In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of
orthogonal polynomials , introduced by
Elwin Bruno Christoffel (
1858 ) and
Jean Gaston Darboux (
1878 ). It states that
∑
j
=
0
n
f
j
(
x
)
f
j
(
y
)
h
j
=
k
n
h
n
k
n
+
1
f
n
(
y
)
f
n
+
1
(
x
)
−
f
n
+
1
(
y
)
f
n
(
x
)
x
−
y
{\displaystyle \sum _{j=0}^{n}{\frac {f_{j}(x)f_{j}(y)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}{\frac {f_{n}(y)f_{n+1}(x)-f_{n+1}(y)f_{n}(x)}{x-y}}}
where f j (x ) is the j th term of a set of
orthogonal polynomials of squared norm h j and leading coefficient k j .
There is also a "confluent form" of this identity by taking
y
→
x
{\displaystyle y\to x}
limit:
∑
j
=
0
n
f
j
2
(
x
)
h
j
=
k
n
h
n
k
n
+
1
f
n
+
1
′
(
x
)
f
n
(
x
)
−
f
n
′
(
x
)
f
n
+
1
(
x
)
.
{\displaystyle \sum _{j=0}^{n}{\frac {f_{j}^{2}(x)}{h_{j}}}={\frac {k_{n}}{h_{n}k_{n+1}}}\left[f_{n+1}'(x)f_{n}(x)-f_{n}'(x)f_{n+1}(x)\right].}
Proof
Let
p
n
{\displaystyle p_{n}}
be a sequence of polynomials orthonormal with respect to a probability measure
μ
{\displaystyle \mu }
, and define
a
n
=
⟨
x
p
n
,
p
n
+
1
⟩
,
b
n
=
⟨
x
p
n
,
p
n
⟩
,
n
≥
0
{\displaystyle a_{n}=\langle xp_{n},p_{n+1}\rangle ,\qquad b_{n}=\langle xp_{n},p_{n}\rangle ,\qquad n\geq 0}
(they are called the "Jacobi parameters"), then we have the three-term recurrence
[1]
p
0
(
x
)
=
1
,
p
1
(
x
)
=
x
−
b
0
a
0
,
x
p
n
(
x
)
=
a
n
p
n
+
1
(
x
)
+
b
n
p
n
(
x
)
+
a
n
−
1
p
n
−
1
(
x
)
,
n
≥
1
{\displaystyle {\begin{array}{l l}{p_{0}(x)=1,\qquad p_{1}(x)={\frac {x-b_{0}}{a_{0}}},}\\{xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n-1}p_{n-1}(x),\qquad n\geq 1}\end{array}}}
Proof:
By definition,
⟨
x
p
n
,
p
k
⟩
=
⟨
p
n
,
x
p
k
⟩
{\displaystyle \langle xp_{n},p_{k}\rangle =\langle p_{n},xp_{k}\rangle }
, so if
k
≤
n
−
2
{\displaystyle k\leq n-2}
, then
x
p
k
{\displaystyle xp_{k}}
is a linear combination of
p
0
,
.
.
.
,
p
n
−
1
{\displaystyle p_{0},...,p_{n-1}}
, and thus
⟨
x
p
n
,
p
k
⟩
=
0
{\displaystyle \langle xp_{n},p_{k}\rangle =0}
. So, to construct
p
n
+
1
{\displaystyle p_{n+1}}
, it suffices to perform Gram-Schmidt process on
x
p
n
{\displaystyle xp_{n}}
using
p
n
,
p
n
−
1
{\displaystyle p_{n},p_{n-1}}
, which yields the desired recurrence.
Proof of Christoffel–Darboux formula:
Since both sides are unchanged by multiplying with a constant, we can scale each
f
n
{\displaystyle f_{n}}
to
p
n
{\displaystyle p_{n}}
.
Since
k
n
+
1
k
n
x
p
n
−
p
n
+
1
{\displaystyle {\frac {k_{n+1}}{k_{n}}}xp_{n}-p_{n+1}}
is a degree
n
{\displaystyle n}
polynomial, it is perpendicular to
p
n
+
1
{\displaystyle p_{n+1}}
, and so
⟨
k
n
+
1
k
n
x
p
n
,
p
n
+
1
⟩
=
⟨
p
n
+
1
,
p
n
+
1
⟩
=
1
{\displaystyle \langle {\frac {k_{n+1}}{k_{n}}}xp_{n},p_{n+1}\rangle =\langle p_{n+1},p_{n+1}\rangle =1}
.
