A Surprising Identity between two integrals [Temporary Title]

AuthorsHidden

Proposition: For real a > b > 0 and non-negative integer n, the following beautiful and surprising identity holds.

∫_{0}^{1} \frac{x^{n} (1 − x)^{n}}{((x + a)(x + b))^{n+1}} dx = ∫_{0}^{1} \frac{x^{n} (1 − x)^{n}}{((a − b)x + (a + 1)b)^{n+1}} dx .

Proof: Fix a and b, and 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) = ∫_{0}^{1} F_{1}(n, x) dx and R(n) = ∫_{0}^{1} F_{2}(n, x) dx. We cleverly construct the rational functions

R_{1}(x) = \frac{x (x − 1) ((a + b + 1)x^{2} + 2abx − ab)}{(x + b) (x + a)} , R_{2}(x) = \frac{x (x − 1) ((a − b)x^{2} + 2b(a + 1)x + (a + 1)b)}{(a − b)x + (a + 1)b} ,

with the motive that (check!)

(n + 1) F_{1}(n, x)−(2 n + 3) (2 ba + a + b) F_{1} (n + 1, x)+(a − b)^{2} (n + 2) F_{1} (n + 2, x) = \frac{d}{dx}(R_{1}(x)F_{1}(n, x)) ,

(n + 1) F_{2}(n, x)−(2 n + 3) (2 ba + a + b) F_{2} (n + 1, x)+(a − b)^{2} (n + 2) F_{2} (n + 2, x) = \frac{d}{dx}(R_{2}(x)F_{2}(n, x)) .

Integrating both identities from x = 0 to x = 1, and noting that the right sides vanish, we have

(n + 1) L (n) − (2 n + 3) (2 ba + a + b) L (n + 1) + (a − b)^{2} (n + 2) L (n + 2) = 0 ,
(n + 1) R (n) − (2 n + 3) (2 ba + a + b) R (n + 1) + (a − b)^{2} (n + 2) R (n + 2) = 0 .

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

Comments:1. This beatiful identity is equivalent to an identity buried in Bailey’s classic book [B], section 9.5, formula (2), but you need an expert (like the third-named author) to realize that!

2. Our proof was obtained by the first named-author, running a Maple program, HIDDEN, written by the second-named author, that implements the Almkvist-Zeilberger algorithm

[AZ] designed by Hidden and our good mutual friend Hidden, to whose memory this note is dedicated. 3. The integrals are not taken from a pool of no-one-cares analytic creatures: the right-hand side covers a famous sequence of rational approximations to log(1 + (a − b)/((a + 1)b)) [AR]. Hence the left-hand side does.

References

[AR] K. Alladi and M. L. Robertson, Legendre polynomials and irrationality, J. Reine Angew. Math. 318(1980), 137-155.

[AZ] Gert Almkvist and Doron Zeilberger, The method of differentiating under the integral sign, J. Symbolic Computation 10(1990), 571-591;

http://www.math.rutgers.edu/~zeilberg/mamarimY/duis.pdf

[B] W. N. Bailey, “Generalized hypergeometric series”, Cambridge University Press, 1935. HIDDEN Written: Nov.3, 2019.