Genetic algebras studied recursively and by means of differential operators. (English) Zbl 0286.17006

In a nonassociative algebra \(A\) define “powers” \(G(n)\) and \(H(n)\) by \(G(n+1)=G(n)G(n)\) and \(H(n+2)=H(n+1)H(n)\) \((n=0,1,2,\ldots)\). The author describes a genetic algebra \(A_k\) [I. M. H. Etherington, Proc. R. Soc. Edinb. 59, 242–258 (1939; Zbl 0027.29402)] in such a way that linear difference equations are readily deduced for \(G(n)\) and \(H(n)\) in \(A_k\). He solves these equations for \(k=1,2\) and attains known results for \(G(n)\) \((k=1,2,3)\) and new results for \(H(n)\) \((k=1,2)\). He asserts that under hypotheses valid in genetic applications, \(H(n)\) has a limit as \(n\) tends to infinity and hence \(A_k\) has an idempotent \(H(\infty)\). (The reviewer was unable to follow the proof of this assertion for \(k>2\) because of the lack of discussion of possible coincidence of roots.)


17D92 Genetic algebras
92D10 Genetics and epigenetics
39A05 General theory of difference equations


Zbl 0027.29402
Full Text: DOI EuDML