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Chapter 1 Sequences and Series
Chapter 1
Sequences and Series
1.1. Order Symbols and Asymptotic Scales, Continuous Variables Let 1/ and 1/' (see Notation) be equipped with pseudometrics d and d', respectively; let n be a cone in 1/ and ¢, IjJ E /T(1/, 1/').
¢=
O(IjJ)
n
in
(1)
means for some M > 0 there is an R(M) > 0 such that d'(¢, O)jd'(IjJ, 0) < M,
Further,
¢ = o(ljJ)
d(x,O) > R.
XEn,
III
n
(2)
(3)
means for any e > 0 there is an R(e) > 0 such that d'(¢, O)jd'(IjJ, 0) < e,
d(x,O) > R.
XEn,
(4)
If ¢, IjJ depend parametrically on a E .sf and (2) holds for all a E .sf, then we shall write "¢ = O(IjJ) in n uniformly in .sf," and similarly for (3). F or the foregoing definitions to apply, the implicit assumption is made that denominators are never zero; for example, there must be some R such that d'(l/J, O) oF 0, X E n, d(x, 0) > R. Thus anytime an order symbol is used, an implicit statement is being made about the zeros of d'(IjJ, 0). The concept of asymptotic equivalence is often useful. This is written in and means both ¢ - IjJ
n
= o(ljJ) and IjJ - ¢ = o(¢) in n.
(5)
2
I. Sequences and Series
Now let cj» E :Y sC'f/, y'), where Y and 'Or' are linear spaces with pseudometrics d and d', cj» is called an asymptotic scale in Q if, for every k ;;::: 0, in
(6)
Q,
and if this holds uniformly in k or uniformly in some parameter space, we speak of a uniform asymptotic scale (properly qualified). See Erdelyi (1956) for many examples. Letf E :Y(Y, y'), A E res and cj» be an asymptotic scale in Q. The statement in
(7)
Q
is to be read "f has the right-hand side as an asymptotic expansion in Q with respect to the scale cj»" and means, for every k ;;::: 0,
f -
k
L A r4>r = O(4)k) r;O
in
Q.
(8)
Often cj» is understood from context, so "with respect to the scale cj»" may be deleted from the definition. Note that 0, 0, and -- are transitive and ~ is symmetric. Clearly no asymptotic scale can contain the zero vector or two identical vectors. If d' is a metric induced by a norm I ·11, the asymptotic expansion (7) is unique (but not otherwise). This is a simple consequence of the fact that 4> = 0(1]), l/J = 0(1]) then imply 4> + l/J = 0(1]). Thus assume another expansion (7) with coefficients A' holds. Setting k = in (8) and its analog and subtracting the two gives
°
(A o - A~)4>o = 0(4)0), or lAo -
A~
I<
s for all e, so that A o =
A~,
(9)
similarly, A j = Aj,j > 0.
1.2. Integer Variables In discussions of sequences, the relevant variable x in 4> or l/J takes values in JO. We write 4>n or l/Jn for 4> or l/J, respectively, or, when there is a possibility of confusion with the index of an asymptotic scale, 4>(n) or l/J(n). 1.1(1) is then written
and means that for some M > 0, there is an N > d'(4)n, O)jd'Cl/Jn' 0) < M
for
°
such that
n> N.
(1)
1.3. Sequences and Transformations in Abstract Spaces
3
A similar modification is made of 1.1(3). An additional complexity occurs when ¢ and t/J depend on a p-tuple with elements in say, n = (ml' m2' ... , m p ) . It is usually important to know exactly how the elements mj become infinite, and it is hardly ever sufficient to say, for instance, that ml + m 2 + ... + mp > N. In fact, the concept of a path in n-space becomes important (see Section 1.3).
r:
1.3. Sequences and Transformations in Abstract Spaces
In this book we shall be concerned with two kinds of sequence transformations. The first is the transformation ofagiven sequences E d sinto a sequence S E.r4's with, generally, a formula given to compute sn in terms of elements of s. (In some situations there is no explicit formula.) The other case is where the given sequence s is mapped into a countable set of sequences S(k), k ~ 0, with a formula given (called a lozenge algorithm) for filling out the array {S~k)}, n, k ~ 0. The whole point is to compare the convergence of the transformed sequence(s) with that of the original sequence. The most useful concepts are formulated in the definitions that follow. Definition 1. (i) (ii) (iii)
Let s, tEAte, a metric space.
t converges as s means d(sn, s) = O(d(t n, t)) and d(t n, t) = O(d(sn's)), t converges more rapidly than s means d(tn' t) = o(d(sn, s)). The convergence of sis pth order if, for some p E
r,
d(sn+ I' s)
= O(d(sn, s)") (I)
and
d(sn' s)"
=
O(d(sn+ I' s)).
It is easy to show that p, if it exists, is unique.
Definition 2.
Let T E !!T(d, JIt s) where d
c
JIt e and T(s)
=
s.
(i) Tis regular for d ifs Ed=> S E Ate and s = s. (ii) T is accelerative for d (or accelerates d) if T is regular for d and S converges more rapidly than s, s e ss, Definition 3. SEd.
Let T E 5"(d, Jlts)whered
c
Jlt s . T sumsdifT(s) E JIt c-
P = {(im,Jm)lim, JmEjO} IS called a path if io
4
I. Sequences and Series n/k (0,0) (0,1)
(O,2)
(1,0)
(0,3)
1,\ )
(0,4)
(1,2)
(2,0)
(0,5)
(1,3)
(2 I)
(3,0)
(1,4)
(4,0)
(2,4)
(I,5)
(3,2)
(3,4)
(4,2)
(5,0)
(5,2)
(44) (4,5)
(5,3)
(6, I)
(54)
(6,2)
(7,0)
(3,5)
(4,3)
(5,1 )
(6,0)
(2,5)
( 3)
(4,1)
(6,3)
(7,1)
(7,2)
(8,0) (8,1)
(9,0)
Fig. l.
ultimately constant are called vertical paths, paths with t; ultimately constant diagonal paths. Figure 1 shows how the (n, k) position on P is labeled for illustrative purposes. Generally the (n, k) position of the diagram itself will
be occupied by the (n + l)th component of the (k + l)th member of the set S(k), i.e., S~k). S~k) may converge as n + k --> 00 along certain paths but not along others. The following definitions contain the key ideas. Let P be a path and
= O(t/!) in P
(2)
means for some M > 0 there is an N > 0 such that d( N. A similar interpretation is made of o.
