Bailey's lemma

Remarks

Proof idea: insert the definition of $\beta_j$ into $\beta_n'$ and exchange the sums. The inner sum over $j$ is a terminating balanced ${}_3\phi_2$ series, which the $q$-Pfaff–Saalschütz summation evaluates in closed form. Example: the unit Bailey pair relative to $a$ is $\beta_n = \delta_{n0}$ with $\alpha_0 = 1$ and $\alpha_n = \frac{(1 - a q^{2n})(a; q)_n (-1)^n q^{\binom{n}{2}}}{(1 - a)(q; q)_n}$ for $n \ge 1$; applying the lemma to it produces new pairs. Letting $\rho_1, \rho_2 \to \infty$ gives $c^n (\rho_1, \rho_2; q)_n \to a^n q^{n^2}$, and with $n \to \infty$ one gets the identity $\sum_{n \ge 0} a^n q^{n^2} \beta_n = \frac{1}{(aq; q)_\infty} \sum_{n \ge 0} a^n q^{n^2} \alpha_n$ for suitable pairs. This limit, applied to a suitable pair with $a = 1$ or $a = q$ and combined with the Jacobi triple product, gives the Rogers–Ramanujan identities.

Going deeper

For a student

Bailey proved the lemma in the 1940s, and Andrews later noticed that it can be iterated, which gives the Bailey chain. The Bailey chain explains the Rogers–Ramanujan identities and their many generalizations, such as the Andrews–Gordon identities, in a uniform way. The proof only needs the $q$-Pfaff–Saalschütz sum, the $q$-analogue of the Pfaff–Saalschütz theorem.