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Full-Text Articles in Physical Sciences and Mathematics
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke
Department of Mathematics: Faculty Publications
Certain well known polynomials have a number of common properties. They arise as coefficients of tn in the expansion of a generating function ; they may be obtained by means of orthogonalization of a set of functions xⁿg(x) when the function ρ(x) = g2(x) and the interval are properly chosen ; they may be regarded as polynomials which become orthogonal when multiplied by a proper factor g(x) ; they satisfy a certain type of difference equation ; they satisfy a certain type of differential equation. The results …
Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce
Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce
Department of Mathematics: Faculty Publications
One of Sylvester's theorems f on matrices states that if the characteristic equation
(1) | M - λI| = f(λ) = 0
of a square matrix M has the roots λ1, λ2, … , λn, then the characteristic equation
(2) | φM - ρI| = = g(ρ) = 0
of any integral function of M, namely, φM, has the roots ρi = φ (λi), i = 1, 2, … , n. In this note an isomorphism is shown to exist between …