symmetry of divided differences

symmetry of divided differences

Theorem 1

.

If

y

0

,

…

,

y

n

is a

permutation

of

x

0

,

…

,

x

n

, then

Δ

n

⁢

f

⁢

[

x

0

,

…

,

x

n

]

=

Δ

n

⁢

f

⁢

[

y

0

,

…

,

y

n

]

.

Proof.

We proceed by

induction

. When

n

=

1

, we have, from the definition,

Δ

1

⁢

f

⁢

[

x

0

,

x

1

]

=

f

⁢

(

x

1

)

-

f

⁢

(

x

0

)

x

1

-

x

0

=

f

⁢

(

x

0

)

-

f

⁢

(

x

1

)

x

0

-

x

1

=

Δ

1

⁢

f

⁢

[

x

1

,

x

0

]

.

Since the only permutations of two

elements

are the

identity

and the

transposition

, we see that the first divided differrence is

symmetric

.

Now suppose that we already know that the

n

-th

divided difference

is symmetric under permutation of its

arguments

for some

n

≥

1

.
We will prove that the

n

+

1

-st divided difference is also symmmetric
under all permutations of its arguments.

The divided difference is symmetric under transposing

x

0

with

x

1

:

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

1

,

x

2

,

…

,

x

n

+

1

]

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

⁢

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

⁢

x

n

+

1

]

x

1

-

x

0

=

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

⁢

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

⁢

x

n

+

1

]

x

0

-

x

1

=

Δ

n

+

1

⁢

f

⁢

[

x

1

,

x

0

,

x

2

,

…

,

x

n

+

1

]

The divided difference is symmetric under transposing

x

1

with

x

2

:

Δ

n

+

1

f

⁢

[

x

0

,

x

1

,

x

2

,

…

,

x

n

+

1

]

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

⁢

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

⁢

x

n

+

1

]

x

1

-

x

0

=

(

x

2

-

x

0

)

⁢

(

Δ

n

-

1

⁢

f

⁢

[

x

2

,

x

3

⁢

…

⁢

x

n

+

1

]

-

Δ

n

-

1

⁢

f

⁢

[

x

1

,

x

3

⁢

…

⁢

x

n

+

1

]

)

-

(

x

2

-

x

1

)

⁢

(

Δ

n

-

1

⁢

f

⁢

[

x

2

,

x

3

⁢

…

⁢

x

n

+

1

]

-

Δ

n

-

1

⁢

f

⁢

[

x

0

,

x

3

⁢

…

⁢

x

n

+

1

]

)

(

x

2

-

x

1

)

⁢

(

x

1

-

x

0

)

⁢

(

x

2

-

x

0

)

=

(

x

1

-

x

0

)

⁢

Δ

n

-

1

⁢

f

⁢

[

x

2

,

x

3

⁢

…

⁢

x

n

+

1

]

-

(

x

2

-

x

0

)

⁢

Δ

n

-

1

⁢

f

⁢

[

x

1

,

x

3

⁢

…

⁢

x

n

+

1

]

+

(

x

2

-

x

1

)

⁢

Δ

n

-

1

⁢

f

⁢

[

x

0

,

x

3

⁢

…

⁢

x

n

+

1

]

(

x

2

-

x

1

)

⁢

(

x

1

-

x

0

)

⁢

(

x

2

-

x

0

)

=

(

x

1

-

x

0

)

⁢

(

Δ

n

-

1

⁢

f

⁢

[

x

1

,

x

3

⁢

…

⁢

x

n

+

1

]

-

Δ

n

-

1

⁢

f

⁢

[

x

2

,

x

3

⁢

…

⁢

x

n

+

1

]

)

-

(

x

1

-

x

2

)

⁢

(

Δ

n

-

1

⁢

f

⁢

[

x

1

,

x

3

⁢

…

⁢

x

n

+

1

]

-

Δ

n

-

1

⁢

f

⁢

[

x

0

,

x

3

⁢

…

⁢

x

n

+

1

]

)

(

x

1

-

x

2

)

⁢

(

x

1

-

x

0

)

⁢

(

x

2

-

x

0

)

=

Δ

n

⁢

f

⁢

[

x

2

,

x

1

,

…

⁢

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

1

,

…

⁢

x

n

+

1

]

x

1

-

x

0

=

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

2

,

x

1

,

…

,

x

n

+

1

]

The divided difference is symmetric under transposing

x

k

with

x

k

+

1

when

k

>

1

:

Δ

n

+

1

f

⁢

[

x

0

,

x

1

,

x

2

,

…

,

x

k

,

x

k

+

1

,

…

,

x

n

+

1

]

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

,

x

k

,

x

k

+

1

,

…

,

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

,

x

k

,

x

k

+

1

,

…

,

x

n

+

1

]

x

1

-

x

0

=

Δ

n

⁢

f

⁢

[

x

1

,

x

2

,

…

,

x

k

+

1

,

x

k

,

…

,

x

n

+

1

]

-

Δ

n

⁢

f

⁢

[

x

0

,

x

2

,

…

,

x

k

+

1

,

x

k

,

…

,

x

n

+

1

]

x

1

-

x

0

=

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

1

,

x

2

,

…

,

x

k

+

1

,

x

k

,

…

,

x

n

+

1

]

Since any permutation of

x

0

,

x

1

,

…

⁢

x

n

+

1

can be
genreated from the transpositions of

x

k

with

x

k

+

1

for

k

between

0

and

n

, it follows that

Δ

n

+

1

⁢

f

⁢

[

x

0

,

x

1

,

…

,

x

k

,

x

k

+

1

,

…

,

x

n

+

1

]

is symmetric under all permutaions of

x

0

,

x

1

,

…

⁢

x

n

+

1

.
∎

Title

symmetry of divided differences

Canonical name

SymmetryOfDividedDifferences

Date of creation

2013-03-22 16:48:29

Last modified on

2013-03-22 16:48:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

22

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 39A70

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