Some Linear and Some Quadratic Recursion Formulas

§ 1. Introduction We shall mainly deal with linear recursion formulas of the type n-1 (l. 1) f(1) = 1 ; /(n) _ i~ ck f(n-k) (n = 2, 3,. . .), k=1 and with quadratic formulas of the type n-1 (1. 2) f(1) = 1 ; /(n) _ d, f(k)/(n-k) (n = 2,3. . .). k=1 We assume that ck > 0, d k > 0 (k = 1, 2,. . .). In a previous paper [1] we discussed (1. 1) under the condition X° ck = 1, and further special assumptions. Presently we deal with it more generally. We shall show that lim {f(n)J-1/11 always exists, and we shall give several sufficient conditions for the existence of lim /(n)//(n + 1). Some of the results can be applied to (1. 2) (see § 6), and some of the methods can be extended to recurrence relations with coefficients c depending on n also (see § 3 and § 7). In [1] as well as in the earlier paper of ERDÖS, FELLER and POLLARD [3], referred to below, the condition on the c k was c k > 0 (k = 1, 2,. . .), whereas the g .c .d. of the k's With ck = 0 was assumed to be 1. For convenience we assume ck > 0 throughout. Consequently we have, both for (1. 1) and for (1. 2), f(n) > 0 (n = 1, 2,. . .). Dealing with the linear relation (1. 1) we put formally (1. 3) C(x) _ 1 1 and we have formally (1 .4) F(x) = x +C(x) F(x). Furthermore, if o is a positive number, and if we put (1 .5) /(n) = `O-n+1 g(n), then we have n-1 (1. 6) g(n) _ b k g(n-k) , g(1) = 1, k=1 where bk = ck o'c ,. Formula (1. 6) is again of the type (1. 1), and b k > 0 for all k .