(n, k)
5
104. Properties of Complex Sequences
Definition 4. Let vIt be a metric space, T and let 7k(s) = S(k), k ~ 0.
E
:Ysed, vIt s) where d
c
vIt c
(i) T is called regularfor d on P if sEd = d(s~kl, s) = 0(1) in P. (ii) T is called accelerative for d on P if T is regular for d on P and if d(s~k),
s)/d(sn' s) = 0(1)
in
P,
s e se.
(3)
If, in the foregoing definitions, d == vIt c- we shall omit the wordsror d and say simply that .r is regular, etc. We now discuss certain computational aspects of the foregoing definitions. Usually To = I, the identity transformation, so s(O) = S and an algorithm that is computationally feasible for filling out the array {S~k)} will start with the values s~O) = s; and assign one and only one value to each (n, k) position in the array. There seems to be no easy characterization of those algorithms that are feasible in this sense. However, several important ones have been discovered recently. Among these are formulas of the kind s~O) =
Sn,
n, k
~
0,
(4)
called a deltoid; and S~k+ I)
=
H(S~k;/), S~k~ I' S~k»,
n, k ~ 0,
S~-I)
= 0,
s~O)
= Sn, n
~ 0;
S~k+l)
=
H'(S~kL, S~k-l), S~k),
n, k
S~-lI
= 0,
s~O)
= Sn' n
~
~
0,
(5)
0, (6)
called rhomboids. There is as yet no general theory for constructing such algorithms. Those that are known have been derived using ad hoc arguments from diverse areas of analysis: Lagrangian interpolation, the theory of orthogonal polynomials, and the transformation theory of continued fractions. Much work remains to be done in this area. For transformations in vector spaces, there are several important concepts that involve the linearity of the underlying space. Definition 5. Let T E :Y(d, 11s) where d c; 11s- T is linear if, for all x, y E d and c l , C2 E '??, T(c1x + e2Y) = C 1 T(x) + C2 T(y); otherwise, Tis nonlinear. T is homogeneous if T(cx) = cT(x) for xEd, c E '?? T is translative if T(d + x) = d + T(x), where d is a constant sequence (dn == d) whenever d + x, XEd. 1.4. Properties of Complex Sequences When the metric space of the previous sections is the complex field, its sequence space possesses elegant properties. Some of these have been long
6
1. Sequences and Series
known, and others are surprisingly recent. This section contains a discussion of some of these results. Definition 1.
Let
S E ~c
and
rn+dr n = (sn+ I
s)/(sn - s) = p
-
+ 0(1).
(1)
(i) If 0 < Ipi < I, s converges linearly and we write s E ~l' (ii) If p = 1, s converges logarithmically and we write S E ~l"
Theorem 1.
Let Ip I i= 0, 1. Then , Sn+1 I1m
n-e co
Remark.
Sn -
S
S
=p
1iff
I'IHl a n +~I = p. n-e oo an
(2)
For the divergent case Ipi> 1, S can be any number.
Proof The validity of either limit implies an i=. O. Assume, without loss of generality, an =f 0 for any n; otherwise delete the finite number of ans that are zero and relabel the members of a and s. =: We have [a n+I
+ (s,
- s)]/(sn - s) ~ p
(3)
or
an+1
~
an = (p - 1)(sn_1 - s).
(p - 1)(sn - s),
(4)
Dividing the former by the latter shows (4')
Note that for this part of the theorem p can be zero. =: We do only the convergent case 0 < Ipl < 1. The other is similar. Since I an converges,
s, -
S
=
00
I
k=n+ 1
ai,
(5)
We can write EE~N'
(6)
Let gn = sup j'2:n
lejl.
(7)
1.4. Properties of Complex Sequences
Then g E
~N'
7
Taking products in (6) gives
= aopn
an
n
n-I
j=O
(1
+ G),
(8)
empty products interpreted as 1. Define
(9)
Thus
.
11m F" ~
n-e co
I Lr pkl k=O
1
>
Ipl,+1
I I
1_
P
-lpr+ 1
- II - pi
(1 + I) . 11 - pi I - Ipl
(10)
For r sufficiently large, the right-hand side is >0. Thus lim F; > 0 and IIFn is bounded. Now S
n+ 1 Sn -
L
-s
'=.
S
au
k=n+1
ak+
/00L
1
k=n+1
a,
=
L00
k=n+1
a k P(1
+ Gk) /00 L
=. p + Un'
k=n+1
ak
(11) (12)
(The foregoing operations are valid since it will turn out that s, of. s.) Thus jUnl~·
00
L
k=n+1
~.Cgn+1
lakGkl/lan+IJFn+1
00
L
k=O
Iplk(1
+ gn+d
Cg n+ 1
(13)
l-p(l+gn+I)'
which actually shows a bit more, namely, Sn+1 Sn -
S S
=
P
+ o(suplak+1 k>n
pak
_
II).•
(14)
8
I. Sequences and Series
Corollary. Proof
Cfl a :=; Cfl/.
This is true since (15)
limlanl 1 / n slim lan+1/anl . •
Another useful result has to do with the order of growth of partial products. Theorem 2.