Now the Christoffel-Darboux formula is proved by induction, using the three-term recurrence.
Specific cases
Hermite polynomials :
∑
k
=
0
n
H
k
(
x
)
H
k
(
y
)
k
!
2
k
=
1
n
!
2
n
+
1
H
n
(
y
)
H
n
+
1
(
x
)
−
H
n
(
x
)
H
n
+
1
(
y
)
x
−
y
.
{\displaystyle \sum _{k=0}^{n}{\frac {H_{k}(x)H_{k}(y)}{k!2^{k}}}={\frac {1}{n!2^{n+1}}}\,{\frac {H_{n}(y)H_{n+1}(x)-H_{n}(x)H_{n+1}(y)}{x-y}}.}
∑
k
=
0
n
H
e
k
(
x
)
H
e
k
(
y
)
k
!
=
1
n
!
H
e
n
(
y
)
H
e
n
+
1
(
x
)
−
H
e
n
(
x
)
H
e
n
+
1
(
y
)
x
−
y
.
{\displaystyle \sum _{k=0}^{n}{\frac {He_{k}(x)He_{k}(y)}{k!}}={\frac {1}{n!}}\,{\frac {He_{n}(y)He_{n+1}(x)-He_{n}(x)He_{n+1}(y)}{x-y}}.}
Associated Legendre polynomials :
(
μ
−
μ
′
)
∑
l
=
m
L
(
2
l
+
1
)
(
l
−
m
)
!
(
l
+
m
)
!
P
l
m
(
μ
)
P
l
m
(
μ
′
)
=
(
L
−
m
+
1
)
!
(
L
+
m
)
!
P
L
+
1
m
(
μ
)
P
L
m
(
μ
′
)
−
P
L
m
(
μ
)
P
L
+
1
m
(
μ
′
)
.
{\displaystyle {\begin{aligned}(\mu -\mu ')\sum _{l=m}^{L}\,(2l+1){\frac {(l-m)!}{(l+m)!}}\,P_{lm}(\mu )P_{lm}(\mu ')=\qquad \qquad \qquad \qquad \qquad \\{\frac {(L-m+1)!}{(L+m)!}}{\big [}P_{L+1\,m}(\mu )P_{Lm}(\mu ')-P_{Lm}(\mu )P_{L+1\,m}(\mu '){\big ]}.\end{aligned}}}
See also
References
Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions , Encyclopedia of Mathematics and its Applications, vol. 71,
Cambridge University Press ,
ISBN
978-0-521-62321-6 ,
MR
1688958
Christoffel, E. B. (1858),
"Über die Gaußische Quadratur und eine Verallgemeinerung derselben." , Journal für die Reine und Angewandte Mathematik (in German), 55 : 61–82,
doi :
10.1515/crll.1858.55.61 ,
ISSN
0075-4102 ,
S2CID
123118038
Darboux, Gaston (1878), "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série", Journal de Mathématiques Pures et Appliquées (in French), 4 : 5–56, 377–416,
JFM
10.0279.01
Abramowitz, Milton; Stegun, Irene A. (1972), Handbook of Mathematical Functions ,
Dover Publications, Inc., New York , p. 785, Eq. 22.12.1
Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (2010),
"NIST Handbook of Mathematical Functions" , NIST Digital Library of Mathematical Functions ,
Cambridge University Press , p. 438, Eqs. 18.2.12 and 18.2.13,
ISBN
978-0-521-19225-5 (Hardback,
ISBN
978-0-521-14063-8 Paperback)
Simons, Frederik J.; Dahlen, F. A.; Wieczorek, Mark A. (2006), "Spatiospectral concentration on a sphere", SIAM Review , 48 (1): 504–536,
arXiv :
math/0408424 ,
Bibcode :
2006SIAMR..48..504S ,
doi :
10.1137/S0036144504445765 ,
S2CID
27519592