Generalization of Ramanujan's formula for sums of half-integer powers of consecutive integers via formal Bernoulli series

TL;DR

This paper extends Ramanujan's formula for sums of half-integer powers of integers using formal Bernoulli series, connecting Bernoulli polynomials to generalize the sum expressions for all positive half-integers.

Contribution

It introduces a novel generalization of Ramanujan's formula for half-integer powers using formal Bernoulli series, broadening the scope of known sum formulas.

Findings

Extended Ramanujan's formula to all positive half-integers

Connected Bernoulli polynomials to sum formulas via formal series

Provided a unified approach to sums of half-integer powers

Abstract

Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$, as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the sum of a polynomial in $\sqrt{n}$ and a certain infinite series. In the present work, we explore the connection to Bernoulli polynomials, and by generalizing those to formal series, we extend the Ramanujan result to all positive half-integers $p$.