Stieltjes continued fractions for polygamma functions; speed of convergence

It is well known that the polygamma functions Ψ k ( z ) ≔ d k + 1 d z k + 1 Log Γ ( z ) , k = 0 , 1 , 2 , … can be represented in the half-plane region | arg z | < π / 2 by Stieltjes continued fractions g ( k , z ) = a 1 ( k ) z 2 + a 2 ( k ) 1 + a 3 ( k ) z 2 + a 4 ( k ) 1 + ⋯ , a m ( k ) > 0 . In the present paper it is shown that the coefficients a m ( k ) have the asymptotic behavior a m ( k ) ∼ m 2 16 , m → ∞ . From this it is deduced that the n th approximant g n ( k , z ) of g ( k , z ) converges at the rate | g ( k , z ) - g n ( k , z ) | ⩽ A n B , n ⩾ 1 , where the positive constants A and B depend upon k and z, but are independent of n.