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

We first show how a special case of Jackson’s 8φ7 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 10φ9 transformation and applying the terminating very-well-poised 6φ5 summation, we find q-analogues for the respective sums of the first n quarts and first n quints. We also derive qanalogues of the alternating sums of squares, cubes and quarts, respectively.