(2026) 6, pp. 69-73. (pdf)
Thad S. Morton
Abstract:
A symmetric q-analogue is found for the classical finite alternating sum of consecutive numbers. Alternating sums of consecutive symmetric q-numbers exhibit parity-dependent closed forms that differ according to whether the truncation contains an even or odd number of terms. Even-length sums reduce to product expressions, whereas odd-length sums contain an additional residual term arising from the unpaired endpoint. These residual terms reveal a half-integer shift in the natural q-number index lattice associated with the alternating pairing structure and a new relevant q-number.
References:
Kupershmidt, B. A. & Morton, T. S., “An alternating sum of quantum integers,” Journal of Scientific and Mathematical Research 5, (2011) pp. 18-19.