Aufgabe VII.47 of Pólya-Szegö Immediately Implies Dave Robbins’s Multi-Integral Evaluation

Doron ZEILBERGER^{1}

Exercise VII.47 of [PS] (brought to my attention by Richard Stanley),

∑_{π∈S_{k}} sgn(π)·π [\frac{x_{1}x_{2}^{2} … x_{k}^{k}}{(1 − x_{k})(1 − x_{k}x_{k−1}) ⋯ (1 − x_{k}x_{k−1} … x_{1})}] = \frac{x_{1} ⋯ x_{k} ∏_{1≤i<j≤k}(x_{j} − x_{i})}{∏_{i=1}^{k}(1 − x_{i}) ∏_{1≤i<j≤k}(1 − x_{i}x_{j})} ,

(that is easily proved by induction on k and Lagrange Interpolation), immediately implies the main result of [R], upon setting x_{i} := q^{a_{i}}, multiplying by (1 − q)^{k}, and letting q → 1.

References

[PS] George Pólya and Gabor Szegö, Aufgaben und Lehrsätze aus der Analysis, Julius Springer, Berlin, 1925.

[R] David Robbins, An application of Okada’s Minor Summation Formula to the Evaluation of a Multiple Integral, xxx archives (http://xxx.lanl.gov), math.CO/9805108, 23 May 1998, (3 pages).

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

http://www.math.temple.edu/~zeilberg . May 28, 1998. Exclusive to the author’s website and the xxx archives.