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arXiv · 2506.05961

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

Max A. Alekseyev·Rafael Gonzalez·Keryn Loor·Aviad Susman·Cesar Valverde
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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 $p$.

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Max A. Alekseyev, Rafael Gonzalez, Keryn Loor, Aviad Susman, Cesar Valverde. 2025-06-06. Generalization of Ramanujan's formula for sums of half-integer powers of consecutive integers via formal Bernoulli series. https://doi.org/10.1007/s11139-025-01259-4

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