Proof of an Identity Conjectured by Iossif Polterovitch that Came Up in the Agmon-Kannai Asymptotic Theory of the Heat Kernel

Doron ZEILBERGER^{1}

Dedicated to my Teacher, Yakar Kannai, who taught me PDEs and much more

Yakar Kannai’s brilliant student, Iossif Polterovitch, who did beautiful work in extending and elaborating the classical Agmon-Kannai theory of the heat kernel, proved, via that theory, the a = 3n case of the following identity:

G(a, n; x) := ∑_{j=0}^{a} \frac{1}{(a − j)!(j + n)!(2j + 1)} ∑_{k=−j}^{j} \frac{(−1)^{k}(x + k)^{2j+2n}}{(j − k)!(j + k)!} = 0 , (a ≥ 2n) ,

that he discovered empirically, using Mathematica. He also conjectured that G(2n − 1, n, x) = −1/(2 · 3^{n}n!). In this note I will prove his conjecture.

Introducing the shift operator Ef(x) = f(x + 1), we have that G(a, n; x) equals

∑_{j=0}^{a} \frac{(−1)^{j}}{(a − j)!(j + n)!(2j + 1)} ∑_{k=−j}^{j} \frac{(−1)^{j+k}E^{k}}{(j − k)!(j + k)!}x^{2j+2n} = ∑_{j=0}^{a} \frac{(−1)^{j}}{(a − j)!(j + n)!(2j + 1)!}(E^{1/2}−E^{−1/2})^{2j}x^{2j+2n} .

Using E = e^{D} (Taylor’s theorem), and writing f(D) = ((2 sinh D/2)/D)^{2} (note that f(D) is an even function (f(D) = 1 + D^{2}/12 + O(D^{4}))), we have that this equals

∑_{j=0}^{a} \frac{(−1)^{j}}{(a − j)!(j + n)!(2j + 1)!}f(D)^{2j}D^{2j}x^{2j+2n} = ∑_{j=0}^{a} \frac{(−1)^{j}}{(a − j)!(j + n)!(2j + 1)!}f(D)^{j} \frac{(2j + 2n)!}{(2n)!} x^{2n}

= \frac{1}{a!n!}_{2}F_{1}(n + 1/2, −a; 3/2; f(D))x^{2n} = \frac{1}{a!n!}_{2}F_{1}(1 − n, 3/2 + a; 3/2; f(D))(1 − f(D))^{a−n+1} x^{2n} ,

by Euler’s transformation: _{2}F_{1}(a, b; c; z) = (1 − z)^{c−a−b}_{2}F_{1}(c − a, c − b; c; z) ([B], p. 2). If a ≥ 2n, then, since 1 − f(D) = O(D^{2}), this equals 0. If a = 2n − 1, then, since f(D) − 1 = D^{2}/12 + O(D^{4}) this equals

\frac{(−1)^{n}}{(2n − 1)!n!}_{2}F_{1}(1 − n, 1/2 + 2n; 3/2; 1)\frac{(2n)!}{12^{n}} ,

which turns out to be −1/(2 · 3^{n}n!), by Gauss’s evaluation ([B], p. 3).

REFERENCE

[B] W.N. Bailey, “Generalized Hypergeometric Series”, Cambridge University Press, 1935.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122. zeilberg@math.temple.edu ;

http://www.math.temple.edu/~zeilberg/ Supported in part by the NSF. Aug. 10, 2000. Exclusive to the ”Personal Journal of Ekhad and Zeilberger”. I thank Amitai Regev for his invitation to stay at the Weizmann Institute between July 24 and Aug. 2, 2000.