If
», = for some t
E Cfl N, Gj
n-l
Il (1 + G),
j=O
n ~ 1,
Vo = 1
i= -1,j ~ 0, then there is an t*
E CflN
(16)
such that (17)
Proof
We have
n-1
Un
= ell (1 + Gj)'
(18)
j=no
(19) but the quantity in square brackets is the Cesaro means of a null sequence, and hence the nth term of a null sequence, say, <5 n • So (20) and this may be extended to all n ~ 0. (s", because of the multiple valuedness of log, is not unique.) • 1.5. Further Properties of Complex Sequences Some unusual convergence properties have recently been demonstrated for complex sequences. These properties are a help in determining whether important sequence transformations are regular or accelerative. Sources for this material are Tucker (1967, 1969). In what follows let s, a E Cfls and be related in the usual way. For all an i= and n ~ 0, define Pn by
°
(1)
and if s is convergent, r, by Tn
Otherwise Pn' r, are undefined.
= (s - sn)/an·
(2)
1.5. Further Properties of Complex Sequences
9
Since we shall in general be concerned only with members of a sequence with large index, the notation "xnR.Yn" (see Notation) will be employed constantly.
Lemma 1. Let
S E C(/C Cn
and an #. O. Let c E
=
C
+ (sn
C(/,
- s),
n
and define
~
(3)
0;
then I - Pn) 1 + C( - an+ 1
C n+ 1 I - Pn ( + -c, - - =. - S-
an
an+ 1
an+ 1
Sn)'
(4)
Proof
Pn) +Cn- - C-n+-1 1-1 +C ( an+ 1 an an+ 1 =.1
+
=. 1 +
c(_I__~) + an+ 1
S -
C
+ Sn
Sn
=. S
an
Sn + 1 _
S -
an+ 1
an
an
-
-
+ Sn+l
S _
C
Sn _
S -
an + 1
an+ 1
an
-
S
Sn
•
(5)
Theorem 1. Let (6)
Then S diverges. Proof Assume s converges. Since (1 - Pn)/an+ 1 is bounded, there is an 8> 0 such that 18(1 - Pn)/a n+ 11 <.l Let C be any complex number satisfying Ic I = 8, so that
Set
Cn
=
C
+ (s,
Re[1
- Re[c(1 - Pn)/a n+ 1]
<·l
(7)
- s). From the previous lemma
+ c(1 -
Pn) an+ 1
+
C
n
an
_
Cn+l] =. Re[1 - Pn (s - Sn)], an+ 1 an + 1
(8)
so (9)
10
1. Sequences and Series
Using (7) and (9) shows
~ + Re c" <. Re C,,+ 1 a"
2
a"+ 1
P,,)] _ ~ <. Re C,,+
Re[C(l a" +1
_
from which it follows that Re c.fa; a,,¢. {z l arg c
-+ 00
+ 3n/4 ~
4
1
a"+1
and so Re c"/a,, >. O. Since C"
argz ~ argc
+ 5n/4}.
(10) -+
c,
(11)
Choosing arg c to be successively 0, n/2, n; 3n/2 shows that a cannot be a complex sequence, a contradiction. • [This beautiful proof is due to Tucker (1967).] We state without proof a similar result for infinite products. Theorem 2.
Let l/a,,+ 1
Then
-
Uo;
= 0(1).
(12)
Il:'=o (1 + a,,) diverges.
Lemma 2.
Let P" be defined ultimately and
Ip,,1 ~.p <
t·
(13)
Then
0<(1 - 2p)/(1 - p)
Proof
~.lr,,/p,,1 ~.
1/(1 - p).
(14)
Note that r" is, ultimately, defined and that
(15)
r" =. P" + P"P,,+ 1 + P"P,,+ 1P"+2 + "',
+ 1 =. r,./p" and the above series converges. We have Ir,,1 ~·lp,,1 + Ip"P"+11 + ... ~·lp"I/(1 - p) ~. p/(1 - p) Thus I»J», I s. 1/(1 - p) and 1iJ», I =.11 + r"+11;::::.ll -lr"+lll =.1 -lr"+11
since r,,+ 1
< 1.
;::::.1 - p/(1 - p) = (1 - 2p)/(1 - p) > 0 . .
(16)
(17)
Theorem 3. Let s, s* be two sequences such that a:/a" = 0(1) and Ip,,1 ~.p < t, Ip:1 ~.p* < 1 for some numbers p, p*. Then s* converges more rapidly than s.
Proof An implication of the hypothesis is a" #-. 0, a: #-. O. The previous lemma shows 0<(1 - 2p)/(1 - p)
~·lr,,/p,,1
(18)
1.5. Further Properties of Complex Sequences
and
II
(19)
One concludes that
I Sn S: -
S* S
I=.1 aa:+n+ II ':/P: I 'n/Pn 1 1
~.la:+ll[(1 an + 1
-
p*)(l - 2p)(1 - p)r 1
=
0(1).
•
(20)
Tucker gives an example (1967, p. 358) to show that 1- cannot be replaced by a larger number.
Lemma 3.
Let b, S, s* E rrls with (21)
Then s* converges more rapidly than s to the same limit if and only if s, = 0(1). Proof Either hypothesis implies the convergence of sand S either case, therefore, bn + 1 ~
bn+tI(s - sn)
+ (s -
and from this the lemma follows.
Theorem 4.
(22)
S -
s:)/(s - sn) =. 1,
-
s, #. O. In (23)
•
Let t, s E rrlc and (24)
and suppose t converges more rapidly than s to the same limit. Then u converges more rapidly than s to the same limit if and only if {In ~ (J.n'
Proof From the previous theorem, a n+ l(J.n+ 1 ~ S - Sn = 0(1). Also u converges more rapidly than s to the same limit if and only if a n+ l{Jn+ 1 ~ S - Sn' Since S - Sn #.0, we have an+l(J.n+l #.0, an+1{Jn+l #.0, and so an' (J.n' {In #. O. By transitivity of ~ we conclude a n+ 1 (J.n+ 1 ~ a n+ l{Jn+ 1 or (J.n ~ {In' This step is reversible, so the theorem follows. • Theorem 5.
