The original Weber action at a distance theory is valid for slowly varying effects, and it in addition to predicting all of the usual electrodynamical results, leads to crucial effects where the Maxwell theory fails. The Weber’s approach is an alternative to Maxwell electrodynamics, where the Coulomb's law becomes velocity dependent [1-6]. Here we prove that the Weber’s theory gives the fine structure energy level splitting for the hydrogen atom without the assumption of mass change with velocity.

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials.

We obtain a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, and the solution of this recurrence gives a closed formula for 𝝉(𝒏) in terms of the complete Bell polynomials