An operational calculus for difference equations



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Operational methods for solving difference equations have been widely used for some time. In this paper, the recently developed operational calculus of Brand (an analogue of Mikusinski's calculus for differential equations) is examined in detail, but in a slightly different setting from that of the original development. Where Brand dealt primarily with the integral domain S of sequences defined on the nonnegative integers (with the operations pointwise addition and Cauchy product), this paper deals with the field of entering sequences, that is, sequences defined on all the integers, but which have at most a finite number of nonzero entries to the left of the origin. The results thus obtained are more general, and yield the results of the original development as a special case. After proving that 9 is a field (under pointwise addition and a generalization of Cauchy product), a table of transforms is constructed, by means of which certain basic sequences given in terms of t may be written as rational polynomials in the shifting sequence s (defined by s(l) = 1; s(t) = 0, t [not equal] 1). A system is then given for obtaining the solution y of the difference equation...