tailieunhanh - Báo cáo khoa học:On the q-analogue of the sum of cubes

A simple q-analogue of the sum of cubes is given. This answers a question posed in this journal by Garrett and their paper Garrett and Hummel commiserate the fact that (1) is not as simple as one might have hoped, and ask for a simpler sum of q-cubes. In response to this I propose the identity | On the q-analogue of the sum of cubes S. Ole Warnaar Department of Mathematics and Statistics The University of Melbourne VIC 3010 Australia warnaar@ Submitted Apr 7 2004 Accepted Aug 17 2004 Published August 23 2004 2000 Mathematics Subject Classification 05A19 Abstract A simple ợ-analogue of the sum of cubes is given. This answers a question posed in this journal by Garrett and Hummel. The sum of cubes and its q-analogues It is well-known that the first n consecutive cubes can be summed in closed form as n X k k 1 I 1 2 n 1 2 Recently Garrett and Hummel discovered the following q-analogue of this result X k-1 1 - qk 2 2 - qk 1 - qk 1 _ Pn 11 2 X 1 - q 2 1 - q2 I 2 . 1 where nl 1 - qn-k 1 1 - qn-k 2 1 - qn k 1 - q 1 - q2 1 - qk is a q-binomial coefficient. In their paper Garrett and Hummel commiserate the fact that 1 is not as simple as one might have hoped and ask for a simpler sum of q-cubes. In response to this I propose the identity X 2n-2k 1 - qk 2 1 - q2k Pn 11 2 2 Ềíq 1 - q 2 1 - q2 2 J Work supported by the Australian Research Council THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N13 1 Proof. Since n 112 n2 ri 2 _ 1 - qn 2 1 - q2n 2 J - q 2 1 - q 2 1 - q2 equation 2 immediately follows by induction on n. The form of 2 should not really come as a surprise in view of the fact that the q-analogue of the sum of squares X k2 7-n n 1 2n 1 6 k 1 is given by XX 2 -2k 1 - f 1 - q3k 1 - q 1 - qn 1 1 - q2n 1 Ỉ 1 q 1 - q 1 - q3 1 - q 1 - q2 1 - q3 and the q-analogue of X k n 1 k 1 X 1 - q n 11 èí 1 - q 1 2 J- References 1 K. C. Garrett and K. Hummel A combinatorial proof of the sum of q-cubes Electron. J. Combin. 11 2004 R9 6pp. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 N13