To our good friend Gert Almkvist (1934–2018) In Memoriam.
FormalPara
Proposition 1.
For real \(a>b>0\)and nonnegative integer n, the following beautiful and surprising identity holds:
$$ \int _0^1 \frac{x^n (1-x)^n}{ ((x+a)(x+b))^{n+1}} \, dx = \int _0^1 \frac{x^n (1-x)^n}{ ((a-b)x +(a+1)b)^{n+1}} \, dx.$$
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
$$ R_1(x) = {\frac{x \left( x-1 \right)((a+b+1) x^2+ 2abx -ab) }{ \left( x+b \right) \left( x+a \right) }}$$
and
$$R_2(x) = {\frac{x \left( x-1 \right) \bigl ( (a-b)x^2+2b(a+1)x-(a+1)b \bigr ) }{(a-b)x - (a+1)b}}$$
motivated by the fact that (check!)
$$\left( n+1 \right) F_1(n,x) - \left( 2 n+3 \right) \left( 2 ba+a+b \right) F_1 \left( n+1,x \right) + \left( a-b \right) ^{2} \left( n+2 \right) F_1 \left( n+2 ,x\right) = \frac{d}{dx} (R_1(x)F_1(n,x))$$
and
$$\left( n+1 \right) F_2(n,x) - \left( 2 n+3 \right) \left( 2 ba+a+b \right) F_2 \left( n+1,x \right) + \left( a-b \right) ^{2} \left( n+2 \right) F_2 \left( n+2 ,x\right) = \frac{d}{dx} (R_2(x)F_2(n,x)).$$
Integrating both identities from \(x=0\) to \(x=1\) and noting that the right-hand sides vanish, we have
$$\begin{aligned}&\left(n+1\right) L \left(n\right) - \left(2n+3\right) \left(2ba+a+b\right) L \left(n+1\right) \\ &\quad +\left(a-b\right)^{2} \left(n+2\right) L \left(n+2\right) = 0\end{aligned}$$
and
$$ \begin{aligned}&\left( n+1 \right) R \left( n \right) - \left( 2 n+3 \right) \left( 2 ba+a+b \right) R \left( n+1 \right) \\ &\quad + \left( a-b \right) ^{2} \left( n+2 \right) R \left( n+2\right) = 0.\end{aligned}$$
Since \(L(0)=R(0)\) and \(L(1)=R(1)\) (check!), the proposition follows by mathematical induction. \(\square \)
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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!
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Remark 2.
Our proof was obtained by the first-named author by running a Maple programFootnote 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.
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Remark 3.
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.
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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
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Remark 6.
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,
$$dx=-\frac{b(1+b)}{(b+u)^2}\, du,$$
and we have
$$\int _0^1 \frac{x^n (1-x)^n}{ ((x+a)(x+b))^{n+1} } \, dx = \int _0^1 \frac{(b(1-u)/(b+u))^n (1- (b(1-u)/(b+u)))^n}{ (b(1-u)/(b+u)+a)(b(1-u)/(b+u)+b))^{n+1} } \frac{b(1+b)}{(b+u)^2}\, du = \int _0^1 \frac{(1-u)^n u^n b^{n+1} (1+b)^{n+1}}{( b(1-u)+a(b+u))(b(1-u)+b(b+u)) )^{n+1}} \, du = \int _0^1 \frac{u^n(1-u)^n}{((a-b)u+(a+1)b)^{n+1}} \, du.$$
Note that n does not have to be an integer.
FormalPara
Remark 7.
Alin Bostan has two further insightful proofs.Footnote 3
References
K. Alladi and M. L. Robinson. Legendre polynomials and irrationality. J. Reine Angew. Math. 318 (1980), 137–155.
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Gert Almkvist and Doron Zeilberger. The method of differentiating under the integral sign. J. Symbolic Computation 10 (1990), 571–591.
MathSciNet
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W. N. Bailey. Generalized Hypergeometric Series. Cambridge University Press, 1935.
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Authors and Affiliations
Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ, 08854-8019, USA
Shalosh B. Ekhad & Doron Zeilberger
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, PO Box 9010, 6500 GL, Nijmegen, The Netherlands
Wadim Zudilin
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