PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 124, Number 7, July 1996
MULTINOMIAL EXPANSIONS
AND THE PYTHAGOREAN THEOREM
ALLAN FRYANT
(Communicated by J. Marshall Ash)
Dedicated to the memory of Professor Morris Marden
Abstract. An addition formula for homogeneous polynomials is used to obtain a generalization of the Pythagorean theorem and a new view of the multinomial expansion.
Let x = (x1 , x2 , . . . , xn ) ∈ Rn , and let P denote the set of all polynomials
m
X
α1 α2
αn
p(x) =
aj x1 j x2 j . . . xnj ,
aj ∈ R.
j=1
Letting ∂/∂x = (∂/∂x1 , ∂/∂x2 , . . . , ∂/∂xn ), we define on P the inner product
∂
(p, q) = p
q(x)
∂x
x=0


(1)
m
α1j
α2j
αn
X
j
∂
∂
∂


=
aj α1 α2 . . . αn q(x)
.
j
j
j
∂xn
∂x1 ∂x2
j=1
x=0
Let Hk ⊂ P denote the vector space of homogeneous polynomials of degree k.
Note that the dimension of Hk is
(n + k − 1)!
dim Hk = dk =
.
(n − 1)! k!
We first obtain a simple analog of the Funk-Hecke theorem [1, p. 247] for homogeneous polynomials.
Theorem 1. If p(x) is a homogeneous polynomial of degree k, then
((x1 y1 + x2 y2 + · · · + xn yn )k , p(y)) = k! p(x).
Proof. Expanding the power,
X
k!
(x1 y1 )α1 (x2 y2 )α2 . . . (xn yn )αn
α
!
α
!
.
.
.
α
!
1
2
n
α
X xα1 xα2 . . . xαn α α
n
1
2
= k!
y1 1 y2 2 . . . ynαn .
α
!
α
!
.
.
.
α
!
1
2
n
α
(x1 y1 + · · · + xn yn )k =
Received by the editors October 7, 1994.
1991 Mathematics Subject Classification. Primary 33E99; Secondary 26C99.
Key words and phrases. Homogeneous polynomials, addition formula.
c
1996
American Mathematical Society
2001
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2002
ALLAN FRYANT
Thus if p(y) =
P
α
aα y1α1 y2α2 . . . ynαn , we have
((x1 y1 + · · · + xn yn )k , p(y))


αn X
X xα1 · · · xαn ∂ α1
∂
n
1
= k!
aβ y1β1 . . . ynβn 
α1 . . .
αn
α
!
.
.
.
α
!
∂y
∂y
n
1
n
1
α
β
X
α1
n
= k!
aα x1 . . . xα
n = k! p(x).
α
Our next result is an addition formula for orthonormal homogeneous polynomials.
k
Theorem 2 (Addition Formula). Let {pj (x)}dj=1
be an orthonormal basis for Hk .
Then
(2)
(x1 y1 + x2 y2 + · · · + xn yn )k = k!
dk
X
pj (x)pj (y).
j=1
Proof. Let x = (x1 , x2 , . . . , xn ) and αj = (α1j , α2j , . . . , αnj ). For convenience we
α1 α2
αn
write xαj = x1 j x2 j . . . xnj . Then expanding the power, we have
X
cj xαj y αj ,
(x1 y1 + · · · + xn yn )h =
k
where xαj and y αj are homogeneous monomials of degree k. Since {pj (x)}dj=1
is a
basis for Hk , there exist constants ajm such that
y αj =
dk
X
ajm pm (y).
m=1
Thus
(x1 y1 + · · · + xn yn )k =
X
cj xαj
X
!
ajm pm (y)
m
j
=
X
gj (x)pj (y),
j
where the last expression is a rearrangement of the previous sum. The gj (x) are
linear combinations of the monomials xαj , and thus are homogeneous polynomials
of degree k in x. Further, since the pj (x) are orthonormal,


X
((x1 y1 + · · · + xn yn )k , pl (y)) = 
gj (x)pj (y), pl (y) = gl (x).
j
But by the identity given in Theorem 1,
((x1 y1 + · · · + xn yn )k , pl (y)) = k! pl (x).
Thus,
gj (x) = k!pj (x),
j = 1, 2, . . . , dk ,
which completes the proof.
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MULTINOMIAL EXPANSIONS AND THE PYTHAGOREAN THEOREM
2003
If αj = (α1j , α2j , . . . , αnj ), for convenience we write αj ! = α1j ! α2j ! . . . αnj !. The
result of Theorem 2 shows that the multinomial expansion
X k!
X xαj y αj
p
p
xαj y αj = k!
(x1 y1 + x2 y2 + · · · + xn yn )k =
αj !
αj ! αj !
p
k
is no more than an addition formula for the orthonormal basis {xαj / αj !}dj=1
of Hk . The significance of the result (2) lies with the fact that
for any
p it holds
k
orthonormal basis {pj (x)}d1k , not merely the monomials {xαj / αj !}dj=1
.
The addition formula of Theorem 2 yields an identity which contains the Pythagorean theorem sin2 θ + cos2 θ = 1 as a special case:
k
is an orthonormal basis
Theorem 3 (Pythagorean Identity). Suppose {pj (x)}dj=1
p
2
2
2
for Hk . Then on the unit sphere |s| = s1 + s2 + · · · + sn = 1,
dk
X
1
[pj (s)]2 = .
k!
j=1
Proof. Letting x = y in the addition formula yields
(x21 + x22 + · · · + x2n )k = k!
dk
X
[pj (x)]2 ,
j=1
from which the result is immediate.
Most inner products are integrals. Although the inner product we have used here
is defined by a derivative, it is far from artificial. Indeed, this inner product allows
us to view Taylor series as Fourier series. That is, suppose we have the Taylor series
f (x) =
∞
X
aj xαj .
j=0
Rewriting the series as
∞
X
xαj
bj p ,
αj !
j=1
p
and noting that the monomials {xαj / αj !}∞
j=1 are orthonormal with respect to
the inner product (1), we have
p
(xαj / αj !, f (x)) = bj .
f (x) =
These simple results appear to have considerable analytic significance. To see this,
one need only consider the kernel
dk
∞ X
X
pjk (x)pjk (y) =
k=0 j=1
∞
X
(x · y)k
k=0
k!
= ex·y ,
and the reproducing formula
(f (y), ex·y ) = f (x).
k
Here {pjk (x)}dj=1
, k = 0, 1, 2, . . . , are arbitrary orthonormal bases for the homogeneous polynomials Hk of degree k.
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2004
ALLAN FRYANT
References
1. A. Erdelyi, Higher transcendental functions, Vol. 2, McGraw-Hill, New York, 1953. MR 15:419i
Department of Mathematics, Greensboro College, Greensboro, North Carolina
27401
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