## Functional equations of sum form.(English)Zbl 0588.39005

The author, the reviewer and several others have studied the functional equation $(1)\sum^{k}_{i=1}\sum^{\ell}_{j=1}f(p_ iq_ j)=\sum_{i}f(p_ i)+\sum_{j}f(q_ j)+\lambda \sum_{i}f(p_ i)\sum_{j}f(q_ j)$ and some of its generalizations. In this paper the author treats the equation $(2)\quad \sum^{k}_{j=1}\sum^{\ell}_{i=1}[f_{ij}(p_ iq_ j)-q_ jg_ i(p_ i)-p_ ih_ j(q_ j)-\lambda g_ i(p_ i)h_ j(q_ j)]=0$ through the study of the general form $$(3)\quad \sum^{k}_{i=1}\sum^{\ell}_{j=1}F_{ij}(p_ i,q_ j)=0$$ (also studied by the reviewer). The author specializes his results for the equation $$(4)\quad \sum^{k}_{i=1}\sum^{\ell}_{j=1}[f_{ij}(p_ iq_ j)-\sum^{N}_{t=1}g_{it}(p_ i)h_{jt}(q_ j)]=0$$ and shows that under measurability conditions (4) can be reduced to the equation (5) $$\bar f_{ij}(xy)-\sum^{N}_{t=1}\bar g_{it}(x)\bar h_{jt}(y)=0,$$ $$i=1,2,...,k$$, $$j=1,2,...,\ell$$.
Reviewer: Pl.Kannappan

### MSC:

 39B99 Functional equations and inequalities 94A17 Measures of information, entropy