A Theorem on the Annamalai’s Binomial Identities

The q-Binomial Theorem and two Symmetric q-Identities

We notice two symmetric q-identities, which are special cases of the transformations of 2φ1 series in Gasper and Rahman’s book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the q-binomial theorem by using conjugation of 2-modular diagrams.

Hybrid Proofs of the q-Binomial Theorem and Other Identities

We give “hybrid” proofs of the q-binomial theorem and other identities. The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities d...

A Treatise on the Binomial Theorem

OF THE DISSERTATION A treatise on the binomial theorem by PATRICK DEVLIN Dissertation Director: Jeff Kahn This dissertation discusses four problems taken from various areas of combinatorics— stability results, extremal set systems, information theory, and hypergraph matchings. Though diverse in content, the unifying theme throughout is that each proof relies on the machinery of probabilistic co...

Binomial Sums and Identities

On Binomial Identities in Arbitrary Bases

We first extend the digital binomial identity as given by Nguyen et al. to an identity in an arbitrary base b, by introducing the b-ary binomial coefficients. Then, we study the properties of these coefficients such as their orthogonality, their link with Lucas’ theorem and their extension to multinomial coefficients. Finally, we analyze the structure of the corresponding b-ary Pascal-like tria...