Vol. 35, No. 1, 1970

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ISSN: 0030-8730
Regular boundary problems for a five-term recurrence relation

Claude Elias Billigheimer

Vol. 35 (1970), No. 1, 23–51
Abstract

We consider in this paper boundary problems for the fiveterm scalar recurrence relation

dnyn+2 + cnyn+1 + (bn − λan)yn + cn−1yn−1 + dn−2yn−2 = 0
(0 ≦ n ≦ m )
(1.1)
where the coefficients an,bn,cn,dn are real, an,dn > 0 and λ is a complex parameter, with boundary conditions of the typical form
y−2 = y−1 = 0
(1.2)

and

ym+1 + k(cmym  + dm−1ym− 1) = 0,ym+2 + hym = 0
(1.3)

for some integer m 0, and real numbers h,k.

We derive oscillation properties, orthogonality relations and associated eigenvector expansion theorems for solutions of (1.1), (1.2), (1.3), and then discuss the solution of boundary problems for the corresponding inhomogeneous recurrence relation in terms of a Green’s function.

Mathematical Subject Classification
Primary: 39.20
Secondary: 34.00
Milestones
Received: 17 September 1968
Revised: 1 April 1970
Published: 1 October 1970
Authors
Claude Elias Billigheimer