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An elementary approach to some analytic asymptotics. (English) Zbl 0828.26003

In J. Math. Anal. Appl. 48, 534-559 (1974; Zbl 0312.65091) M. L. Fredman and D. E. Knuth derived the asymptotic behaviour of sequences fulfilling conditions of the type \[ M(0)= 1,\;M(n+ 1)= \min_{0\leq k\leq n} (\alpha M(k)+ \beta M(n- k)) \] using the Wiener- Ikehara Tauberian theorem in the case where \(\log\alpha/\log \beta\) is irrational.
In the present paper the author gives an elementary derivation that in the irrational case one only makes use of the fact that for irrational \(\vartheta\) the sequence \(n\vartheta\) is uniformly distributed modulo 1.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
05A10 Factorials, binomial coefficients, combinatorial functions
05A16 Asymptotic enumeration
11B37 Recurrences

Citations:

Zbl 0312.65091
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