In this note, we provide analytic expressions for the R\'enyi common information of orders in $(1,\infty)$ for the doubly symmetric binary source (DSBS). Until now, analytic expressions for the R\'enyi common information of all orders in $[0,\infty]$ have been completely known for this source. We also consider the R\'enyi common information of all orders in $[-\infty,0)$ and evaluate it for the DSBS. We provide a sufficient condition under which the R\'enyi common information of such orders coincides with Wyner's common information for the DSBS. Based on numerical analysis, we conjecture that there is a certain phase transition as the crossover probability increasing for the R\'enyi common information of negative orders for the DSBS. Our proofs are based on a lemma on splitting of the entropy and the analytic expression of relaxed Wyner's common information.

翻译：本文针对双重对称二元源（DSBS），给出了阶数在$(1,\infty)$范围内的Rényi公共信息的解析表达式。迄今为止，对于该信源，阶数在$[0,\infty]$范围内的所有Rényi公共信息的解析表达式已完全明确。我们还考虑了阶数在$[-\infty,0)$范围内的Rényi公共信息，并针对DSBS进行了计算。我们给出了一个充分条件，在此条件下，此类阶数的Rényi公共信息与DSBS的Wyner公共信息一致。基于数值分析，我们推测对于DSBS的负阶Rényi公共信息，随着交叉概率的增加，存在某种相变。我们的证明基于一个关于熵分裂的引理以及松弛Wyner公共信息的解析表达式。