Table 1. A framework for Hill numbers, functional Hill numbers, mean functional diversity and (total) functional diversity of a single assemblage.

	Abundance vector/matrix	weights	q-th power sum ( q≠1)	Equating the two q-th power sums
(1) Hill numbers
Actual assemblage	S species with relative abundance vector:	Unity weight for each species
		(1, 1, …., 1)
Idealized reference assemblage	D equally-abundant species	Unity weight for each species		(Hill number of order q)
		(1, 1, …., 1)
(2) Functional Hill number, mean functional diversity and (total) functional diversity
Actual assemblage	matrix of the product of relative abundances for pairs of species	distance matrix as weight
Idealized reference assemblage	D×D matrix of the product of equal relative abundances for pairs of species	D×D idealized distance matrix as weights	Or
				(Functional Hill number  =  number of rows or columns in the idealized distance matrix)
		or
				(Mean functional diversity  =  column/row sum in the idealized distance matrix)
		Q* = QD /(D −1)
				(Total functional diversity =  grand sum of the idealized distance matrix)