Zur Definition der Nörlundschen Hauptlösung von Differenzengleichungen

Abstract

φ: R→R. Nörlund [4] defined the principal solution f_N of the difference equation

$$V (x, y) \varepsilon R \times R_ + : \frac{1}{y}\left[ {g(x + y, y) - g(x, y)} \right] = \phi (x)$$

by V (x, y) ɛ [b, ∞) ×R₊:

$$f_N (x, y) : = \mathop {\lim }\limits_{s \to 0 + } ( \int\limits_a^\infty {\phi (t) e^{ - st} dt} - y \sum\limits_{\nu = 0}^\infty { \phi (x + \nu y) e^{ - s(x + \nu y )} } )$$

with suitable a,bεR and proved the existence of f_N under certain restrictions onφ. In this paper, another way of defining a principal solution of the difference equation above, which includes Nörlund's, is gone. As an application, we construct in an easy manner a class of limitation methods for getting a principal solution, generalizing results from Nörlund [5].¹)