Let
S
be convergent and (25)
Then the three conditions below are all equivalent: (i) s* converges more rapidly than s to the same limit; (ii) (J.n+ 1 ~ 'n/Pn; (iii) (J.n ~ 1 + Tn'
I2
1. Sequences and Series
Proof
From Lemma 3, s* converges more rapidly than s iff a n + 1(Xn+ 1
~
(Xn+1
~
1.6. Totally Monotone and Totally Oscillatory Sequences Definition.
s is totally monotone (written s E
«i
(-I)kNsn~O,
~TM)
:«
if (I)
s is totally oscillatory (written s E 9fT O ) if {( -I)nsn} E 9fT M • Here n
~
0,
k
~
1
n, k ~ O.
If s E 9fT M , one has (2)
so s converges since it is monotone decreasing and bounded. On the other hand, if s E 9fTQ, S is alternating and so converges to O. Examples. s~1)
= I/(n +
The sequences S(k)
E ~TM'
where
2),
(X
> 0,
(3)
since
S~k) = ft dl/lit) n
(4)
for I/Ik bounded and nondecreasing
1/12 = (I - tin t),
(5)
(see Theorem 3). ~TM is an important regularity space for certain nonlinear transformations in ff(~s, ~s).
Theorem 1. Let s E ~TM' Then the sequences whose nth elements are given below are also E ,'3iP T M • (Empty products are interpreted as 1.)
(so < 1);
13
1.6. Totally Monotone and Totally Oscillatory Sequences
nj:6 (1 -
(ii)
s)
(iii)
A.(-I)k+l~ksn
(iv)
A.J:~lSj
(so:s: 1); (0 < A.:s: 1, k
(0
:s: A.:s:
> 0, a, k fixed);
1).
Proof The proofs are straightforward. We prove here only (i); the reader is referred to Wynn's paper (1972) for the others. Write
1/(1 - sn) = tn·
(6)
Multiplying both sides by 1 - s; and using the difference formula 1.6(40) gives
(-~)ktn = (1 -
°
sn)-I
±(~)(-~)k-jtn+i-~)jSn.
(7)
j= 1 )
Now s, - Sn+ 1 ?: 0, so :S: s, < 1 for all n, and thus Eq. (7) provides an immediate induction argument on k. •
Theorem 2. Proof
Let
S,
Obvious.
t
Bf T M • Then {sntn}, {asn + bt n} E BfT M , a, b > 0.
E
•
Theorem 3. S is totally monotone if and only if there is a function t/J(t) bounded and nondecreasing on [0, 1] that satisfies s;
=
f
n t dt/J(t),
(8)
n ?: 0.
Proof
Write
<=:
( -1)k~ksn = For all k and
=:
°:S: n :S: k,
f
(1 - t)ktn dljJ(t) ?: 0.
(-It-n~k-nsn
or
?: 0,
k-"(k L - n) (-I)'sn+r = r-.
r=O
where r _
sn -
E Bf~.
r
(10)
°
(11)
:S: n :S: k,
It is easily seen that this system of equations has the solution
(k - n) r; -_ L m(m L m=n m - n m=n k(k k
(9)
k
1)··· (m - n + 1) L m, 1)··· (k - n + 1)
°
:S: n :S: k,
(12)
14
I. Sequences and Series
where (13) This can be written
°::; n s k,
(14)
where cDk n(t) .
t(t - llk)(t - 21k) ... [t - (n - 1)lk) 1(1 - llk)(1 - 2Ik)··· [1 - (n - l)jk]
=
-,---:-:---,---;c:-,---:-:---=-:-:-,.------=cc----,-----~_=_
= t" + O(k- 1)
(15)
uniformly in t, and l/Jk(t) is the step function defined by
°
t ::; 0,
< t ::; 11k, 11k < t s 21k, Lo + L 1 + Lo + L 1 +
+ Lk + Lk ,
(k - 1)lk < t < 1, 1 ::; t.
1,
Because of (12) with n = 0, So
(16)
= l/Jk(1) :2: l/Jk(t) :2: l/Jk(O) = 0,
°s t ::; 1.
(17)
But any sequence of bounded nondecreasing functions on [0, 1] contains a subsequence converging to a bounded nondecreasing function [see Wall (1948, p. 246)]. It is easy to justify taking the limit over this subsequence inside the integral sign (Wall, 1948, p. 245), so for some bounded nondecreasing l/J(t),
s, = For
SE
fr
dl/J(t),
n :2: 0.
•
(18)
fJil s , the determinants sn
H~k)(s)
= Sn+ 1 Sn+k-l
Sn+ 1 Sn+ 2 Sn+k
are called Hankel determinants.
+k- 1 Sn+k
Sn
Sn+ 2k - 2
n, k :2: 0,
(19)
1.7. Birkhoff-Poincare Logarithmic Scales
Theorem 4. If s E 3l TM ,
H~k)(S) 2:
O. If s
E ~TO, (_l)nkH~k)(s)
15
2: O.
Let
Proof Q~k) =
L sn+i+A¢j =
k-1
I1 0
i.j~O
tn(~o
+ ~lt + ... + ¢k_ltk-1)Z dt/J(t)
2: O.
(20)
But this means by a known result on quadratic forms (Bellman, 1970, p. 75) that the determinant of the coefficients of Qin) must be nonnegative, which gives the first part of the theorem. The second is similar. •
Theorem 5 (Brezinski). Let f(x) = Lk~O CkXk be a power series with nonnegative coefficients and radius of convergence p > O. Let s E 3l TM and So < p. Then {f(sn)} E 3l T M • Proof
Obvious, since the sequence S(k) S~k) =
Co
E ~TM
when
+ C 1Sn + '" + CkS~
by Theorem 2, and any limit of totally monotone sequences is totally monotone. [This also provides a proof of Theorem l(i).] •
Theorem 6. If S E ~TM' then H~k)(~2rs)
If S E
~TO'
2: 0
and
then
( - 1)kn H~k)(LlZrs) 2: 0
Proof
Obvious.
