Preface

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .

     

Sets generated by arithmetic sequences

We discuss necessary and sufficient conditions for the partition of integer arithmetic sequences by sets generated by the multiples of irrational numbers by members of other arithmetic sequences.     

A note on cascade theory with ionisation loss

The cascade theory of cosmic ray showers including ionisation loss is dealt with on the basis of the new approach suggested by us in an earlier contribution and an explicit Mellin transform solution is obtained for the mean number of particles produced in an infinite thickness of matter.     

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.