Explicit inverses of some tridiagonal matrices - Estudo Geral ...

20 C.M. da Fonseca, J. Petronilho / Linear Algebra and its Applications 325 (2001) 7–21
( ) ( )
a b c
a b c
Q 3i
α β γ ; x := P i
α β γ ; x
( )
a b c
+γ(x− b) P i−1
α β γ ; x ,
( )
( )
a b c
a b c
Q 3i+1
α β γ ; x := (x − a)P i
α β γ ; x
( )
a b c
+αγ P i−1
α β γ ; x ,
( )
( )
a b c
a b c
Q 3i+2
α β γ ; x := [(x − a)(x − b) − α] P i
α β γ ; x ,
where a, b, c, α, β and γ are some parameters, subject to the restriction αβγ > 0.
Under these conditions,
⎧
⎨(−1) i+j β i β i+1 ···β j θ i−1 φ j+1 /θ n if i j,
(B −1 ) ij =
(19)
⎩
(−1) i+j γ j γ j+1 ···γ i θ j−1 φ i+1 /θ n if i>j,
where
β 3l+s+1 = b s+1 , γ 3l+s+1 = c s+1 (s = 0, 1, 2; l = 0, 1, 2,...),
and the θ i ’s and φ i ’s are explicitly given by
( )
θ i = (−1) i a1 a
Q 2 a 3
i ; 0
(20)
b 1 c 1 b 2 c 2 b 3 c 3
and
⎧
( )
(−1) n+1−i a3 a
Q 2 a 1
n+1−i ; 0 if n ≡ 0, (mod 3),
b 2 c 2 b 1 c 1 b 3 c 3
⎪⎨
( )
φ i = (−1) n+1−i a1 a
Q 3 a 2
n+1−i ; 0
b 3 c 3 b 2 c 2 b 1 c 1
( )
⎪⎩ (−1) n+1−i a2 a
Q 1 a 3
n+1−i ; 0
b 1 c 1 b 3 c 3 b 2 c 2
if n ≡ 1, (mod 3),
if n ≡ 2,(mod 3).
Acknowledgments
This work was supported by CMUC (Centro de Matemática da Universidade de
Coimbra). The authors thank the editor as well as the unknown referees for their
helpful remarks which helped to improve the final version of the paper.