Special functions Basic hypergeometric functions Theorem

q-Vandermonde identity

Builds on Basic hypergeometric series q-Pochhammer symbol

Statement

Let $n \ge 0$ be an integer, let $q \in \C \setminus \set{0}$ with $q^k \ne 1$ for $1 \le k \le n$ , and let $a, c \in \C$ with $a \ne 0$ and $(c; q)_n \ne 0$ . Then the terminating basic hypergeometric series satisfy ${}_2\phi_1 \left( \begin{matrix} a, q^{-n} \\ c \end{matrix} ; q, q \right) = \frac{(c/a; q)_n}{(c; q)_n}\, a^n$ and ${}_2\phi_1 \left( \begin{matrix} a, q^{-n} \\ c \end{matrix} ; q, \frac{c q^n}{a} \right) = \frac{(c/a; q)_n}{(c; q)_n},$ where $(\cdot\,; q)_n$ is the q-Pochhammer symbol.

Remarks

Both sides are rational functions, and the series stop at index $n$ since $(q^{-n}; q)_k = 0$ for $k > n$ . Proof idea: the second form is the case $b = q^{-n}$ of the $q$ -Gauss sum, which follows from Heine's transformation. The first form is the second one with the order of summation reversed. Example: $n = 1$ . The first sum is $1 + \frac{(1 - a)(1 - q^{-1})}{(1 - c)(1 - q)}\, q = 1 - \frac{1 - a}{1 - c} = \frac{a - c}{1 - c}$ , and the right side is $\frac{(1 - c/a)\, a}{1 - c} = \frac{a - c}{1 - c}$ . In terms of Gaussian binomial coefficients it reads $\binom{m+n}{k}_q = \sum_{j} \binom{m}{j}_q \binom{n}{k-j}_q\, q^{(m-j)(k-j)}$ . At $q = 1$ this becomes Vandermonde's identity.

Going deeper

For a student

Its Gaussian binomial form can also be proved, for $q$ a prime power, by counting subspaces of $\F_q^{m+n}$ . It is the $q$ -analogue of the Chu–Vandermonde identity for ${}_2F_1$ , and it is a terminating case of the $q$ -Gauss sum. It is used to evaluate connection coefficients between $q$ -orthogonal polynomials.

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Tag	`MBRSCQRGGH`
URI	`m2://MBRSCQRGGH`
Kind	statement, theorem
Writing	W0 stub
MSC	33D15, 05A30
Status	This copy of the site has no build data. Run `m2 status MBRSCQRGGH`.
About	Basic hypergeometric series
Uses	Basic hypergeometric series, q-Pochhammer symbol