Sum form equations on an open domain. II. (English) Zbl 0602.39005
[For part I see C. R. Math. Acad. Sci., Soc. R. Can. 7, 85-90 (1985; Zbl 0568.39005).]
Let k,\(\ell \geq 3\) be fixed integers. The functional equation \[ \sum^{k}_{i=1}\sum^{\ell}_{j=1}f(p_ iq_ j)=\sum^{k}_{i=1}p_ i^{\alpha}\sum^{\ell}_{j=1}f(q_ j)+\sum^{\ell}_{j=1}q_ j^{\beta}\sum^{k}_{i=1}f(p_ i) \] for all positive \(p_ i\), \(q_ j\) with \(\sum p_ i=\sum q_ j=1\) is solved. It leads to a characterization of entropies of degree (\(\alpha\),\(\beta)\) [M. Behara and P. Nath, Kybernetika 10, 491-503 (1974; Zbl 0295.94046)].
Let k,\(\ell \geq 3\) be fixed integers. The functional equation \[ \sum^{k}_{i=1}\sum^{\ell}_{j=1}f(p_ iq_ j)=\sum^{k}_{i=1}p_ i^{\alpha}\sum^{\ell}_{j=1}f(q_ j)+\sum^{\ell}_{j=1}q_ j^{\beta}\sum^{k}_{i=1}f(p_ i) \] for all positive \(p_ i\), \(q_ j\) with \(\sum p_ i=\sum q_ j=1\) is solved. It leads to a characterization of entropies of degree (\(\alpha\),\(\beta)\) [M. Behara and P. Nath, Kybernetika 10, 491-503 (1974; Zbl 0295.94046)].
Reviewer: C.T.Ng
