Remark B.2. The expression (B.6) shows that our definition generalizes the definition of [k]p,q due to Wachs and White [Wachs and White 91].

In his original introduction to basic numbers in 1847 [Heine 47], E. Heine replaced the parameters α, β, γ in a Gaussian hypergeometric series

1 + (1− qα)(1− qβ) (1− q)(1− qγ) x +

(1− qα)(1− qα+1)(1− qβ)(1− qβ+1) (1− q)(1− q2)(1− qγ)(1− qγ+1) x

2 + · · · ,

by 1− qα 1− q ,

1− qβ 1− q ,

1− qγ 1− q ,

respectively, in order to obtain a continuous generalization from α, as

1− qα 1− q → α,

etc., for q → 1. He also noticed the similarity between q and r = 1q . We have replaced q by Q/R to obtain

1− qα 1− q =

Rα −Qα R−Q R

1−α = {α}R1−α,

hardly a generalization at all. Nevertheless, some of our formulas include two of Heine’s by choosing R = q and Q = 1 or R = 1 and Q = q.