From al at univ-mlv.fr Thu Mar 29 04:46:19 2007

29 03 2007

Dear Doron,

my method is not too different from yours, when proceeding by induction on the number of variables. Indeed, there is essentially only one operator associated to the symmetric group: the alternation, followed by division by the Vandermonde. When the input has already some symmetry (like in the case of Lagrange interpolation, which should be written as an operator on the ring of polynomials in $n$ variables which are symmetrical in the last $n-1$ ones), one can reduce the summation to take into account this symmetry. Writing Lagrange interpolation with divided differences allows to decompose the operation into elementary steps, and this is specially suited for computing rational generating series generalizing the Cauchy kernel.

In the case of your "intriguing identity", one needs only considerations of degrees, once taken into account the symmetry that you found by recognizing the interpolation points $[y_1,...,y_{2n+2}] = [z_1,..., a,b]$ Introducing a variable $y=(-T-z)$, and another alphabet $U=\{ u_1,..,u_n\}$ not to interfere with the divided differences associated to the interpolation points, I would rewrite your polynomial
$P(z)$ as $ (2z+T) \prod_i(1-z u_i)(1-yu_i)$
By Cauchy formula this develops into
$(2z+T) \sum \pm S_\mu(z+y) S_{\mu'}(U)$
sum over all partitions $\mu: \mu_1 \leq\mu_2\leq n$ , $\mu'$ being the conjugate partition. Now your Lagrange interpolation can be written as taking the image of $P(z_1)$ under the product of divided differences $d_1 ...d_{2n+1}$, which decreases degrees by $2n+1$ Therefore, only the term for $\mu=[n,n]$ survives :
$ 2 z_1 (z_1*(-T-z_1)^n S_{2^n}(U)$ gives
$2* (-1)^n* S_{2^n}(U) = 2* (-1)^n*(u_1...u_n)^2 $

In fact, you should put another parameter in your formula, so that one can also use residues to prove the formula (this is not the change of variables x_i \to Bx_i ):

Intriguing6:=proc(n)   local i,j;
 if nargs>1 then RETURN( (1-B*x1^2)*convert([seq( 1-B*x1*x.i,i=2..n),
   seq( (1+x.i*T+B*x1*x.i)/(B*x.i+T+x1*B),i=2..n)],`*`))
 fi;
 factor(NcaOnPol(A[n,$1..n-1],
    (1-B*x1^2)*convert([seq( 1-B*x1*x.i,i=2..n),
       seq( (1+x.i*T+B*x1*x.i)/(B*x.i+T+x1*B),i=2..n)],`*`)))
end:

 Intriguing6(3,see);
         2
(1 - B x1 ) (1 - B x1 x2) (1 - B x1 x3) (1 + x2 T + B x1 x2)

    (1 + x3 T + B x1 x3)/((B x2 + T + x1 B) (B x3 + T + x1 B))

ACE> Intriguing6(3);
                                2  3   2   2
                             -x1  B  x3  x2  + 1

                     Alain.Lascoux at univ-mlv.fr
                http://phalanstere.univ-mlv.fr/~al
                http://combinatorics.net/Lascoux  
                 C.N.R.S., Institut Gaspard Monge 
                  Universite de Marne-la-Vallee,
                77454 Marne La Vallee Cedex 2 FRANCE 
                    tel: (33) 01 60 95 77 18
                    Fax: (33) 01 60 95 75 57