and
•
1.7. Birkhoff-Poincare Logarithmic Scales Let p E J+, Ill' Ilz, ... , II p, ebe complex constants, Ilo an integral multiple (positive or negative) of lip. Define Q(w) = lloW In w + 1l1W + IlZW(p-1)/P + ... + IlpW1/ p, WE 3l+. (1) Consider the sequence of functions t/J;jw) = eQ(W)w6 -
j/P(ln
wy,
i,j = 0, 1,2, ... ,
co
E ~+,
(2)
and let F = {t/Ji,j}' It is easily verified that F is strictly (nonreflexively) well ordered under the operation written"
16
1. Sequences and Series
bounded, i = 0, 1, 2, ... , p, then one may define a unique asymptotic scale on F, say, {
I/Ii", k"'
(3)
This scale is called the Birkhoff-Poincare logarithm scale (B-P log scale); p is called the index of the scale. If p = 0, it is called simply the BirkhoffPoincare scale. The ~pecial case J1; = () = 0, p = 1 is called the Poincare scale. Any function satisfying a fairly general difference equation or differential equation is known to possess an asymptotic expansion in a B-P log scale, or, more precisely, the function can be written as a linear combination of such expansions, once it is decided how to interpret sums of asymptotic expansions. [In fact, this can easily be done; see Wimp (1974b).] For difference equations, this is called the Birkhoff-Trjitzinsky theory (1930, 1932),and for differential equations, the theory of subnormal forms. [See Wasow (1965) and the references given there.]
Theorem 1 (Birkhoff-Trjitzinsky). Ao(w)y(w)
+ Al(w)y(w +
1)
Consider the difference equation
+ '" +
Am(w)y(w
+ m) =
0,
(4)
where Ai is defined for w E ~o and Am(w) i:- 0. Let Ai have an asymptotic expansion with respect to some B-P scale F with Q = J1j = () = in ~+. Then there is a B-P log scale G and a basis of solutions Yl' Y2'" ., Ym of the equation such that Yj has an asymptotic expansion gi with respect to G in ~ +. The general form of these solutions is
°
+ aOlw- l/ p + ...) + (a l O + allw- l/ P + .. ·)lnw + '" + (a mo + amlw- l/ P + .. -)(In co)"], (5)
y(w) = eQ(W)w 8[(aoo
Proof The proof is the subject of two papers. The first (Birkhoff, 1930) treats the formal (constructive) theory of the question; the second (Birkhoff and Trjitzinsky, 1932) treats the analytic theory. •
While the theorem is simple to state, in the construction ofthe asymptotic expansions there are many complexities. For instance, p for G and F need not be the same. Further, once certain expansions gi are obtained others may be found from these by formal manipulations, thus vastly simplifying the work involved. For example, the difference equation () yw
+
(w (w (w
+
+ b + c + 1) + A] yw+ ( 1) (w + b)(w + c) + 1)(w + 2) + b)(w + c) y(w + 2) = 0, b, c > 1)[(2w
°
(6)
17
1.7. Birkhoff-Poincare Logarithmic Scales
has solutions
h1(w) =
r(b
r(w hz(w) = f(w
+ w)r(C + w) r(w + 1) 'P(b + w, b + + b) + 1)
1 - c; A), (7)
1 - c; A)
(see Wimp, 1974a). There is a formal basis of solutions
gj(w) = exp[( -IY+ lAl/2Wl/2]W9
LE 00
k=O
k
(
_ly k w - k / 2,
() =
-!(b
+ c)
-
i (8)
and
h1(w) '" fiA(e-b)/2-1/4e~/2g1(W)
in
(9)
~+.
Here the A i are rational; p = 1 for F; for G, P = 2. Let us see how the construction proceeds for the important case m = 1. For the formal computations, assume all series are convergent for w > R, say. Then Eq. (4) can be written yew + 1) = w/i/P(aoe/i/P + a1w- 1/ P + .. -)y(w), f.1 E J, ao # O. (10) Let
yew)
Then
z(w
+
1) = b(w)z(w),
Finally, let u(w)
u(w
+
= w/iw/Pa~z(w).
=
b(w) = 1
In z(w). Then u satisfies
(11)
+ b1w- 1 / P + .... w- 2 / P
1) - u(w) = In b(w) = b1w- 1/P + (2b 2 - bi) 2
=
C1W- 1/ P
In this equation write
+ C2W-2/p + ....
+ d2_pWl-2/P + ... + d_1W 1 / P + d1w- 1/ p + d 2w- 2/P + "',
(12)
+ ... (13)
u(w) = dl_pWl-l/P
+ de In w
and it is immediately found that d 1 with
P'
d2 -
P"
' "
(14)
do are uniquely determined,
(15)
18
I. Sequences and Series
by comparison of terms w- I / P, w-z/p, ... ,w- I • On comparison of terms w- k / p , (k > p), one gets equations of the form
+ I,
k :::: p
(16)
in which IY.k is a known polynomial in d ,_ p, dz-p, ... ,dk-I-P' Thus all the d, are determined in succession and uniquely. Then, writing (17) and exponentiating the series for u(w) gives the desired formal series. This construction, of course, shows that a unique formal asymptotic series always exists, and also that, for the first-order case, p for F = P for G and the index of G is zero. The next result shows how the partial sums of a series grow when the general term of the series has an asymptotic expansion of the form eQs.
Theorem 2.
Let (18)
with
S E
Cfic . Let an'" eQ(W)wlJp(w),
where
~
co = n
+(
J+
(19)
+ IY.IW- I/ P + IY.zW- z/ P + ....
(20)
In
is arbitrary and complex and where pew) = IY. 0
Then
s, where ()*,
(;(6 are as
Case I.
S '"
eQ (W)w 8' p*((I))
(21)
follows:
Q"¥= O. Denote the first nonzero fl j in the seq uence
by u., Then
+ flo, - () + (r -
()* _ {()
r=O I)/p,
S r
Case II.
Q == O. ()*
=
8
+
I,
IY.6 =':1.0/(8
p;
r=O = I;
T
s
r
+
I).
2
s s
(22)
(23)
p.
(24)
1.7. Birkhoff-Poincare Logarithmic Scales
19
Proof. A straightforward application of Theorem 1; see Wimp (1974b) (whose r = 0 values are incorrect). •
Also of interest is a related result for the partial sums of divergent series.
