For a community \(i\), the Hill's diversity numbers are defined by the expression
\(J(\lambda) = \left(\sum \limits_{k\in S_i} p_{ki}^\lambda\right)^{\frac{1}{1-\lambda}}\)
with the restriction \(\lambda \geq 0\) where \(p_{ki}\) represents the relative frequency of species \(k\) and \(S_i\) are all species at the community, species richness, and \(\lambda\) is a free parameter. (This is equivalent to the exponential of Renyi's generalised entropy). The Renyi entropy of order \(\lambda\), where \(\lambda \geq 0\) and \(\lambda \neq 1\), is defined as
\(\mathrm{H}_{\lambda}(X)=\frac{1}{1-\lambda} \log \left(\sum \limits_{i=1}^{n} p_{i}^{\lambda}\right)\)
Here, \(X\) is a discrete random variable with possible outcomes in the set \(\mathcal{A}=\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\) and corresponding probabilities \(p_{i} \doteq \operatorname{Pr}\left(X=x_{i}\right)\) for \(i=1, \ldots, n\). The logarithm is conventionally taken to be base 2, especially in the context of information theory where bits are used. If the probabilities are \(p_{i}=1 / n\) for all \(i=1, \ldots, n\), then all the Renyi entropies of the distribution are equal: \(\mathrm{H}_{\lambda}(X)=\log n\). In general, for all discrete random variables \(X, \mathrm{H}_{\lambda}(X)\) is a non-increasing function in \(\lambda\)..
Particular cases of \(\lambda\) values: \(\lambda = 0, J(0)=S_i\), it corresponds species richness; \(\lambda = 1, J(1)=e^{H_{t}}\), it corresponds the exponential of Shannon's entropy; and \(\lambda = 2, J(2)= D_{S_i}\), it corresponds the 'inverse' Simpson index.
In onomastic context, \(p_{ki}\) denotes the relative frequency of surname \(k\) in region (\(\approx\) community diversity context) \(i\) and \(S_i\) are all surnames in region \(i\).