$𝑞$ -analogue

Let $𝑛 \in ℤ +$ . The $𝑞$ -analogue $(𝑛) 𝑞$ of $𝑛$ is defined by

(𝑛) 𝑞 = 𝑞 𝑛 - 1 𝑞 - 1 = 𝑛 - 1 \sum 𝑖 = 0 𝑞 𝑖

where we note that $l i m 𝑞 \to 1 (𝑛) 𝑞 = 𝑛$ . The $𝑞$ -analogue of other functions is given by replacing $𝑛$ with $(𝑛) 𝑞$ , see e.g. Gaußian binomial coëfficient.

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𝑞-analogue

$𝑞$ -analogue