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 language | English (US) |
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Article number | 109041 |
Journal | Advances in Mathematics |
Volume | 423 |
DOIs | |
State | Published - Jun 15 2023 |
Keywords
- Bernoulli functions
- Franel integrals
ASJC Scopus subject areas
- General Mathematics