binomial theorem for q-binomial coefficients, known as the Cauchy binomial theorem: Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such, corresponding to Newton's generalized binomial theorem for negative powers, is In the limit , these formulas yield and . Setting gives the generating functions for distinct and any parts respectively. (See also Basic hypergeometric series.) Central q-binomial identity With the ordinary binomial coefficients, we have: With q-binomial coefficients, the analog is: Applications Gaussian binomial coefficients occur in the counting of symmetric polynomials and in the theory of partitions. The coefficient of qr in is the number of partitions of r with m or fewer parts each less than or equal to n. Equivalently, it is also the number of partitions of r with n or fewer parts each less than or equal to m. Gaussian binomial coefficients also play an important role in the enumerative theory of projective spaces defined over a finite field. In particular, for every finite field Fq with q elements, the Gaussian binomial coefficient counts the number of k-dimensional vector subspaces of an n-dimensional vector space over Fq (a Grassmannian). When expanded as a polynomial in q, it yields the well-known decomposition of the Grassmannian into Schubert cells. For example, the Gaussian binomial coefficient is the number of one-dimensional subspaces in (Fq)n (equivalently, the number of points in the associated projective space). Furthermore, when q is 1 (respectively −1), the Gaussian binomial coefficient yields the Euler characteristic of the corresponding complex (respectively real) Grassmannian. The number of k-dimensional affine subspaces of Fqn is equal to . This allows another interpretation of the identity as counting the (r − 1)-dimensional subspaces of (m − 1)-dimensional projective space by fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplane; these latter subspaces are in bijective correspondence with the (r − 1)-dimensional affine subspaces of the space obtained by treating this fixed hyperplane as the hyperplane at infinity. In the conventions common in applications to quantum groups, a slightly different definition is used; the quantum binomial coefficient there is . This version of the quantum binomial coefficient is symmetric under exchange of and . References Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, , , (undated, 2004 or earlier). Ratnadha Kolhatkar, Zeta function of Grassmann Varieties (dated