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To the theory of linear difference equations. (English) Zbl 0586.39002

This paper concerns the application of Borůvka’s theory of the central dispersion of the first kind for second order linear differential equations of Jacobi type [O. Borůvka, Lineare Differentialtransformationen 2. Ordnung (1967; Zbl 0153.112)] to solutions of the functional equation \((1)\quad \sum^{n}_{j=0}a_ jf[\phi_{n-j}(t)]=0,\) \(a_ j\) real, \(a_ 0\neq 0\), \(a_ n\neq 0\), \(\phi_ k\) the kth central dispersion of the first kind of the two-sided oscillatory equation (2) \(Y''=Q(t)Y\). If \(\phi\) is the basic first-kind central dispersion of (2), X an increasing solution of \(X[\phi (t)]=X(t)+1,\) \(\lambda_ 0\) a simple root of \(\sum^{n}_{j=0}a_ j\lambda^{n-j}=0,\) then \(f=\lambda_ 0^{X(t)}\) is a solution of (1). The paper also includes similar results for the case of multiple roots \(\lambda_ 0\), a theorem relating solutions of two equations (1) with the same coefficients \(a_ j\) but two different functions f and g, where X(t) satisfies two equations (2), results on the number of zeros of certain solutions of \(y''=(1-m^ 2)y/(1+t^ 2)^ 2,\) \(m=2,3,...\), \(t\in R\), and some consequences of it.
Reviewer: E.Kreyszig

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis

Citations:

Zbl 0153.112
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References:

[1] Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB OVW, Berlin 1967 · Zbl 0153.11201
[2] Barvínek E.: О свойстве заменительности дисперсий в решении дифференциального уравнения \(\sqrt{|x'|}\cdot(1/ \sqrt{|x'|})'' + q(x)\cdot x'^2 =Q(t). Publ. Fac. Sci. Univ. Masaryk, Brno, No. 393 (1958), 141-155.\) · Zbl 0082.07502
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