Theorem 3. Let s
~
Then
rcc and let an' p, to be as in (19), for some constant c. r:
in where lJ*,
(25)
are as in (22) and (23) except in the following cases:
O(~
llo;6 O. Then O(~ =
Case I.
lJ* = lJ.
0(0'
Q == 0 and p(w) contains a term w -
Case I I.
S,,-
Corollary.
C
+ din w
I.
Then for some c, d,
-- W6+ lp*(W)
(26)
Let 0(0;6
(27)
O.
Then for some constants /3j' 'Ij, h j ,
An+ Ino
(
/31
/32
(
n+n
)
- - IXo+-+-+"', 2 n
(A-I)
Sn -
S --
and
l
6
1
-n + (8 + 1)
OCo +
'II
n
'12
2 o In n + lJdn + lJ 2/n
< I (28)
)
A = 1, Rc lJ < - 1,
+ ... ,
2
Sn -- iX
1..1.1
+ ... ,
A = I, lJ = -I.
(29)
As examples of the use of these formulas, consider the computation of
e' and (s) from their defining series. Let Sn
(u) = s~ - r~,
For
L k'
k=O n
=
L
k=1
•
1 k'"
(30)
Re a > I.
Sn'
p
= 1,
and for p
s~
=
" xk
= t,
IIp = -I,
III
= In ex,
= III = 0,
() =
-u,
0=
0(0
= I/J"~, (31)
s~,
JLo
0(0
= l,
IXj = 0,
.i >
0,
(32)
20
I. Sequences and Series
and (for both may be taken to be O. We have (33) and the series on the right can be rewritten (34) Higher coefficients are easily determined by formal series manipulations [see Smith (1978)]:
Pl =
PI = 1,
P3 = Xl -
X,
2x,
(35)
For the Riemann zeta function, ((a) -
~ L.
k= I
a1 '" n I - " ((Xo + -(Xl + 2(Xl + ... ) , n
k
n
(36)
where (Xo
(37)
= 1/(a - 1),
and (Xlk+ I
(Xlk
= 0,
k '2. 1,
(ahk-l = (2k)!
k '2. 1,
Blk>
(38)
the higher coefficients being obtained by the formula in Wimp (1974b, 2.42) or from the integral representation for ((a). Equation (36) is known to hold for all complex a =I- 1 [see Olver (1974, p. 292)]. In what follows let (39) be a formal asymptotic series. In future sections we shall need to know the effect of certain difference operators on this series.
Lemma 1
L~Ju(n)v(n) = Proof
±
r=O
(j)l.l.ru(n)!.l.j-rv(n r
See Milne-Thomson (1960, p. 35).
•
+ r).
(40)
1.7. Birkhoff-Poincare Logarithmic Scales
21
Lemma 2
An(A- 1)PnO( IJ( O+ Pdn + "'), kid, APy(n)= { (-OM-l)Pno-P(lJ(o+Ydn+ ... ), A=l,
0#0,1,2, ... ,p-1, (41)
for some Pi' Yi' Proof
Left to the reader.
•
Theorem 4
L dijA)AP+ry(n) = i
r=O
An(A - 1)P( -l)i( - O)ino- i(lJ(o
+ lJ('dn + ."), (42)
A # I. 0 # 0, 1,2,,,. ,j - 1,
where IJ('J'
IJ(~, .•••
depend on j and p and di.rCA)
Proof
= C)A-i(1 - A)i- r.
The first part of the previous lemma gives A-nAPy(n) = (A - l) PnO(lJ(o + Pdn +
(43)
" -).
(44)
Then letting v(n) = A-n and u(n) = APy(n)and using the two previous lemmas gives the theorem. • The paper by Wimp (1974b) includes a number of applications of the previous results, particularly to the problem of finding asymptotic formulas for the remainder terms in expansions in orthogonal polynomials. Brezinski has shown that something like Theorems 2 and 3 is true for series of arbitrary real terms provided the terms are ultimately positive and that their differences are ultimately of one sign. First, note that Un ~ V n is equivalent to the statement that u; #.0, Vn #.0, and (45) Lemma 3.
Let an >.0, s diverge to
+ 00, and b; =
n
L akbk = o(sn)'
k=O
0(1). Then (46)
Proof n
L
k=O
f.lnkbk = 0(1),
(47)
22
l. Sequences and Series
where (48) by the Toeplitz limit theorem, Theorem 2.1(3). Note b may be a complex sequence. •
Theorem 5.
Case I.
Let
S E ~s,
an >.0 and h; = an/l1an with I1h n = 0(1).
Sa; <. O. Then s converges and (49)
Case I I.
l1a n >. O. Then s diverges and (50)
Proof
Case I. h; - ho
n-I
= k~O I1hk =
0
(n-I ) k~O = o(n), 1
(51)
by the lemma (with s = {n}). This means
1)r 1 = 0(1),
[n(an+ dan -
(52)
or, since an+ dan <. 1, the sequence {n(an+ dan - I)} is definitely divergent to - 00. By Raabe's test s is convergent. Thus 00
00
L ak = L hkl1ak =
k=n
k=n
00
-hna n -
L
k=n+1
akl1hk-t·
(53)
Now
f akl1hkIk=n+1
l !
S sup Il1h kl(s - Sn) s.sup/l1h kl(s - Sn-I)'
(54)
L
(55)
k~n
k~n
Thus
00
k=n+1 where
~ E ~N'
so (53) may be written S -
or, since
S -
akl1hk_1 =(s - Sn-I)¢n,
Sn =1=.0,
Sn-I
= -hnan - (s -
-hnan/(s - Sn-I) and letting n ->
CJJ
gives the result.
=.
Sn-I)~n,
1 + ~n
(56) (57)
23
1.7. Blrkhoff-Poincare Logarithmic Scales
Case II.
Sincean+1 >.an'Sn--' +00. Also, n
L hk~ak =
Sn =
hnan+ I
k=O
-
hoao -
n
L ak~hk-I'
(58)
- ho a o .
