The nonhomogeneous linear difference equation with variable difference interval

Abstract

Certain ordinary linear difference equations whose coefficients can be expressed as factorial series have been treated by Norlund [1]. The corresponding nonhomogeneous equation can then be solved by the difference analogue of the Lagrange variation of parameter method or possibly in other ways. The whole process is very complicated in execution and is at most little more than an existence theorem. The present note treats a general linear difference equation where the constant difference interval is replaced by a variable difference interval. So far as the author knows this is the first time that this has been done. Results include the case where the difference interval is constant as a special case. Also the series used, called general factorial series, include ordinary factorial series as used by Norlund [1] as a special case. It is believed that the author was the first to study such series [2].