{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:25Z","timestamp":1753893805954,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $$ S_{m,n}(q):=\\sum_{k=1}^{n}\\frac{1-q^{2k}}{1-q^2} \\left(\\frac{1-q^k}{1-q}\\right)^{m-1}q^{\\frac{m+1}{2}(n-k)}. $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\\in{\\Bbb Z}[q]$ such that $$ S_{2m+1,n}(q) =\\sum_{k=0}^{m}(-1)^kP_{m,k}(q) \\frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}} {(1-q^2)(1-q)^{2m-3k}\\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ 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: $$ T_{m,n}(q):=\\sum_{k=1}^{n}(-1)^{n-k} \\left(\\frac{1-q^k}{1-q}\\right)^{m}q^{\\frac{m}{2}(n-k)}. $$<\/jats:p>","DOI":"10.37236\/1876","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:17:49Z","timestamp":1578719869000},"source":"Crossref","is-referenced-by-count":11,"title":["A $q$-Analogue of Faulhaber's Formula for Sums of Powers"],"prefix":"10.37236","volume":"11","author":[{"given":"Victor J. W.","family":"Guo","sequence":"first","affiliation":[]},{"given":"Jiang","family":"Zeng","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,8,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:47:50Z","timestamp":1579322870000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i2r19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,8,30]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2004,6,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1876","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,8,30]]},"article-number":"R19"}}