 4 ( 5, Smar to tranguar propertes n the snge bnoma coeffcents form we aso we have the foowng denttes: Theorem 2.2. Let r, s,, p be non-negatve ntegers. Then ( ( ( r, s r, s r, s (rght shftng property, (2.4 ( ( ( r, s r, s r, s (eft shftng property, (2.5 ( r, s ( r, s (rght shftng expanson, (2.6 0 ( r, s p ( ( r, s p p (Vandermonde convouton, (2.7 0 ( ( ( r, s r, s r, s 2 and (2.8 ( ( ( r, s r, s r, s 2. (2.9 Proof. Frst we sha prove that equaton (2.4 hods. Then Snce ( r, s ( r, s ( ( r s ( ( ( r s ( 0 0 ( s 0, then 0 ( r ( 0 ( r ( 0 ( ( r s ( r ( ( 0 [( ( ] s s ( s. ( s ( r, s ( r, s.

7 506 A. Kııçman, C. Öze and E. Yımaz Let be even. Then we have ( ( ( ( r, s s, r s, r s, r ( ( r, s r, s ths s the equaton (2.5. Now et be odd. Then we have the equaton (2.6 as foows: ( ( ( r, s s, r s, r ( ( r, s r, s. ( s, r If we tae the sum (dfference of both sdes of equatons (2.4 and (2.5, then we obtan equatons (2.8 and (2.9. Equatons (2.6 and (2.7 can be aso obtaned from equaton (2.4. Theorem 2.3. Let r, s, be non-negatve ntegers. Then ( ( ( r, s r, s r, s s ( r r, (2.0 ( ( ( r, s r, s r, s r ( s s, (2. ( ( ( r, s r, s r, s (r s r s. (2.2 Proof. Let us begn the proof of the frst equaton (2.0. Then ( ( r, s r, s ( r r ( ( r s ( ( r s ( r ( r ( 0 0 ] ( [( r! s! r!(r! (!(s! r! s!!(r! (!(s! 0 ( r!s! ( r r!(r!(!(s! 0 s ( r! (s! ( ( r s!(r! (!(s! s ( 0 0 ( ( r s s ( 0

8 Note On Some Identtes of New Combnatora Integers 507 ( s snce 0. Therefore we have ( ( ( r, s r, s r, s ( r r s. If s odd then we have ( ( ( ( r, s s, r s, r s, r r r ( s s ( ( r, s r, s ( s s. If s even then we have ( ( ( ( r, s s, r s, r s, r r r ( s s ( ( r, s r, s ( s s. By usng the equatons (2.0 and (2. we obtan the equaton (2.2 as foows ( ( ( ( ( ( r, s r, s r, s r, s r, s r, s r s s ( r (s r ( ( ( r, s r, s r, s 2(r s s r and hence we have { ( ( } ( r, s r, s r, s 2 r s 2(r s. Lemma 2.. Let n be a non-negatve nteger and 0,, 2,..., n then (, n 2 n f 0 0 f 0. Proof. For 0, Let 0. Snce for x, then we have 0 (, n 0 ( x n ( x ( n 2 n. (, n x, (, n.

9 508 A. Kııçman, C. Öze and E. Yımaz Smary we have the foowng resut. Lemma 2.2. Let n be a non-negatve nteger. For 0,, 2,..., n we have ( (, n 2 n f n 0 f n. 0 Thus we can state ths resut as the foowng coroary. Coroary 2.. (Twn pars Let n be a non-negatve nteger. Then we have ( 2n, 2n 2n 2 0 ( ( 2n 2 ( 2n ( n. n Note that n partcuar we can easy show that ( n, n ( ( n 2 n n f n even, 2 n 0 f n odd. Now et us consder addton of the twn bnoma coeffcents as foows: ( 0, 0 ( 0 ( 0,, 0 ( 0 ( ( 0, 2, 2, 0 ( 0 ( ( 2 ( 0, 3, 2 2, 3, 0 ( 0 ( ( 2 ( 3 ( 0, 4, 3 2, 2 3, 4, 0 ( 0 ( ( 2 ( 3 ( 4 ( 0, 5, 4 2, 3 3, 2 4, 5, and so on. Thus n genera case we have the foowng theorem Theorem 2.4. (Dagona formua Let n be a non-negatve nteger. Then we have (, n 2 n 2 f n even, 0 f n odd. Proof: We prove ths n two cases: Case. Let n be odd. Then by usng the theorem (2. we have (, n ( n, n

