Stieltjes continued fractions for polygamma functions; speed of convergence

摘要

It is well known that the polygamma functions≔Ψk(z)≔dk+1dzk+1LogΓ(z),k=0,1,2,…can be represented in the half-plane region |argz|<π/2 by Stieltjes continued fractionsg(k,z)=a1(k)z2+a2(k)1+a3(k)z2+a4(k)1+⋯,am(k)>0.In the present paper it is shown that the coefficients am(k) have the asymptotic behavioram(k)∼m216,m→∞.From this it is deduced that the nth approximant gn(k,z) of g(k,z) converges at the rate|g(k,z)-gn(k,z)|⩽AnB,n⩾1,where the positive constants A and B depend upon k and z, but are independent of n.