Sum of cubes: Old proofs suggest new q analogues

Transcription

1 Sum of cubes: Old proofs suggest ew aalogues Joha Cigler Faultät für Mathemati, Uiversität Wie Abstract We prove a ew aalogue of Nicomachus s theorem about the sum of cubes ad some related results Itroductio I [], [5] ad [8] some aalogues of the well-ow formula () have bee foud Some iformatio about this formula which is sometimes called Nicomachus s theorem is give i [7] I this ote I propose aother aalogue which is ispired by a old result of C Wheatstoe [6] He observed that the odd umbers ca be grouped i such a way that the idetities () hold, which implies that () follows from the well-ow formula () ()

2 Idetity () is usually stated i the form (4) This is a immediate coseuece of (5) A simple proof without words of (4) has bee give i [4], which I reproduce here: The simplest computatioal proof of () uses the trivial idetity gives the telescopig sum which

3 aalogues [ ] [ ] [ ] As usual we let [ ] ad [] [] [ ] 0 It is clear that lim ad lim for KC Garrett ad K Hummel [] derived a combiatorial proof of the aalogue SO Waraar [5] proposed the idetity [ ] [ ], [] (6) [ ] [ ] (7) ad G Zhao ad H Feg [8] gave a combiatorial iterpretatio of 4( ) [] 0 (8) For a proof by iductio observe that implies a a a a a a a,

4 ad 4 A similar idetity is, (9) which follows from This idetity gives the telescopig sum [ ] [ ] Thus we have obtaied our mai result Theorem [ ] [ ] (0) 4

5 We ow give two further proofs which geeralize the beautiful proofs which we have setched i the itroductio ad which origially led to this aalogue We start with the followig well-ow (cf []) aalogue of () [] [ ] () A computatioal proof uses the fact that ( ) [ ] [] But formula () has also a ice combiatorial iterpretatio Cosider the suares S i, 0, wi (, ) i ad associate with each poit ( i, ) S the weight i The weight of S is i i0 0 ws [ ] () The suare S is the uio of the hoos h,0,,,,,,,,, 0,, The weight of the hoo h is 0 w h The poit,0 will be called the base-poit of the suare As a example cosider the case 6 i matrix otatio () Here we have wh 5 ( ), wh 4 ( ) [], wh ( ) [5], 5

6 Secod proof of Theorem We first observe the ice aalogues [] [ ] [] [5] [] [ ] [7] [9] [] [] [ ] ad more geerally [ ] [ ] (4) of formulae () They follow from the idetity 0 0 [ ] [ ] [ ] [ ] Usig () we get the desired result [ ] [ ] [ ] Third proof of Theorem A combiatorial proof ca also be give alog the lies of the above proof without words For odd the uio R of the hoos h,, h is the uio of suares of sidelegth whose base poits have weight m, 0 m 6

7 The weight of these suares is ( ) [ ] [ ] [ ] [ ] For the uio R of the hoos h,, h is the uio of suares whose m base-poits have weights, 0 m, ad of two rectagles with side legths ad as i the blue regio i the above figure The weight of the uppermost rectagle is rectagle is [ ] [ ] [ ] [ ] ad the weight of the leftmost Thus the total weight of this regio is m [ ] [ ] [ ] [ ] m0 m 4 m0 [ ] [ ] 4 4 [ ] [ ] [ ] [ ] 4 Thus we get wr ws ad thus agai (0) 4 Related results Let us ote some related results For ay seuece of positive itegers a ( ) the sums ai () i [ a( )] a( i) i (5) 7

8 ad ai () i [ a( )] ai ( ) i (6) are aalogues of i ai () The proofs are obvious because () a() i a i a() i ( ) () i a a i i i i ai () [ a ( )] ai () i i ad ai () ai () a( ) i a( ) i a( ) a i a a i i i () [ ( )] () By choosig a ( ) we get the followig aalogues of (): Theorem [ ] (7) ad i i [ ] (8) 8

9 From the recurrece relatios for the biomial coefficiets we get the well-ow formulae (9) ad ( )( ) (0) For these sums ca be used to compute we get For example from ( ) Ufortuately i geeral the sums [ ] do t have a simple expressio For example from [ ] [] [] [] [] [ ] we get [ ] [] [] [] [] [ ] 5 ( ) 4 Now ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] ( )( ) [][] 9

10 Thus we get 5 [ ] [] [] [] [] 4 () This formula caot be simplified A curious geeralizatio of () is due to P Luthy [] He observed that () ad more geerally () ( ) To see this observe that Similar results also hold for the aalogues of () for (4) For 0 0 ( ) 0

11 4 More aalogues of idetities by C Wheatstoe [6] 4 A aalogue of ( ) is 0 For [] [ ] [ ] (5) ( )[ ] [ ] ( ) 4 A similar idetity is The left-had side is m m (6) 0 [ ] m m (m ) (m) C Wheatstoe observed that (7) implies 0 (8)

12 A aalogue is Theorem [] (9) ad therefore (0) 0 Proof Observe that (cf []) [ ] 0 () This is the special case m 0 of (6) For a ( ) the left-had side of (9) becomes a ( ) a( ) a( ) a ( ) ( )( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) a( ) This gives )( ) ( )( ) ( ( )( ) ( )( ) [] ad therefore, which by () implies (0) 0

13 44 is 0 A aalogue of ( ) ( ) ( ) () 0 A aalogue of () is 0 ( ) ( ) () 0 More geerally we get for m m m m ( ) m ( ) m (4) 0 The proofs are straightforward ad will be omitted 45 C Wheatstoe also observed that More geerally this gives 0 We will ow prove three differet aalogues of this idetity

14 Theorem 4, (5) 0 0 ( ), (6) 0 ad [ ] (7) 0 Proof (5) follows from For the secod idetity the left-had side is, which implies the right-had side After dividig (7) by [ ] ad multiplyig with ( )( ) the left-had side is ( ) ( ) 0 0 ( ) This proves (7) 4

15 46 Fially we loo for a aalogue of Theorem 5 Proof (8) 0 () 8 84 () () () ( ) 4 Refereces [] K C Garrett ad K Hummel, A combiatorial proof of the sum of -cubes, Electro J Comb (004), #R9 [] P Luthy, Odd sums of cosecutive odds, [] M Schlosser, -Aalogues of the sums of cosecutive itegers, suares, cubes, uarts ad uits, Electro J Comb (004), #R7 [4] B R Sears, Problem collectio, [5] S O Waraar, O the -aalogue of the sum of cubes, Electro J Comb (004), #N [6] C Wheatstoe, O the formatio of powers from arithmetical progressios, Proc Royal Soc Lodo 7 (854), 45-5 [7] Wiipedia, Suared triagular umber, [8] G Zhao ad H Feg, A ew aalogue of the sum of cubes, Discrete Math 07 (007),