The Coefficients of Certain Infinite Products

Three Term Identities for the Coefficients of Certain Infinite Products

The Periodicity of the Signs of the Coefficients of Certain Infinite Products

Ramanujan, Richmond and Szekeres, Ramanathan, Andrews, Hirschhorn, Alladi and Berndt, and Bressoud have studied certain infinite products arising from continued fractions from the point of view of the periodicity of the signs of their Taylor series coefficients. In this paper we prove more general theorems of the same sort.

Structure of Certain Banach Algebra Products

Let  and  be Banach algebras, ,  and . We define an -product on  which is a strongly splitting extension of  by . We show that these products form a large class of Banach algebras which contains all module extensions and triangular Banach algebras. Then we consider spectrum, Arens regularity, amenability and weak amenability of these products.

On the transcendence of certain Petersson inner products

‎We show that for all normalized Hecke eigenforms $f$‎ ‎with weight one and of CM type‎, ‎the number $(f,f)$ where $(cdot‎, ‎cdot )$ denotes‎ ‎the Petersson inner product‎, ‎is a linear form in logarithms and‎ ‎hence transcendental‎.

Infinite Products of Infinite Measures

Let (Xi, Bi, mi) (i ∈ N) be a sequence of Borel measure spaces. There is a Borel measure μ on ∏ i∈N Xi such that if Ki ⊆ Xi is compact for all i ∈ N and ∏ i∈N mi(Ki) converges then μ( ∏ i∈N Ki) = ∏ i∈N mi(Ki)