Two Definite Integrals That Are Definitely (and Surprisingly!) Equal
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Two Definite Integrals That Are Definitely (and Surprisingly!) Equal SHALOSH B. EKHAD, DORON ZEILBERGER, AND WADIM ZUDILIN
Sergei Tabachnikov, Editor
To our good friend Gert Almkvist (1934–2018) In Memoriam.
PROPOSITION 1. For real a [ b [ 0 and nonnegative integer n, the following beautiful and surprising identity holds: Z
1
0
x n ð1 xÞn dx ¼ ððx þ aÞðx þ bÞÞnþ1
Z
1 0
x n ð1 xÞn dx: ðða bÞx þ ða þ 1ÞbÞnþ1
PROOF. Fix a and b, let L(n) and R(n) be the integrals on the left and right sides respectively, and let F1 ðn; xÞ and F2 ðn; xÞ be the corresponding integrands, so that LðnÞ ¼ R1 R1 0 F1 ðn; xÞ dx and RðnÞ ¼ 0 F2 ðn; xÞ dx: We cleverly construct the rational functions
This column is a place for those bits of contagious
R1 ðxÞ ¼
x ðx 1Þðða þ b þ 1Þx 2 þ 2abx abÞ ðx þ bÞðx þ aÞ
mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.
and x ðx 1Þ ða bÞx 2 þ 2bða þ 1Þx ða þ 1Þb R2 ðxÞ ¼ ða bÞx ða þ 1Þb
Contributions are most welcome. motivated by the fact that (check!) ðn þ 1ÞF1 ðn; xÞ ð2n þ 3Þð2ba þ a þ bÞF1 ðn þ 1; x Þ þ ða bÞ2 ðn þ 2ÞF1 ðn þ 2; x Þ d ðR1 ðxÞF1 ðn; xÞÞ ¼ dx and ðn þ 1ÞF2 ðn; xÞ ð2n þ 3Þð2ba þ a þ bÞF2 ðn þ 1; x Þ þ ða bÞ2 ðn þ 2ÞF2 ðn þ 2; x Þ d ðR2 ðxÞF2 ðn; xÞÞ: ¼ dx Integrating both identities from x ¼ 0 to x ¼ 1 and noting that the right-hand sides vanish, we have ðn þ 1ÞLðnÞ ð2n þ 3Þð2ba þ a þ bÞLðn þ 1Þ þ ða bÞ2 ðn þ 2ÞLðn þ 2Þ ¼ 0 and ðn þ 1ÞR ðnÞ ð2n þ 3Þð2ba þ a þ bÞRðn þ 1Þ þ ða bÞ2 ðn þ 2ÞR ðn þ 2Þ ¼ 0:
â
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Since Lð0Þ ¼ Rð0Þ and Lð1Þ ¼ Rð1Þ (check!), the proposition follows by mathematical induction. (
Ó 2020 The Author(s) https://doi.org/10.1007/s00283-020-09972-2
REMARK 1. This beautiful identity is equivalent to an
OPEN ACCESS
identity buried in Bailey’s classic book [3, Section 9.5, formula (2)], but you need an expert (like the third-named author) to realize that!
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REMARK 2. Our proof was obtained by the first-named author by running a Maple program1 written by the secondname
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