$q$-Analogues of the Sums of Consecutive Integers, Squares, Cubes, Quarts and Quints

Abstract

We first show how a special case of Jackson's ${}_8\phi_7$ summation immediately gives Warnaar's $q$-analogue of the sum of the first $n$ cubes, as well as $q$-analogues of the sums of the first $n$ integers and first $n$ squares. Similarly, by appropriately specializing Bailey's terminating very-well-poised balanced ${}_{10}\phi_9$ transformation and applying the terminating very-well-poised ${}_6\phi_5$ summation, we find $q$-analogues for the respective sums of the first $n$ quarts and first $n$ quints. We also derive $q$-analogues of the alternating sums of squares, cubes and quarts, respectively.