Algebraic addition theorems

Addition Theorems via Continued Fractions

We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several addition theorems for basic hypergeometric functions. Applications to the evaluat...

Addition Theorems in Acyclic Semigroups

We give a necessary and sufficient condition on a given family A of finite subsets of integers for the Cauchy-Davenport inequality |A+ B| ≥ |A|+ |B| − 1, to hold for any family B of finite subsets of integers. We also describe the extremal families for this inequality. We prove this result in the general context of acyclic semigroups, which contain also the semigroup of sequences of elements in...

On linear versions of some addition theorems

Let K ⊂ L be a field extension. Given K-subspaces A,B of L, we study the subspace 〈AB〉 spanned by the product set AB = {ab | a ∈ A, b ∈ B}. We obtain some lower bounds on dimK〈AB〉 and dimK〈B 〉 in terms of dimK A, dimK B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.

Algebraic proofs of some fundamental theorems in algebraic K - theory

We present new proofs of the additivity, resolution and cofinality theorems for the algebraic K-theory of exact categories. These proofs are entirely algebraic, based on Grayson’s presentation of higher algebraic K-groups via binary complexes. 2000 Mathematics Subject Classification: 19D99

Basic Elements: Theorems from Algebraic Avtheory