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:*

**Proof.**

Fix *a* and *b*, let *L*(*n*) and *R*(*n*) be the integrals on the left and right sides respectively, and let \(F_1(n,x)\) and \(F_2(n,x)\) be the corresponding integrands, so that \(L(n)=\int _{0}^{1} F_1(n,x)\, dx\) and \(R(n)=\int _{0}^{1} F_2(n,x)\, dx.\) We cleverly construct the rational functions

and

motivated by the fact that (check!)

and

Integrating both identities from \(x=0\) to \(x=1\) and noting that the right-hand sides vanish, we have

and

Since \(L(0)=R(0)\) and \(L(1)=R(1)\) (check!), the proposition follows by mathematical induction. \(\square \)

FormalPara Remark 1.This beautiful identity is equivalent to an 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!

FormalPara Remark 2.Our proof was obtained by the first-named author by running a Maple program^{Footnote 1} written by the second-named author that implements the Almkvist–Zeilberger algorithm [2] designed by Zeilberger and our good mutual friend Gert Almkvist, to whose memory this note is dedicated.

Our integrals are not taken from a pool of no-one-cares-about-them analytic creatures: the right-hand side covers a famous sequence of rational approximations to \(\log (1+(a-b)/((a+1)b))\) [1], and hence the left-hand side does, too.

FormalPara Remark 4.We thank Greg Egan for spotting a sign error in an earlier version.

FormalPara Remark 5.Watch Greg Egan’s beautiful animation.^{Footnote 2}

To our surprise, the identity turned out to be not as surprising as we had believed. Mikael Sundquist noticed that the change of variable \(x=b(1-u)/(b+u)\) gives a “Calculus 1 proof.” Indeed,

and we have

Note that *n* does not have to be an integer.

Alin Bostan has two further insightful proofs.^{Footnote 3}

## Notes

## References

K. Alladi and M. L. Robinson. Legendre polynomials and irrationality.

*J. Reine Angew. Math.*318 (1980), 137–155.Gert Almkvist and Doron Zeilberger. The method of differentiating under the integral sign.

*J. Symbolic Computation*10 (1990), 571–591.W. N. Bailey.

*Generalized Hypergeometric Series*. Cambridge University Press, 1935.

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Ekhad, S.B., Zeilberger, D. & Zudilin, W. Two Definite Integrals That Are Definitely (and Surprisingly!) Equal.
*Math Intelligencer* **42, **10–11 (2020). https://doi.org/10.1007/s00283-020-09972-2

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DOI: https://doi.org/10.1007/s00283-020-09972-2