Determinants and Their Applications in Mathematical Physics

A.9 The Euler and Modified Euler Theorems on Homogeneous Functions 331

The first is due to Euler. The second is similar in nature to Euler’s and can

be obtained from it by means of a change of variable.

The functionfis said to be homogeneous of degreesin its variables if

f(λx 0 ,λx 1 ,λx 2 ,...,λxn)=λ

s

f. (A.9.2)

Theorem A.5 (Euler). If the variables are independent andfis differ-

entiable with respect to each of its variables and is also homogeneous of

degreesin its variables, then

n

∑

r=0

xr

∂f

∂xr

=sf.

The proof is well known.

The functionfis said to be homogeneous of degreesin thesuffixesof

its variables if

f(x 0 ,λx 1 ,λ

2

x 2 ,...,λ

n

xn)=λ

s

f. (A.9.3)

Theorem A.6 (Modified Euler). If the variables are independent andf

is differentiable with respect to each of its variables and is also homogeneous

of degreesin the suffixes of its variables, then

n

∑

r=1

rxr

∂f

∂xr

=sf.

Proof. Put

ur=λ

r

xr, 0 ≤r≤n [in (A.9.3)].

Then,

f(u 0 ,u 1 ,u 2 ,...,un)≡λ

s

f.

Differentiating both sides with respect toλ,

n

∑

r=0

∂f

∂ur

dur

dλ

=sλ

s− 1

f,

n

∑

r=0

∂f

∂ur

rλ

r− 1

xr=sλ

s− 1

f.

Putλ= 1. Then,ur=xrand the theorem appears.

A proof can also be obtained from Theorem A.5 with the aid of the

change of variable

vr=x

r

r

.

Both these theorems are applied in Section 4.8.7 on double-sum relations

for Hankelians.