An -sequence is defined by , with initial conditions a0 = 0, a1 = 1. These -sequences play a remarkable role in partition theory, allowing -generalizations of the Lecture Hall Theorem [Bousquet-Mélou and Eriksson, Ramanujan J. 1: 101–111, 1997, Bousquet-Mélou and Eriksson, Ramanujan J. 1: 165–185, 1997] and Euler's Partition Theorem [Bousquet-Mélou and Eriksson, Ramanujan J. 1: 165–185, 1997, Savage and Yee, J. Combin. Theory Ser. A 115: 967–996, 2008]. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The -sequence gives rise to the , which is known to be an integer [Lucas, Amer. J. Math. 1: 197–240, 1878]. In this paper, we use algebraic and combinatorial properties of -sequences to interpret the -nomial coefficients in terms of weighted lattice paths, integer partitions, and probablility distributions. We show how to use these interpretations to uncover -generalizations of familiar hypergeometric identities involving binomial coefficients. This leads naturally to an -analogue of the q-binomial coefficients (Gaussian polynomials) and a corresponding generalization of the “partitions in a box” interpretation of ordinary q-binomial coefficients.
© de Gruyter 2010