# zbMATH — the first resource for mathematics

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.)

##### MSC:
 17D92 Genetic algebras 92D10 Genetics and epigenetics 39A05 General theory of difference equations
##### Keywords:
genetic algebra; powers; linear difference equations
Full Text: