converse of Proposition 15, showing that 1-summability and summation in

form of a factorial series are equivalent.

Proposition 16 Let d ∈ R be given. For an arbitrary formal power series

∞

ˆ

f (z) = 0 fn z n , let ω = reid , r > 0, and de¬ne bn (ω) as in (6.2). For

± < d < β, assume existence of r0 = r0 (d) > 0 so that for 0 < r < r0 the

factorial series (6.3) converges absolutely for Re (ω/z) > c, with suitably

ˆ

large c = c(r) ≥ 0. Then, f ∈ E {z}1,d for every d as above.

Proof: Observe that absolute convergence of (6.3) implies f (z) ’ 0 as

z ’ 0, uniformly in the region Re (ω/z) ≥ c + µ, for every µ > 0. Suppose

that we had shown f to be independent of ω “ then all that were left to

show would be that g = B1 f is holomorphic near the origin. Both, however,

can be obtained at once as follows:

For ¬xed ω, we use (5.4) (p. 82) to compute g(u) = g(u; ω) = (B1 f )(u),

for u = xeid , x > 0. Moreover, we can interchange summation and limit

∞

y ’ ∞ to obtain g(u) = n=1 bn (ω) (1’e’u/ω )n /n!, for u as above. This is

a power series in t = 1 ’ e’u/ω , and consequently we obtain convergence in

some disc |t| < ρ. This shows holomorphy of g at the origin. Re-expanding

the above series as a power series in u and using (6.2) then shows that

ˆˆ

g = S (B1 f ), so g does not depend upon ω (and then so does f ), completing

2

the proof.

While above we treated the case of k = 1, we shall now brie¬‚y discuss a

more general situation of rational k = p/q, with p, q ∈ N being co-prime.

As we shall see later, this is good enough when dealing with systems of

ODE, since there all “levels” will indeed be rational. So assume for such k

ˆ fn z n is k-summable in direction d, and let g(u) stand for

that f (z) =

6.6 Factorial Series 113

ˆˆ ˆ

the sum of Bk f , and f (z) for the k-sum in direction d of f (z). Perhaps, the

¬rst idea that comes to mind is to observe that then f (z 1/k ) is the Laplace

transform of index one of g(u1/k ), so that one might think of applying the

above propositions. However, g(u1/k ) will in general not be analytic at the

origin, but instead has a branch-point of order p, or in other words, has a

representation in terms of a power series in u1/p . To remedy this, we form

the series

∞

ˆ (z) = fpn+j z pn , 0 ¤ j ¤ p ’ 1,

fj (6.6)

n=0

p’1

ˆ ˆ

z j fj (z). According to Exercise 6 below, the series

so that f (z) = j=0

ˆ

fj (z) are going to be k-summable in direction d if and only if the original

ˆ

series f (z) is k-summable in all directions of the form d + 2jπ/p, 0 ¤

j ¤ p ’ 1. Then we can indeed apply the above propositions to each of

the series fj (z 1/k ), obtaining representations of their sums as convergent

ˆ

factorial series in the variable ω/z k . Combining these representations, one

ˆ

then ¬nds a corresponding formula for the sum of f (z). Since we are not

going to use this, we shall not work out the details of this approach here, but

mention that this approach has already been taken by Nevanlinna [201].

Here we have restricted ourselves to investigating convergence of factorial

series. In the context of di¬erence equations, divergent factorial series also

arise, and their summability properties may be investigated. For partial

results in this direction, see [28, 48]. Instead of factorial series, several

authors have given representations of solutions of ODE in terms of higher

transcendental functions. These series have the advantage of converging

“globally” and can be used in the context of the central connection problem.

We here mention Schmidt [244, 245], Kurth and Schmidt [165], Dunster and

Lutz [90], and Dunster, Lutz, and Sch¨fke [91].

a

Exercises:

1. Setting “n = 0 for m ≥ n, show the recursion

m

n ≥ m ≥ 0.

“n+1 = n “n n

m’1 + “m ,

m

2. Let g(u) be analytic near the origin. For n ≥ 1, show the existence of

numbers bnm , independent of g, such that

n’1

dn

bnm g (n’m) (’ log[1 ’ t]) .

(1 ’ t) g (’ log[1 ’ t]) =

n

n

dt m=0

Applying this to g(u) = ezu , conclude bnm = “n .

m

∞

0 gn u , |u| < ρ, with ρ > 0. Conclude that then

n

3. Let g(u) =

∞

g(’ log[1 ’ t]) = 0 hn tn , |t| < ρ, with h0 = g0 and

˜

n

n ≥ 1.

“n gm m!,

hn n! = n’m

m=1

114 6. Summable Power Series

4. Under the assumptions of Proposition 15, use the previous exercise

to show for n ∈ N that the coe¬cients hn , de¬ned in the proof, are

equal to bn (ω)/n!, with bn (ω) given by (6.2).

5. Show that the Laplace transform of order 1 of (1 ’ e’u/ω )n equals

[(ω/z + 1) · . . . · (ω/z + n)]’1 n!.

ˆ ˆ

6. Given f (z) ∈ E [[z]], let fj (z) be de¬ned as in (6.6). Show that

p’1

’j

e’2πijµ/p f (ze2πiµ/p ),

ˆ ˆ 0 ¤ j ¤ p ’ 1,

fj (z) = p z

µ=0

ˆ

and use this to show that all fj (z) are k-summable in direction d if

ˆ

and only if f (z) is k-summable in all directions of the form d + 2jπ/p,

0 ¤ j ¤ p ’ 1.

