tailieunhanh - Báo cáo toán học: " A q-Analogue of Faulhaber’s Formula for Sums of Powers"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A q-Analogue of Faulhaber’s Formula for Sums of Powers. | A Ợ-Analogue of Faulhaber s Formula for Sums of Powers Victor J. W. Guo and Jiang Zeng Institut Camille Jordan Universite Claude Bernard Lyon I F-69622 Villeurbanne Cedex France jwguo@ zeng@ Submitted Jan 25 2005 Accepted Aug 16 2005 Published Aug 30 2005 Mathematics Subject Classifications 05A30 05A15 Dedicated to Richard Stanley on the occasion of his 60th birthday Abstract Let q - n-k 1 - q2k 1 - qk w-1 Smn i I 2 1 1 - q2 1 - q J k 1 Generalizing the formulas of Warnaar and Schlosser we prove that there exist polynomials Pm k q G Z q such that S2ra 1 q X -1 kPmk qg. 1 - qf-f T -iL . Gi 1 - q2 1 - q 2- 3 nk o 1 - q 1- and solve a problem raised by Schlosser. We also show that there is a similar formula for the following q-analogue of alternating sums of powers Tmn q X -1 n-k k 1 1 - q qm n-k 1-q 1 Introduction In the early 17th century Faulhaber 1 computed the sums of powers 1m 2m nm up to m 17 and realized that for odd m it is not just a polynomial in n but a polynomial in the triangular number N n n 1 2. A good account of Faulhaber s work was given by Knuth 7 . For example for m 1 . 5 Faulhaber s formulas read as follows 11 21 n1 N N n2 n 2 12 22 . n2 i N 3 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 R19 1 13 23 n3 N2 14 24 n4 2 -2 V - 3N 15 25 n5 1 4N3 - N2 . 3 Recently the problem of q-analogues of the sums of powers has attracted the attention of several authors 2 9 8 who found in particular -analogues of the Faulhaber formula corresponding to m 1 2 . 5. More precisely setting XL 1 _ n2k Smn q L i i 1 - q 1 - qk m-1 1 - q q - 1 n-k Warnaar 9 for m 3 and Schlosser 8 found the following formulas for the q-analogues of the sums of consecutive integers squares cubes quarts and quints S 1 - qn 1 - qn 1 S1n q 1 - q 1 - q2 1 - qn 1 - qn 1 1 - qn 1 2 nq 1 - q 1 - q2 1 - q3 S 1 - qn 2 1 - qn 1 2 S3 n q 1 - q 2 1 - q2 2 S _ 1 - qn 1 - qn 1 1 - qn 1 r 1 - qn 1 - qn 1 1 - q1 S4 n q 5 2 3 q 1 - q 1 - q2 1 - q2 1 - q 2 1 - q2 .