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- W2012142308 abstract "Very important progress has recently been made in the analytic theory of homogeneous linear diSerence equations. GALBRUN t has used the Laplace transformation to derive important existence theorems, and has investigated the nature of certain principal solutions for large values of the variable. At about the same time NORLUND t applied the theory of factorial series to linear difference equations, showed the existence of the same solutions, and gave their asymptotic form. In addition to the case of polynomial coefficients he has considered the case where the coefficients can be expressed in factorial series. In a more recent paper he has investigated an equation with still more general coefficients.§ BY means of a method of successive approximation, and a suitable extension of a contour integral due to GUICHARD, CARMICEAEL 1t has independently shown the existence of solutions, and has found the bounds of increase and decrease of the solutions in a direction parallel to the real axis. The latest important contribution to the general theory is by BIRKHOFF*1T He employs a matrix notation, and shows in a direct manner the existence of certain intermediate solutions and of the principal solutions. The asymptotic form of these solutions is determined throughout the complex plane. A modification of the integral used by CARMICHAEL plays an important r81e." @default.
- W2012142308 created "2016-06-24" @default.
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- W2012142308 date "1913-02-01" @default.
- W2012142308 modified "2023-09-26" @default.
- W2012142308 title "The solutions of non-homogeneous linear difference equations and their asymptotic form" @default.
- W2012142308 doi "https://doi.org/10.1090/s0002-9947-1913-1500945-5" @default.
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