(2026) 6, pp. 74-76. (pdf)
Thad S. Morton
Abstract:
The product of two symmetric q-numbers is known to admit an exact expansion in terms of ordinary symmetric q-numbers. This paper extends that result to products of an arbitrary number of generalized symmetric q-numbers. The resulting identities express each product as a finite sum of ordinary symmetric q-numbers whose indices are generated by centered arithmetic progressions associated with the individual factors. The formula provides a unified representation of products of factors involving different base q and demonstrates that products of generalized symmetric q-numbers remain within the algebra generated by ordinary symmetric q-numbers.
References:
Fraenkel, A. A., Integers and Theory of Numbers, Scripta Mathematica, NY (1955).
Morton, T. S., “On a product of two quantum integers,” Journal of Scientific and Mathematical Research 3 (2009) pp. 5-6.
Nathanson, M. B., “A functional equation arising from multiplication of quantum integers,” Journal of Number Theory 103 (2003) pp. 214–233.
Nathanson, M. B., “Formal power series arising from multiplication of quantum integers,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science 64 (2004) pp. 145-167.