10 Note On Some Identtes of New Combnatora Integers 509 then by mang substtuton n we have (, n (, n (, n 0 (, n (, n (, n Case 2. If n s even nteger then we w show that 2 n 2. The proof w be done by usng the nducton. ( 0, 0 Case s trva for n 0 snce we have (, For 0 n wth s even and assume that 2 2. n2 (, n 2 Our am s now to show 2 n2 2. ( n, n (, n Now f n snce 2 n/2 then t foows that 0. Now n2 ( n, n 2 ( (, n, n n ( n, n (, n [( ( ], n, n 2 (, n (, n on usng shftng propertes n ( 2 n, n 2 2 ( n 2 ( n 2 2 ( n ( n ( n 2 2, n 2 for 0 n 2. On usng the geometrc seres summaton we obtan that competes the proof. n2 (, n 2 2 ( n2 2

11 50 A. Kııçman, C. Öze and E. Yımaz Theorem 2.5. (Orthogonaty formua Let n be a non-negatve nteger. For, 0,, 2,..., n we have ( (, n, n f, 2 n δ 0 f. Proof: Smar to the prevous theorem we aso use the nducton methods on n. The case s very obvous for n. Suppose that ( (, n, n f, 2 n δ 0 f. s true. Then we w show that n ( (, n, n f, 2 n δ 0 f. There are two dfferent cases: ( Let then t foows that n ( ( ( (, n, n n n ([( ( ] [( ( ], n, n, n, n ( n ( (, n, n ( (, n, n 2 ( (, n, n ( (, n, n ( n ( (, n, n 2 n ( ( (, n, n n ( 2 2 n 2 n 2(2 n 2 n ( Let. Then there are two subcases: Case. If. Then we have

12 Note On Some Identtes of New Combnatora Integers 5 n ( ( (, n, n n [( ( ] [(, n, n, n ( ( ( n n n ( n ( (, n, n ( n n ( (, n, n ( ( n n ( ( 0 ( (, n, n on notng that 0. ( n ( ( ( n ( ], n Case 2. If then the proof s smar to the case on notng that ( (, n, n 0 and (, n (, n 2 n. Acnowedgement: A part of ths wor was competed whe the second author was vstng the Mathematca Research Insttute, Unversty Putra Maaysa. Therefore the second author thans the staff of the Mathematca Research Insttute (INSPEM for ther hosptaty. The frst author aso acnowedge that ths research was partay supported by the Unversty Putra Maaysa under the Research Unversty Grant Scheme RU. References [] M. Agrawa, N. Kaya and N. Saxena, PRIMES n P, Ann. of Math., 60(2004, [2] P. Erdös, R. L. Graham, I. Z. Ruzsa and E. G. Straus, Math. Comput., 29 (975,

13 52 A. Kııçman, C. Öze and E. Yımaz [3] R. L. Graham, D. E. Knuth, and O. Patashn, Concrete Mathematcs: A Foundaton for Computer Scence (Second Edton, Addson-Wesey, Readng, Massachusetts, 989. [4] J. Rordan, Combnatora Identtes, John Wey & Sons, New Yor, 968. [5] C. Öze and E. Yımaz, Int. J. Math. Math. Sc., 2006 (2006, do:0.55 /IJMMS /2006/ Adem Kcman s a fu professor at the Department of Mathematcs, Facuty of Scence, Unversty Putra Maaysa. He receved hs B.Sc. and M.Sc. degree from Department of Mathematcs, Hacettepe Unversty, Turey and Ph.D from Lecester Unversty, UK. He has oned Unversty Putra Maaysa n 997 snce then worng wth Facuty of Scence and he s aso actve member of Insttute for Mathematca Research, Unversty Putra Maaysa. Hs research areas ncudes Functona Anayss and Topoogy. snce 998. C. Öze has got PhD from Unversty of Gasgow, Scotand, 998. He s the recpent of Sedat Smav Foundaton Natona Scence Laudabe prze from Tursh Journasts Socety 2005, and Young Outstandng Scentst prze from Tursh Scence Academy Hs nterests are Agebrac Topoogy especay compex cobordsm of nfnte dmensona manfods, Le theory, Quantum Mechancs and Mathematca Physcs. He s a member of Abant Izzet Baysa Unversty AIBU, Bou Turey

14 Note On Some Identtes of New Combnatora Integers 53 Ero Ymaz graduated from Istanbu Technca Unversty n 992 wth B.Sc degree n Mathematca engneerng. He receved ht M.Sc and Ph.D from The Unversty of Texas at Arngton n 996 and 999 respectvey. Hs researches ncude the ftng probem of poynoma deas, computng prmary decompostons, radcas and enveopes of submodues of Noetheran modues, computatona agebrac approach to center and cyccty probems of dnamca systems, and probems about homogeneous deas and syzyges. He generay used Gr?bner Bass Technques n hs wors. Ero Ymaz s member of Mathematc department of Abant Izzet Baysa Unversty, Bou, Turey snce 2000.