that they sum the same formal power series to the same analytic functions.

However, observe that we do not claim that two such kernels of the same

order represent the sum on exactly the same sectorial region. Moreover, the

two summability methods may be inequivalent for series other than power

series.

All the summability methods we consider, aside from some minor modi-

¬cations, ¬t into the following classical family of methods:

Moment Methods

Let e(x) be positive and continuous on the positive real axis and

asymptotically zero as x ’ ∞, so that all moments m(n) =

108 6. Summable Power Series

∞

xn e(x) dx, n ≥ 0, exist and are positive. A series xn

0

then is said to be me -summable if the power series x(t) =

∞n

0 t xn /m(n) converges for every t ∈ C , and

∞

T

lim e(t) x(t) dt = e(t) x(t) dt

T ’∞ 0 0

exists.

Applied to formal power series, it is not clear in general whether summa-

bility for some z = 0 implies summability for other values, in particular

for such values closer to the origin. Moreover, in applications to ODE, it is

tn xn /m(n) for every t, but

more natural not to require convergence of

instead be content with a positive radius of convergence plus existence of

holomorphic continuation along the positive real axis. In detail, this leads

to the following modi¬ed de¬nition for summability of formal power series

by means of methods de¬ned by kernel functions introduced in Sections 5.5

and 5.6. For further details, see [25].

Moment Summability of Power Series

Let a kernel function e(z) of order k > 0, with corresponding

integral operator T , be given. We say that a formal power series

ˆ

f (z) is T -summable in direction d if the following holds:

(S1) The series g = T ’ f has positive radius of convergence.

ˆˆ

ˆ

(S2) For some µ > 0, the function g = S g can be holomor-

ˆ

phically continued into S = S(d, µ) and is of exponential

growth at most k there.

ˆ

Obviously, (S1) holds if and only if f ∈ E [[z]]1/k . Condition

(S2) implies applicability of the integral operator T to g, and

ˆ ˆ

we call f = T g the T -sum of f , and write f = ST,d f . The set

ˆ

of all such f shall be denoted by E {z}T,d .

Due to the results of Section 5.5 it is immediately seen that the above

summability method is equivalent to k-summability in the following sense:

Theorem 38 Let an arbitrary kernel function e(z) of order k > 0 be given.

ˆ ˆ

Then E {z}T,d = E {z}k,d , and for every f ∈ E {z}T,d we have (ST,d f )(z) =

ˆ

(Sk,d f )(z) on some sectorial region of bisecting direction d and opening

more than π/k.

ˆ

Proof: For f ∈ E {z}T,d , we conclude from Theorem 27 (p. 91) that

ST,d f (z) ∼1/k f (z) in a sectorial region G with bisecting direction d and

ˆ ˆ

=

ˆ ˆ

opening more than π/k, so by de¬nition f ∈ E {z}k,d and ST,d f (z) =

6.5 General Moment Summability 109

ˆ ˆ

Sk,d f (z) in G. Conversely, f ∈ E {z}k,d implies, owing to Theorem 28

(p. 91), that g = T ’ (Sk,d f ) is analytic and of exponential growth not

ˆ

more than k in a sector S(d, µ), and g(u) ∼0 g (u) = T ’ f (u) there. An

ˆˆ

=ˆ

asymptotic of Gevrey order zero implies convergence of g to g, for small

ˆ

|u|. Hence (S1), (S2) follow. 2

The equivalence of these summability methods is used in the following

exercises:

Exercises: Let k > 0 be ¬xed and observe the following terminology: A

sequence (»n )n≥0 is called a summability factor for k-summability, if for

every Banach space E and every fn z n ∈ E {z}k , we have » n fn z n ∈

E {z}k .

1. Show that if (»n ) is a summability factor for k-summability, then

»n z n ∈ C {z}k .

necessarily

2. Let e(z) be a kernel function of order k, with corresponding mo-

ment function m(u). Show that the sequences (“(1 + n/k)/m(n)) and

(m(n)/“(1 + n/k)) both are summability factors for k-summability.

3. Let e1 (z), e2 (z) be kernel functions of the same order k, with corre-

sponding moment functions m1 (u), m2 (u). Show that the sequence

m1 (n)/m2 (n) is a summability factor for k-summability.

4. Show that the following sequences are summability factors for k-

summability:

“(±+n/k)

(a) »n = “(β+n/k) , for ±, β > 0.

“(1+s1 n)(1+s2 n)

for sj ≥ 0, s1 + s2 = s3 + s4 .

(b) »n = “(1+s3 n)(1+s4 n) ,

5. Show that the sequence »n = »n , for any ¬xed » ∈ C , is a summa-

bility factor for k-summability.

6. For ¬xed p ∈ N, show that the sequence »n = 1 (resp. = 0) whenever

n is (resp. is not) a multiple of p, is a summability factor for k-

summability.

7. For m ∈ Z, show that the sequence »n = (1 + n)m is a summability

factor for k-summability.

8. Show that the sequence »n = 1 (resp. = 0) whenever n is (resp. is

not) of the form 2m with m ∈ Z, is not a summability factor for

k-summability.

9. For s > 0, show that (1/“(1 + sn)) is a summability factor for k-

summability if and only if s ≥ 1/k.

110 6. Summable Power Series

6.6 Factorial Series

In this section, we brie¬‚y investigate in¬nite series of the form

∞

bn

bn ∈ E .

