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Simultaneous Linear Recurrence Relations with Variable Coefficients

Published online by Cambridge University Press:  20 January 2009

H. D. Ursell
H. D. Ursell
Affiliation:
The University, Leeds.
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Our subject is a set of equations

where the uj(n)(j = 1, 2, …, k) are k “unknown” functions of the integer variable n, the zi(n) (i = 1, 2, … h) are h “known” functions of n, and the Aij(n) are hk “known” operators

which are polynomials in E, each of fixed order pij but with coefficients which may vary with n. E is the usual operator defined by

Our first task is to determine whether the equations (1) are self-consistent. Secondly, if they are self-consistent, we ask what follows from them for a given subset of the unknowns, e.g. for (uj+1, …, uk) in other words we wish to eliminate (u1, …, uj). In particular we wish to eliminate all the variables but one, say uk. We shall in fact find that either uk is arbitrary or else that it has only to satisfy a single linear recurrence relation : and the order of that relation is of interest to us. Thirdly, we ask that reduction to standard form is possible, with or without a transformation of the unknowns themselves.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 4 , May 1958 , pp. 183 - 206
Copyright
Copyright © Edinburgh Mathematical Society 1958

References

page 183 note 1 We shall keep to this application. But, as a referee has noted, much of our work applies to any elements A,ij belonging to a (non-commutative) ring which possesses a right (and for some purposes a left) Euclidean algorism : in particular it applies to differential operators.

In the purely algebraic context, we would draw attention to the definition of the row-rank ρ (column-rank κ) of a matiix when the ring possesses a right (left) algorism, to the reduction to ρ rows (κ columns), and to the equality of ρ and κ when both algorisms are available.

In the context of differential equations, the H.C.K.F. process appears to be known, but not well-known. The order of a system “in general” (and therefore also of the eliminant “in general”) is given already by Jacobi (Crelle, 64, 297; Werke, V, 193) as p of our (45·1), §6. There is no apparent analogue in this case of the ε of our (45·2). Jaoobi states the result for non-linear simultaneous differential equations. His first two steps, reducing the problem to one of homogeneous linear equations with constant coefficients, seem to the writer not entirely satisfying in these days : that he finds no need of such considerations as are involved in our two lemmas arises partly from this, partly from the fact that he concerns himself with the order of the system, not that of the eliminant.

page 194 note 1 From (35) one easily gets p nq n–1—q np n–1 = (—1)n, p nq n–2—q np n–2 = (—1)n–1αn, well-known formulae from which (38) follows trivially. The proof given above is merely an example of the process of elimination between simultaneous recurrence relations.

page 198 note 1 In the figure the rows and columns are rearranged so that the maximum block is in the north-east corner of the plan.