(59)
~n)
(60)
k=1
or ao
+
n
L ak(l
k=1
+ ~hk - I) = h; an+ I
But, by the lemma, n
L ak(l + ~hk-I) =
k=1
where; E
fYtN .
sn(l
+
Since s; =1-.0, hnan+ Js,
=. 1 +
and letting n --. 00 gives the result.
ao(l + ho)/sn + ~n,
(61)
•
Sometimes the variable appearing in a Poincare series is n + f3 rather than n. This is immaterial, however, as the following result indicates. Theorem 6.
Let
L crCn + a;o-" 00
Un '"
Then, for any f3
E
,=0
e, «e «.
(62)
'fl,
L c~(n + f3)O-', 00
Un '"
(63)
,=0
Proof
k k ( a _ f3)O-' L c,(n + at-' =. L c,(n + f3)O-' 1 + --f3 ,=0 ,=0 n+ k
=. ,=0 LcrC n +
0 00 (f3 - ay-'(r - (})s-, f3) s=, L (s-r )'( . n + f3y
=. i>,(n + f3t ,=0
±
se r
(f3 - aY-,'(r - (}):-, (s - r).(n + f3)
k
=. L c:(n + f3)o-S + O(nO- k- I ), s=o
and from this the theorem follows immediately.
•
+
O(nO- k- l )
(64)
Sequences and Series
1.1. Order Symbols and Asymptotic Scales, Continuous Variables Let 1/ and 1/' (see Notation) be equipped with pseudometrics d and d', respectively; let n be a cone in 1/ and ¢, IjJ E /T(1/, 1/').
¢=
O(IjJ)
n
in
(1)
means for some M > 0 there is an R(M) > 0 such that d'(¢, O)jd'(IjJ, 0) < M,
Further,
¢ = o(ljJ)
d(x,O) > R.
XEn,
III
n
(2)
(3)
means for any e > 0 there is an R(e) > 0 such that d'(¢, O)jd'(IjJ, 0) < e,
d(x,O) > R.
XEn,
(4)
If ¢, IjJ depend parametrically on a E .sf and (2) holds for all a E .sf, then we shall write "¢ = O(IjJ) in n uniformly in .sf," and similarly for (3). F or the foregoing definitions to apply, the implicit assumption is made that denominators are never zero; for example, there must be some R such that d'(l/J, O) oF 0, X E n, d(x, 0) > R. Thus anytime an order symbol is used, an implicit statement is being made about the zeros of d'(IjJ, 0). The concept of asymptotic equivalence is often useful. This is written in and means both ¢ - IjJ
n
= o(ljJ) and IjJ - ¢ = o(¢) in n.
(5)
2
I. Sequences and Series
Now let cj» E :Y sC'f/, y'), where Y and 'Or' are linear spaces with pseudometrics d and d', cj» is called an asymptotic scale in Q if, for every k ;;::: 0, in
(6)
Q,
and if this holds uniformly in k or uniformly in some parameter space, we speak of a uniform asymptotic scale (properly qualified). See Erdelyi (1956) for many examples. Letf E :Y(Y, y'), A E res and cj» be an asymptotic scale in Q. The statement in
(7)
Q
is to be read "f has the right-hand side as an asymptotic expansion in Q with respect to the scale cj»" and means, for every k ;;::: 0,
f -
k
L A r4>r = O(4)k) r;O
in
Q.
(8)
Often cj» is understood from context, so "with respect to the scale cj»" may be deleted from the definition. Note that 0, 0, and -- are transitive and ~ is symmetric. Clearly no asymptotic scale can contain the zero vector or two identical vectors. If d' is a metric induced by a norm I ·11, the asymptotic expansion (7) is unique (but not otherwise). This is a simple consequence of the fact that 4> = 0(1]), l/J = 0(1]) then imply 4> + l/J = 0(1]). Thus assume another expansion (7) with coefficients A' holds. Setting k = in (8) and its analog and subtracting the two gives
°
(A o - A~)4>o = 0(4)0), or lAo -
A~
I<
s for all e, so that A o =
A~,
(9)
similarly, A j = Aj,j > 0.
1.2. Integer Variables In discussions of sequences, the relevant variable x in 4> or l/J takes values in JO. We write 4>n or l/Jn for 4> or l/J, respectively, or, when there is a possibility of confusion with the index of an asymptotic scale, 4>(n) or l/J(n). 1.1(1) is then written
and means that for some M > 0, there is an N > d'(4)n, O)jd'Cl/Jn' 0) < M
for
°
such that
n> N.
(1)
1.3. Sequences and Transformations in Abstract Spaces
3
A similar modification is made of 1.1(3). An additional complexity occurs when ¢ and t/J depend on a p-tuple with elements in say, n = (ml' m2' ... , m p ) . It is usually important to know exactly how the elements mj become infinite, and it is hardly ever sufficient to say, for instance, that ml + m 2 + ... + mp > N. In fact, the concept of a path in n-space becomes important (see Section 1.3).
r:
1.3. Sequences and Transformations in Abstract Spaces
In this book we shall be concerned with two kinds of sequence transformations. The first is the transformation ofagiven sequences E d sinto a sequence S E.r4's with, generally, a formula given to compute sn in terms of elements of s. (In some situations there is no explicit formula.) The other case is where the given sequence s is mapped into a countable set of sequences S(k), k ~ 0, with a formula given (called a lozenge algorithm) for filling out the array {S~k)}, n, k ~ 0. The whole point is to compare the convergence of the transformed sequence(s) with that of the original sequence. The most useful concepts are formulated in the definitions that follow. Definition 1. (i) (ii) (iii)
Let s, tEAte, a metric space.
t converges as s means d(sn, s) = O(d(t n, t)) and d(t n, t) = O(d(sn's)), t converges more rapidly than s means d(tn' t) = o(d(sn, s)). The convergence of sis pth order if, for some p E
r,
d(sn+ I' s)
= O(d(sn, s)") (I)
and
d(sn' s)"
=
O(d(sn+ I' s)).
