Faulhaber’s formula expresses the sum of the first n positive integers, each raised to an integer power <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, as a polynomial in n of degree <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Ramanujan expressed this sum for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\in \{\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> as the sum of a polynomial in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sqrt{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>n</mi> </msqrt> </math></EquationSource> </InlineEquation> 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.

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

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2025年11月18日

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Max A. Alekseyev,
Rafael Gonzalez,
Keryn Loor,
Aviad Susman,
Cesar Valverde

摘要

Faulhaber’s formula expresses the sum of the first n positive integers, each raised to an integer power $p\ge 0$ $p \geq 0$ , as a polynomial in n of degree $p+1$ $p + 1$ . Ramanujan expressed this sum for $p\in \{\frac{1}{2},\frac{3}{2},\frac{5}{2},\frac{7}{2}\}$ $p \in {\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}}$ as the sum of a polynomial in $\sqrt{n}$ $\sqrt{n}$ 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.