7. Using the Stirling numbers, show that every formal power series with

a zero constant term can be formally rewritten as a, usually divergent,

factorial series of the form (6.3), and vice versa.

7

Cauchy-Heine Transform

Let ψ(w) be a continuous function for w on the straight line segment from

0 to a point a = 0. Then the function

a

1 ψ(w)

f (z) = dw

w’z

2πi 0

obviously is holomorphic for z in the complex plane with a cut from 0 to

a, but f in general will be singular at points on this cut. If ψ is holomor-

phic, at least at points strictly between 0 and a, then one can use Cauchy™s

integral formula to see that f is holomorphic at these points, too. At the

endpoints, however, f will be singular, even if ψ is analytic there; for this,

see the exercises at the end of the ¬rst section. For our purposes, it will

be important to assume that ψ, for w ’ 0, decreases faster than arbitrary

powers of w, since then we shall see that f will have an asymptotic power

series expansion at the origin. We shall even show that this expansion is

of Gevrey order s, provided that ψ(w) ∼s ˆ Hence integrals of the above

= 0.

type provide an excellent tool for producing examples of functions with

asymptotic expansions, or even of series that are k-summable in certain

directions. Much more can be done, however: For arbitrary functions, ana-

lytic in a sectorial region and having an asymptotic expansion at the origin,

we shall obtain a representation that is the analogue to Cauchy™s formula

for functions analytic at the origin. As a special case, we shall obtain a

very useful characterization of such functions f that are the sums of k-

summable series in some direction d. In other words, we shall characterize

the image of the operator Sk,d , for k > 0 and d ∈ R. As another appli-

cation of this representation formula, we shall prove several decomposition

116 7. Cauchy-Heine Transform

theorems. For example, we shall show that a divergent k-summable series

can be decomposed into ¬nitely many such series that all have exactly

one singular direction. An even more important decomposition theorem for

multisummable series will be proven in Section 10.4.

7.1 De¬nition and Basic Properties

Let s > 0 and a sectorial region G = G(d, ±) be given. We shall write

As,0 (G, E ) for the set of ψ ∈ As (G, E ) with J(ψ) = ˆ From Proposition 11

0.

(p. 75) we conclude that for ± > sπ the space As,0 (G, E ) only contains the

zero function. Hence we may restrict our discussion to regions with ± ¤ sπ,

but everything we say will be trivially correct in other cases, too.

Let ψ ∈ As,0 (G, E ) and ¬x a ∈ G. Then the function

a

1 ψ(w)

f (z) = (CHa ψ)(z) = dw

w’z

2πi 0

will be called Cauchy-Heine transform of ψ(w). Clearly, f (z) is holomorphic

for z, on the Riemann surface of the logarithm, in the sector S = {z :

arg a < arg z < 2π + arg a}, and vanishes as z ’ ∞. The function f (z)

even is holomorphic at ∞, if we consider z in the complex plane instead of

the Riemann surface, but that is of no importance right now. By deforming

the path of integration, we can holomorphically continue f (z) into the

region G = Ga ∪ S ∪ e2πi Ga , where Ga denotes G with points xa (x ≥ 1)

˜

deleted, and e2πi Ga stands for the “same” region as Ga on the next sheet

˜

of the Riemann surface. Hence G is a sectorial region of bisecting direction

˜

d = d + π and opening ± = ± + 2π.

˜

Proposition 17 Let G, ψ, G, f be as above. Then f (z) ∼s f (z) in G, with

=ˆ

˜ ˜

∞

ˆ

f (z) = 0 fn z n given by

a

1 ψ(w)

n ≥ 0.

fn = dw, (7.1)

wn+1

2πi 0

Moreover, if z ∈ G and |z| < |a|, hence both z and ze2πi in G, then

˜

f (z) ’ f (ze2πi ) = ψ(z). (7.2)

N ’1

Proof: We have (w ’ z)’1 = z N w’N (w ’ z)’1 + z n w’n’1 for

n=0

N ≥ 0. Hence, with fn as in (7.1), we ¬nd

a

1 ψ(w) ˜

z ∈ G,

rf (z, N ) = dw,

wN (w ’ z)

2πi 0

¯

if we choose the path of integration so that w ’ z = 0. Let S be a closed

˜

subsector of G. In case its opening is 2π or more, we may split it into

7.1 De¬nition and Basic Properties 117

¬nitely many closed sectors of smaller opening; thus, we may assume the

opening to be strictly less than 2π. For such a sector, we can choose a path

¯ ¯

of integration from 0 to a, independent of z ∈ S and so that c = c(S) > 0

¯

exists for which |w ’ z| ≥ c|w|, for every w on the path and every z ∈ S.

Since ψ ∈ As,0 (G, E ), we have for su¬ciently large c, K > 0, independent

˜

’N

of w, that w ψ(w) ¤ c K “(1 + sN ), for every N ≥ 0 and every w

N

˜

on the path of integration. This implies, with L denoting the length of the

path of integration, rf (z, N ) ¤ c’1 c L (2π)’1 K N +1 “(1 + [N + 1]s), for

˜

every N ≥ 0 and z ∈ S. From this follows f (z) ∼s f (z) in G. To prove

=ˆ

¯ ˜

(7.2), we observe

1 ψ(w)

f (z) ’ f (ze2πi ) = dw

w’z

2πi γ

with a closed path of positive orientation around z, and use Cauchy™s for-

2

mula.

Remark 9: Under the assumptions of Proposition 17, and setting k = 1/s,

ˆ

we see from the de¬nition on p. 100 that f (z) is k-summable in every