Whittaker functions on orthogonal groups of odd degree

We give explicit formulas for Whittaker functions for the class one principal series representations of the orthogonal groups $ SO_{2n+1}(\R) $ of odd degree. Our formulas are similar to the recursive formulas for Whittaker functions on $SL_n(\R)$ given by Stade and the author \cite{ISt}. Some parts of our results are announced in \cite{I3}.