Proofs of McIntosh's conjecture on Franel integrals and two generalizations

Bruce C. Berndt, Likun Xie, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, ∫01((ax))((bx))((cx))((ex))dx, where ((x)) is the first Bernoulli function defined in (1.1) below. Secondly, we extend our ideas to prove a similar theorem for ∫01((a1x))((a2x))⋯((anx))dx. Lastly, we prove a further generalization in which ((x)) is replaced by any particular Bernoulli function with odd index.

Original languageEnglish (US)
Article number109041
JournalAdvances in Mathematics
Volume423
DOIs
StatePublished - Jun 15 2023

Keywords

  • Bernoulli functions
  • Franel integrals

ASJC Scopus subject areas

  • General Mathematics

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