L The Determinantal Formulas for Divided Differences

The Determinantal Formulas for Divided Differences 1. The general divided difference o f f is linear in the corresponding values off:

where the Uvare rational expressions in xl, obtained immediately from (IA.6):

urn=

...,x,, the values of which are

1 (xm-xJ(xm-xd

(xm-xm-1)

(La

*

2. To write (L.l) as a quotient of determinants, introduce two column vectors Z ( x ) and N ( x ) by Z ( x ) = (lyxy...y~-~,f(x)~y N ( x ) = (l,x, ...,xm-l,.P)’.

(L.3)

Then we easily see that

Indeed, since both sides in (L.4) are symmetric, it is sufficient to compare the coefficient of f ( x , ) . This coefficient on the right is the quotient of two Vandermonde’s determinants,

and this is just the above value of Urn.

3. In order to obtain the corresponding representation of the divided difference in the conjluent case, consider k systems of variables xl,...yx~~;Yl,...yYm~;

-**;

z1,..-,Zmk

336

(rn,+...+rnk

= n),

THE DETERMINANTAL FORMULAS FOR DIVIDED DIFFERENCES

337

Subtracting in both the numerator and the denominator the first column from the second and dividing by xZ-x1 we see that the second columns can be replaced respectively by Applying the same procedure to every x, column (1 < v replace each x, column respectively by

< ml) we can

Cxv, X l l Z(Xl), Cx,, xi1 N(x1). Operating in the same way and starting from the second column, we can replace the third column in the numerator and denominator by

CX3, Xz, xi1 z(xi), CX3, xz, xi1 N(xi). Applying the same procedure repeatedly we can finally replace the first m, columns in the numerator and denominator respectively by

Z(Xl), Cxz,x11 Z(X,), . * * , CXm1,...,xz,x,lz(x1); N(x1),

C x Z 9 ~ 1 N(x,), 1

Cxm, * * * , ~ 2 N(x1),~11 9

The same procedure can be applied to the columns depending on the yv,...,zvand we obtain finally

. *.*,~1,.*.,zmklf . [xI,...,xmI;yl,...,~m*,

~ ) . ***,zmkIZ(z1)l .. - ( Z ( ~ , ) C X , , X ~ I ~ ( X , ) ... . . , x ~ , I ~ ( x , > Z ( Y Cz1, . IN(x~)CXI,X~I N(x1) [XI, ..*>xm1IN(x1) C Z ~ ,.-.,zmkI N(z1) * a *

(L.6) 4. Assume now k distinct variables c,, . ..,t k and in (L.6) let all x, tend to t,, all yv tend to t,, ..., all z, tend to f k ; then we obtain [$?I,

z*= IZ(t])Z'(t,)

..., t,""] f = Z * / N * , z(ml-l)(t1)Z(fz)

N * = I N ( f , ) N ' ( t l ) ZV(ml-l)(cl) provided N* is #O.

.**

..' Z(m*-l)(tk)I,

N(mk-l)(fk)l,

(L.7)

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APPENDIX L

5. We are now going to show that indeed N * is # O if the tk are all distinct,

as we will prove that

1

A(x) = x1 $1-1

... ... ...

1 =

n

C'V

(X,-x")

Xml-l ml

and define similarly A(y), ..., A(z). Then, if we denote the denominator in (L.5) by Nand that in (L.6) by No, we have obviously N N o - A(x)A(JJ)*--A(z)' Put, further, R(x,y) = n(x,,-y,)

( p = 1,...,m,; v = I,

..., mz)

and define similarly R ( x , z ) , R ( y , z ) , etc. Then, writing in (L.9) the Vandermonde determinant N as the difference product of the arguments, we obtain No = R ( x , y ).*.R ( x , z ) R ( y , z ) * a .

a * *

and this tends indeed to the right-hand product in (L.8). 6. The above gives a second proof of the existence of the confluent divided difference (1B.4)and also a formula for this difference, formally different from (1B.4).