q-Series

A Series involving coefficients of the form

$$(a)_n\equiv (a;q)_n \equiv \prod_{k=0}^\infty {(1-aq^k)\over (1-aq^{k+n})}$$ (1)

$$= \prod_{k=0}^{n-1} (1-aq^k)$$ (2)

(Andrews 1986). The symbols

$${[}n] \equiv 1+q+q^2+\ldots+q^{n-1}$$ (3)

$${[}n]! \equiv [n][n-1]\cdots[1]$$ (4)

are sometimes also used when discussing $q$-series.

There are a great many beautiful identities involving $q$-series, some of which follow directly by taking the q-Analog of standard combinatorial identities, e.g., the q-Binomial Theorem

$$\sum_{n=0}^\infty {(a;q)_n z^n\over (q;q)_n}={(az;q)_\infty\over(z;q)_\infty}$$ (5)

($\vert z\vert<1$, $\vert q\vert<1$; Andrews 1986, p. 10) and q-Vandermonde Sum

$${}_2\phi_1(a,q^{-n};c;q,q)={a^n(c/a,q)_n\over(c;q)_n},$$ (6)

where ${}_2\phi_1(a,b;c;q,z)$ is a Heine Hypergeometric Series. Other $q$-series identities, e.g., the Jacobi Identities, Rogers-Ramanujan Identities, and Heine Hypergeometric Series identity

$${}_2\phi_1(a,b;c;q,z)={(b;q)_\infty(az;q)_\infty\over (c;q)_\infty(z;q)_\infty} {}_2\phi_1(c/b,a;az;q,b),$$ (7)

seem to arise out of the blue.

See also Borwein Conjectures, Fine's Equation, Gaussian Coefficient, Heine Hypergeometric Series, Jackson's Identity, Jacobi Identities, Mock Theta Function, q-Analog, q-Binomial Theorem, q-Cosine, q-Factorial, Q-Function, q-Gamma Function, q-Sine, Ramanujan Psi Sum, Ramanujan Theta Functions, Rogers-Ramanujan Identities

References

q-Series

Andrews, G. E. $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.

Berndt, B. C. ``$q$-Series.'' Ch. 27 in Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 261-286, 1994.

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.

Gosper, R. W. ``Experiments and Discoveries in $q$-Trigonometry.'' Unpublished manuscript.