, (6.1)

(z + 1) · . . . · (z + n)

n=1

The form of the terms requires the variable z to be di¬erent from negative

integers; for convenience, we shall always restrict z to the right half-plane.

Series of this or a very similar form have frequently been studied in con-

nection with Laplace or Mellin transform, and are usually called (inverse)

factorial series. A number of authors have used them to represent solutions

of ODE, or of di¬erence equations. For an application to even more general

equations, see Braaksma and Harris Jr. [72], or G´rard and Lutz [107].

e

Unlike power series, factorial series converge in half-planes: From the

functional equation of the Gamma function (B.9) and (B.13) (p. 231), it is

easy to conclude that

n! “(1 + n) “(1 + z) “(1 + z)

∼

= ,

(z + 1) · . . . · (z + n) nz

“(1 + z + n)

in the sense that the quotient of the two sides tends to one as n ’ ∞.

Hence, if (6.1) converges for some z = z0 in the right half-plane, then

the terms necessarily tend to 0. So we obtain that |bn | ¤ nRe z0 n! for

large n. Consequently, we have absolute and locally uniform convergence

of (6.1) for all z with Re z > c = 1 + Re z0 . We also observe that the rate

of convergence of factorial series, in general, is rather slow, limiting their

usefulness for numerical purposes.

It is well understood (see [204], or Wasow [281]) how factorial series are

related to Laplace integrals, and we here are going to show corresponding

relations with k-summability. To do so, we shall make use of the following

numbers:

Stirling Numbers

The numbers “n , de¬ned by

m

n

ω (ω + 1) · . . . · (ω + n ’ 1) = n ≥ 1, ω ∈ C ,

“n m

n’m ω ,

m=1

are called Stirling numbers, or factorial coe¬cients. For a re-

cursive de¬nition of these numbers and other details, see the

exercises below.

To establish the said connection between k-summability and factorial

series, we ¬rst treat the case of k = 1:

6.6 Factorial Series 111

Proposition 15 Let d ∈ R be given. For an arbitrary formal power series

∞

ˆ

f (z) = 0 fn z n ∈ E {z}1,d , let ω = reid , r > 0, and de¬ne

n

n ≥ 1.

ω m fm “n ,

bn (ω) = (6.2)

n’m

m=1

Then for every su¬ciently small r > 0, there exists a number c > 0 so that

the factorial series

∞

bn (ω)

f (z) = (6.3)

(ω/z + 1) · . . . · (ω/z + n)

n=1

converges absolutely for Re (ω/z) > c. The function f (z) then does not

depend upon r, and

ˆ

f0 + f (z) = (S1,d f )(z), (6.4)

for such z where both sides give sense.

ˆˆ

Proof: According to our assumptions, the function g = S (B1 f ) is an-

alytic in G = D(0, ρ) ∪ S(d, µ), for su¬ciently small ρ, µ > 0. With r

small enough, the function g(uω) is holomorphic in the region de¬ned by

|1 ’ e’u | < 1, and continuous up to its boundary. In other words, the func-

tion h(t) = g(’ω log[1 ’ t]) is analytic in the unit disc, and continuous

along its boundary except for a singularity at t = 1. In the sector S(d, µ),

the function g(u) is of exponential growth at most one. We may assume

without loss of generality that µ ¤ π/2, so that this fact can be expressed

as |g(u)| ¤ c| exp[K u e’id ]| in G, for su¬ciently large c, K > 0. This then

implies

c

|h(t)| ¤ , |t| ¤ 1, (6.5)

|1 ’ t|rK

and we may take 0 < r < 1/K, so that h(t) has an integrable singularity

at t = 1. As follows from Exercise 4 below, the coe¬cients hn of the power

series expansion of h(t) about the origin are equal to bn (ω)/n!, with bn (ω)

given by (6.2), for n ≥ 1, while h0 = f0 . Using Cauchy™s Formula and (6.5),

∞

we ¬nd for the remainder term hN (t) = n=N hn tn :

2π

tN h(u) du c dφ

|hN (t)| = ¤ ,

uN (u ’ t) 2π|1 ’ t| |1 ’ eiφ |Kr

2πi |u|=1 0

for 0 ¤ t < 1. In the original variable u, this implies

∞

bn (ω)

(1 ’ e’u/ω )n , |gN (u)| = |hN (1 ’ e’u/ω )| ¤ c|eu/ω |,

g(u) = f0 + ˜

n!

n=1

for arg u = d. Hence, by means of Lebesgue™s dominated convergence theo-

rem, we conclude that we may insert the expansion for g(u) into Laplace

112 6. Summable Power Series

transform and integrate termwise. Using the exercise below to ¬nd the

Laplace transform for (1 ’ e’u/ω )n , we obtain (6.4), completing the proof.

2

The above proposition implies for fn = δnm , m ∈ N, that

∞

“n n’m

m m

z =ω ,

(ω/z + 1) · . . . · (ω/z + n)

n=m

where ω can be any complex number. An analysis of the proof shows that

this series converges whenever ω/z is is the right half-plane. This implies

∞

ˆ m

that for an arbitrary formal power series f (z) = m=1 fm z we may

m

replace z by the above expansion and formally interchange the two sums

in order to obtain a formal factorial series. Thus, the above proposition

may be viewed as saying that for 1-summable series this process leads to

ˆ ˆ

a convergent factorial series. Since f ∈ E {z}1,d implies f ∈ E {z}1,„ , for „

su¬ciently close to d, we obtain that (6.3) remains convergent for ω = rei„ ,