New formulas for Whittaker functions on GL(n,R)

We derive new recursive formulas for principal series Whittaker functions, of both fundamental and class one type, on GL(n,R). These formulas relate such Whittaker functions to their counterparts on GL(n−1,R), instead of on GL(n−2,R) as has been the case in numerous earlier studies. In the particular case n=3, and for certain special values of the eigenvalues, our formulas are seen to resemble classical summation formulas, due to Gegenbauer, for Bessel functions. To illuminate this resemblance we also derive, in this work, formulas for certain special values of GL(3,R) fundamental Whittaker functions. These formulas may be understood as analogs of expressions derived by Bump and Friedberg for class one Whittaker functions on GL(3,R).