# A Binomial Congruence by Mike Swarbrick Jones By Mike Swarbrick Jones

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The argument is rather generally applicable, and is not confined to the secondorder case. We work in terms of the concept of an eigenvalue as a non-zero (k + 1)-tuple, and topologize the set of such (k + 1)-tuples by component-wise convergence. We give sufficient conditions that such a (k + 1)-tuple not be a limit-point of eigenvalues. We take the latter in real terms, as we confine attention to formally self-adjoint definite cases. 1. 2) be formally self-adjoint and deﬁnite. Let (µ0 , . . 8) are of ﬁxed order, with no singular points, for all sets (λ0 , .

Yk ; z1 , . . 1) r = 1, . . , k, s = 0, . . , k, s = t. 1), multiplied by a signfactor (−1)t . 1) should be defined. 1. 2) be formally self-adjoint and deﬁnite. Let y1 , . . 2) for the eigenvalue λ0 , . . , λk , and let z1 , . . 7) replaced by µs , where µ0 , . . , µk is a distinct eigenvalue. Then χt (y1 , . . , yk ; z1 , . . , zk ) = 0, t = 0, . . , k. 8) by zr , respectively, and integrate over (ar , br ). This gives k λs Ψrs (yr , zr ) = 0, r = 1, . . , k. 3) µs Ψrs (zr , yr ) = 0, r = 1, .

In some cases, this is quite appropriate. 4) s=1 over some interval [a, b], in which we ask for there to exist a non-trivial solution that vanishes at a, at b, and also at k − 1 assigned points in (a, b); the ktuples (λ1 , . . , λk ) for which this is possible constitute the spectrum. 6), with separate independent variables. 9. Certain singular cases, in particular those involving semi-infinite intervals, will be considered later. These, of course, cannot be re-formulated in terms of a single independent variable.