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Many further numerical examples are given in the literature [39,41â€“ 44,50,84].

Appendix A. Stieltjes series and functions

A Stieltjes series is a formal expansion
âˆž
(âˆ’1) j j z j
f(z) = (A.1)
j=0

with partial sums
n
(âˆ’1) j j z j :
fn (z) = (A.2)
j=0

The coe cients n are the moments of an uniquely given positive measure (t) that has inÃ¿nitely
many di erent values on 06t Â¡ âˆž [4, p. 159]:
âˆž
t n d (t);
= n âˆˆ N0 : (A.3)
n
0
142 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

Formally, the Stieltjes series can be identiÃ¿ed with a Stieltjes integral
âˆž
d (t)
f(z) = ; |arg(z)| Â¡ : (A.4)
1 + zt
0

If such an integral exists for a function f then the function is called a Stieltjes function. For every
Stieltjes function there exist a unique asymptotical Stieltjes series (A.1), uniformly in every sector
|arg(z)| Â¡ Ã‚ for all Ã‚ Â¡ . For any Stieltjes series, however, several di erent corresponding Stieltjes
functions may exist. To ensure uniqueness, additional criteria are necessary [88, Section 4.3].
In the context of convergence acceleration and summation of divergent series, it is important that
for given z the tails f(z) âˆ’ fn (z) of a Stieltjes series are bounded in absolute value by the next
term of the series,
n+1
|f(z) âˆ’ fn (z)|6 n+1 z zÂ¿0: (A.5)
Hence, for Stieltjes series the remainder estimates may be chosen as
!n = (âˆ’1)n+1 n+1
n+1 z : (A.6)
fn (z), i.e., to a tËœ variant.
This corresponds to !n =

Appendix B. Derivation of the recursive scheme (148)
(k) (k)
We show that for the divided di erence operator = n [{{x n }}] the identity
n
(k) (k)
(x n+k+1 + â€˜) n+1 ((x)â€˜ g(x)) âˆ’ (x n + â€˜) n ((x)â€˜ g(x))
(k+1)
((x)â€˜+1 g(x)) = (B.1)
n
x n+k+1 âˆ’ x n
holds. The proof is based on the Leibniz formula for divided di erences (see, e.g., [69, p. 50]) that
(k)
yields upon use of (x)â€˜+1 = (x + â€˜)(x)â€˜ and n (x) = x n k; 0 + k; 1
k+1
(k+1âˆ’j)
(k+1) (k+1) ( j)
((x)â€˜+1 g(x)) = â€˜ n ((x)â€˜ g(x)) + n (x) n+j ((x)â€˜ g(x))
n
j=0
(k)
(k+1)
= (x n + â€˜) ((x)â€˜ g(x)) + n+1 ((x)â€˜ g(x)): (B.2)
n

Using the recursion relation of the divided di erences, one obtains
(k) (k)
n+1 ((x)â€˜ g(x)) âˆ’ n ((x)â€˜ g(x)) (k)
(k+1)
((x)â€˜+1 g(x)) = (x n + â€˜) + n+1 ((x)â€˜ g(x)): (B.3)
n
x n+k+1 âˆ’ x n
Simple algebra then yields Eq. (B.1).
Comparison with Eq. (140) shows that using the interpolation conditions gn = g(x n ) = sn =!n and
â€˜ = k âˆ’ 1 in Eq. (B.1) yields the recursion for the numerators in Eq. (148), while the recursion
for the denominators in Eq. (148) follows for â€˜ = k âˆ’ 1 and using the interpolation conditions
gn = g(x n ) = 1=!n . In each case, the initial conditions follow directly from Eq. (140) in combination
with the deÃ¿nition of the divided di erence operator: For k = 0, we use (a)âˆ’1 = 1=(a âˆ’ 1) and obtain
(k)
n (x n )kâˆ’1 gn = (x n )âˆ’1 gn = gn =(x n âˆ’ 1).
H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147 143

Appendix C. Two lemmata

Lemma C.1. DeÃ¿ne
k n+j
â—¦ (k)
A= ; (C.1)
j
(n + j)r+1
j=0

â—¦ â—¦ (k)
k
(k) j
where is a zero of multiplicity m of (z) = j=0 j z . Then
â—¦
(âˆ’1)m d m (k)
r+m
n+m
Aâˆ¼ () (n â†’ âˆž): (C.2)
r nr+m+1 d xm

Proof. Use
âˆž
1 1
exp(âˆ’at)t r dt;
= aÂ¿0 (C.3)
ar+1 r! 0

to obtain
k
âˆž n âˆž
1 â—¦
â—¦ (k) n+j r (k)
( exp(âˆ’t))t r dt:
A= exp(âˆ’(n + j)t)t dt = exp(âˆ’nt) (C.4)
j
r! r!
0 0
j=0

Taylor expansion of the polynomial yields due to the zero at
â—¦
(âˆ’ )m d m (k) (x)
â—¦
(k)
t m (1 + O(t)):
( exp(âˆ’t)) = (C.5)
m! d xm
x=

Invoking Watsonâ€™s lemma [6, p. 263 ] completes the proof.

â—¦ (k)
(k)
Lemma C.2. Assume that assumption (C-3 ) of Theorem 13 holds. Further assume â†’ for
n; j j
n â†’ âˆž. Then; Eq. (280) holds.

Proof. We have
jâˆ’1
!n+j n+t
j j
âˆ¼ exp âˆ¼ exp(j n ) (C.6)
n
!n n
t=0

for large n. Hence,
k k
!n â—¦
â—¦ (k)
(k)
exp( n ))âˆ’j = (k)
âˆ¼ j( (1= + n) (C.7)
n; j
!n+j
j=0 j=0

â—¦
(k)
Since the characteristic polynomial (z) has a zero of order at z = 1= according to the
assumptions, Eq. (280) follows using Taylor expansion.
144 H.H.H. Homeier / Journal of Computational and Applied Mathematics 122 (2000) 81â€“147

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