On the Multiplication of Recurrences
نویسنده
چکیده
(1) An+2 = An+1+An , with initial values AQ and A-j. The special case AQ = 0, AI= 1 in (1) is the well known Fibonacci series (Fn), and AQ = 2, AJ=1 is the Lucas series (Ln), The integer N(A) = A\ -A0A2 is the norm (also known as the characteristic number) of (1).\Nben recurrences (An) and (Bn) are multiplied (the multiplication of recurrences, which is defined below, was developed in [5]), we have that N(A)N(B) = N(AB). This multiplicative property is the justification of the use of the word norm. In this paper, we shall derive some basic properties of recurrences under multiplication. Our main result will be that recurrences can be factored uniquely (up to order and sign) into recurrences whose norms are prime. Let AQ = AO, A*I= AQAp The recurrence (A„), obtained by using AQ and /S/ as initial values in (1),will be called the dual recurrence of (An), and the asterisk will be used to denote dual recurrences. The notion of dual recurrences was introduced in [3]. It may be shown by induction that
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