Page 45 - Textos de Matemática Vol. 34

CHRISTOFFEL-DARBOUX IDENTITY 35
(a) {Pn} is a sequence of matrix orthonormal polynomials.
(b) {Pn} verifies, for all n ∈ N , a Christoffel-Darboux identity
n
(2.1) P∗(t)A P (u)−P∗ (t)A∗ P (u) = (u−t)P∗(t)P (u), t,u ∈ C.
n n+1 n+1 n+1 n+1 n
k k
k=0 (c) {Pn} verifies, for all n ∈ N , the confluent formula
n
(2.2) P∗(t)A P′ (t)−P∗ (t)A∗ P′(t) = P∗(t)P (t),
k=0
n n+1 n+1 n+1 n+1 n
(2.3) P∗(t)A P (t)−P∗ (t)A∗ P (t) = θ, t∈C.
n n+1 n+1 n+1 n+1 n
Proof. Firstly we show that condition (a) implies (b). In fact, by (1.1) we have,
because Bn is hermitean
uPn(u) = An+1Pn+1(u) + BnPn(u) + A∗nPn−1(u)
tP∗(t) = P∗ (t)A∗ +P∗(t)B +P∗ (t)A , n n+1 n+1 n n n−1 n
and subtract the result of multiplying, the first equation on the left by Pn∗(t), and the second on the right by Pn(u) to get
(P∗(t)A P (u)−P∗ (t)A∗ P (u)) n n+1 n+1 n+1 n+1 n
− (Pn∗−1(t)AnPn(u) − Pn∗(t)A∗nPn−1(u)) = (u − t)Pn∗(t)Pn(u)
and applying the telescoping rule we get (2.1).
In order to show that, (b) implies (c), note that if we take limits when u tends
to t and u = t in (2.1), we get (2.2) and (2.3), respectively.
To complete the proof we need to show that (c) implies (a). We rewrite the
equation (2.2) in the form
P∗(t)A P′ (t)−P∗ (t)A∗ P′(t)=P∗(t)P (t)+P∗(t)P (t),
k k
= Pn∗(t)Pn(t) + (Pn∗−1(t)AnPn′ (t) − Pn∗(t)A∗nPn′ −1(t)) . Because, only for a finite number of values Pn is a singular matrix, we rewrite this
n n+1n+1 n+1 n+1n n n and so, applying (2.2) for n − 1 instead of n we get
P∗(t)A P′ (t) − P∗ (t)A∗ P′ (t) n n+1 n+1 n+1 n+1 n
equation in the form
(A P +A∗P )′P−1 −(P∗)−1(P∗ D∗ +P∗ D )(P−1)′ =I
n+1 n+1 n n−1 and using (2.3) we get
n
n n+1 n+1 n−1 n n
((A P n+1 n+1
+A∗P )P−1)′ =I, n∈N. n n−1 n
k k
n−1 k=0