On generating functions of Hausdorff moment sequences

Abstract

The class of generating functions for completely monotone sequences (moments of finite positive measures on [ 0 , 1 ] [0,1] ) has an elegant characterization as the class of Pick functions analytic and positive on ( − ∞ , 1 ) (-\infty ,1) . We establish this and another such characterization and develop a variety of consequences. In particular, we characterize generating functions for moments of convex and concave probability distribution functions on [ 0 , 1 ] [0,1] . Also we provide a simple analytic proof that for any real p p and r r with p > 0 p>0 , the Fuss-Catalan or Raney numbers r p n + r ( p n + r n ) \frac {r}{pn+r}\binom {pn+r}{n} , n = 0 , 1 , … n=0,1,\ldots , are the moments of a probability distribution on some interval [ 0 , τ ] [0,\tau ] if and only if p ≥ 1 p\ge 1 and p ≥ r ≥ 0 p\ge r\ge 0 . The same statement holds for the binomial coefficients ( p n + r − 1 n ) \binom {pn+r-1}n , n = 0 , 1 , … n=0,1,\ldots \, .