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Highlights • A new nabla Taylor formula expanded at b ≥ n is derived for a discrete function. • Two kinds of series representations of are derived for nabla discrete fractional sum/difference. • Some essential properties of nabla discrete fractional calculus are investigated. • Besides fixed initial instant case, nabla fractional calculus with fixed memory step are introduced. Abstract This paper addresses the description and analysis problems of nabla discrete fractional calculus. The series representation framework is developed first, including two Taylor series expanded at the initial instant and the current time, respectively. Under this framework, several essential properties of fractional sum/difference are presented and investigated. Notably, the short memory principle is introduced for nabla discrete fractional calculus, along with which two corresponding series are studied. [ABSTRACT FROM AUTHOR]