Special functions Basic hypergeometric functions Theorem

Ramanujan's 1psi1 summation

Builds on Basic hypergeometric series q-Pochhammer symbol

Statement

Let $a, b, q, z \in \C$ with $0 < \abs{q} < 1$ , $a \ne 0$ , $b \notin \set{q^{-m} : m \ge 0}$ , and $\abs{b/a} < \abs{z} < 1$ . For an integer $n \ge 1$ put $(a; q)_{-n} = 1 / (a q^{-n}; q)_n$ , so that $(a; q)_n$ is defined for all $n \in \Z$ through the q-Pochhammer symbol. Then $\sum_{n=-\infty}^{\infty} \frac{(a; q)_n}{(b; q)_n} z^n = \frac{(q; q)_\infty\, (b/a; q)_\infty\, (az; q)_\infty\, (q/(az); q)_\infty}{(b; q)_\infty\, (q/a; q)_\infty\, (z; q)_\infty\, (b/(az); q)_\infty}.$ In the notation of bilateral basic hypergeometric series, the left side is ${}_1\psi_1(a; b; q, z)$ .

Remarks

The condition $\abs{z} < 1$ makes the part with $n \ge 0$ converge, and $\abs{b/a} < \abs{z}$ makes the part with $n < 0$ converge. Proof idea: for $b = q$ the terms with $n < 0$ vanish and the identity is the q-binomial theorem; then $b = q^{m+1}$ follows by a shift of the summation index. Both sides are analytic in $b$ in a disc, and they agree on the points $q^{m}$ , $m \ge 1$ , which accumulate at $0$ , so they agree everywhere by the identity theorem. Example: with $b = 0$ (a limit) and $z$ replaced by $-z/a$ , letting $a \to \infty$ turns the identity into the Jacobi triple product. The identity is due to Ramanujan, who stated it without proof; Hahn and Jackson gave the first published proofs around 1950.

Going deeper

For a student

Many product formulas of the theory of partitions and of theta functions are special cases. It is the starting point for the bilateral series ${}_r\psi_r$ , of which Bailey's ${}_6\psi_6$ summation is the main generalization.

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Tag	`RK6DCJVJCR`
URI	`m2://RK6DCJVJCR`
Kind	statement, theorem
Writing	W0 stub
MSC	33D15
Status	This copy of the site has no build data. Run `m2 status RK6DCJVJCR`.
About	Basic hypergeometric series
Uses	Basic hypergeometric series, q-Pochhammer symbol