Dirichlet eta and beta functions at negative integer arguments: Exact results from anti-limits

2019/09/30 by Kamal Bhattacharyya, Bhattacharyya, Kamal
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1910.04682

openalex publication_date 2019/09/30 · openalex created_date 2019/10/18 · openalex updated_date 2026/07/28

Abstract

A route to evaluate exact sums represented by Dirichlet eta and beta functions, both of which are alternating and divergent at negative integer arguments, is advocated. It rests on precise polynomial extrapolations and stands as a generalization of an early endeavor on lattice sums. Apart from conferring a physical meaning to anti-limits, the scheme advanced here is direct, independent and computationally appealing. A new interpretation of summability is also gained.

Citations

• Dynamical Theory of Crystal Lattices

Related

• A New Formula for The Values of Dirichlet Beta Function at Odd Positive Integers Based on The WZ Method
• Diophantine properties for q-analogues of Dirichlet's beta function at positive integers
• Fundamental relations between the Dirichlet beta function, euler numbers, and Riemann zeta function for positive integers
• Strong q-analogues for values of the Dirichlet beta function
• An Alternative Approach to Computing β(2k+1)