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corresponding to kernels of a ¬xed order k > 0 are equivalent in the sense
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

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 :

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„ ,

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