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

TL;DR

This work extends Ramanujan’s half-integer power sums to all positive half-integers by employing formal Bernoulli series in place of classical Bernoulli polynomials. It proves that for odd $k$, $\sum_{i=1}^n i^{k/2}$ admits a finite exact expression of the form $C_k + \sqrt{n}\,P_k(n) + \sum_{i=3,\ i\text{ odd}}^{k+2} A_i^k\,\tau(n,i)$, with $P_k(n)$ a Bernoulli-derived polynomial, and $A_i^k$ and $C_k$ given by explicit Bernoulli-series and Catalan-function representations. The coefficients are characterized via generating functions and Lagrange inversion, and the proof leverages Chebyshev polynomial representations and coefficient-identity lemmas. The results provide a precise, finite Ramanujan-type generalization of Faulhaber’s formula for positive half-integer powers, revealing a deep Bernoulli-Catalan structure behind the sums. Practically, this framework yields exact formulas for sums of half-integer powers and clarifies the connection between Bernoulli series, Catalan numbers, and Ramanujan’s tau-sums.

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 $p$.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5