It is easy to show that p, if it exists, is unique.
Definition 2.
Let T E !!T(d, JIt s) where d
c
JIt e and T(s)
=
s.
(i) Tis regular for d ifs Ed=> S E Ate and s = s. (ii) T is accelerative for d (or accelerates d) if T is regular for d and S converges more rapidly than s, s e ss, Definition 3. SEd.
Let T E 5"(d, Jlts)whered
c
Jlt s . T sumsdifT(s) E JIt c-
P = {(im,Jm)lim, JmEjO} IS called a path if io
4
I. Sequences and Series n/k (0,0) (0,1)
(O,2)
(1,0)
(0,3)
1,\ )
(0,4)
(1,2)
(2,0)
(0,5)
(1,3)
(2 I)
(3,0)
(1,4)
(4,0)
(2,4)
(I,5)
(3,2)
(3,4)
(4,2)
(5,0)
(5,2)
(44) (4,5)
(5,3)
(6, I)
(54)
(6,2)
(7,0)
(3,5)
(4,3)
(5,1 )
(6,0)
(2,5)
( 3)
(4,1)
(6,3)
(7,1)
(7,2)
(8,0) (8,1)
(9,0)
Fig. l.
ultimately constant are called vertical paths, paths with t; ultimately constant diagonal paths. Figure 1 shows how the (n, k) position on P is labeled for illustrative purposes. Generally the (n, k) position of the diagram itself will
be occupied by the (n + l)th component of the (k + l)th member of the set S(k), i.e., S~k). S~k) may converge as n + k --> 00 along certain paths but not along others. The following definitions contain the key ideas. Let P be a path and
= O(t/!) in P
(2)
means for some M > 0 there is an N > 0 such that d(
(n, k)
5
104. Properties of Complex Sequences
Definition 4. Let vIt be a metric space, T and let 7k(s) = S(k), k ~ 0.
E
:Ysed, vIt s) where d
c
vIt c
(i) T is called regularfor d on P if sEd = d(s~kl, s) = 0(1) in P. (ii) T is called accelerative for d on P if T is regular for d on P and if d(s~k),
s)/d(sn' s) = 0(1)
in
P,
s e se.
(3)
If, in the foregoing definitions, d == vIt c- we shall omit the wordsror d and say simply that .r is regular, etc. We now discuss certain computational aspects of the foregoing definitions. Usually To = I, the identity transformation, so s(O) = S and an algorithm that is computationally feasible for filling out the array {S~k)} will start with the values s~O) = s; and assign one and only one value to each (n, k) position in the array. There seems to be no easy characterization of those algorithms that are feasible in this sense. However, several important ones have been discovered recently. Among these are formulas of the kind s~O) =
Sn,
n, k
~
0,
(4)
called a deltoid; and S~k+ I)
=
H(S~k;/), S~k~ I' S~k»,
n, k ~ 0,
S~-I)
= 0,
s~O)
= Sn, n
~ 0;
S~k+l)
=
H'(S~kL, S~k-l), S~k),
n, k
S~-lI
= 0,
s~O)
= Sn' n
~
~
0,
(5)
0, (6)
called rhomboids. There is as yet no general theory for constructing such algorithms. Those that are known have been derived using ad hoc arguments from diverse areas of analysis: Lagrangian interpolation, the theory of orthogonal polynomials, and the transformation theory of continued fractions. Much work remains to be done in this area. For transformations in vector spaces, there are several important concepts that involve the linearity of the underlying space. Definition 5. Let T E :Y(d, 11s) where d c; 11s- T is linear if, for all x, y E d and c l , C2 E '??, T(c1x + e2Y) = C 1 T(x) + C2 T(y); otherwise, Tis nonlinear. T is homogeneous if T(cx) = cT(x) for xEd, c E '?? T is translative if T(d + x) = d + T(x), where d is a constant sequence (dn == d) whenever d + x, XEd. 1.4. Properties of Complex Sequences When the metric space of the previous sections is the complex field, its sequence space possesses elegant properties. Some of these have been long
6
1. Sequences and Series
known, and others are surprisingly recent. This section contains a discussion of some of these results. Definition 1.
Let
S E ~c
and
rn+dr n = (sn+ I
s)/(sn - s) = p
-
+ 0(1).
(1)
(i) If 0 < Ipi < I, s converges linearly and we write s E ~l' (ii) If p = 1, s converges logarithmically and we write S E ~l"
Theorem 1.
Let Ip I i= 0, 1. Then , Sn+1 I1m
n-e co
Remark.
Sn -
S
S
=p
1iff
I'IHl a n +~I = p. n-e oo an
(2)
For the divergent case Ipi> 1, S can be any number.
Proof The validity of either limit implies an i=. O. Assume, without loss of generality, an =f 0 for any n; otherwise delete the finite number of ans that are zero and relabel the members of a and s. =: We have [a n+I
+ (s,
- s)]/(sn - s) ~ p
(3)
or
an+1
~
an = (p - 1)(sn_1 - s).
(p - 1)(sn - s),
(4)
Dividing the former by the latter shows (4')
Note that for this part of the theorem p can be zero. =: We do only the convergent case 0 < Ipl < 1. The other is similar. Since I an converges,
s, -
S
=
00
I
k=n+ 1
ai,
(5)
We can write EE~N'
(6)
Let gn = sup j'2:n
lejl.
(7)
1.4. Properties of Complex Sequences
Then g E
~N'
7
Taking products in (6) gives
= aopn
an
n
n-I
j=O
(1
+ G),
(8)
empty products interpreted as 1. Define
(9)
Thus
.
11m F" ~
n-e co
I Lr pkl k=O
1
>
Ipl,+1
I I
1_
P
-lpr+ 1
- II - pi
(1 + I) . 11 - pi I - Ipl
(10)
For r sufficiently large, the right-hand side is >0. Thus lim F; > 0 and IIFn is bounded. Now S
n+ 1 Sn -
L
-s
'=.
S
au
k=n+1
ak+
/00L
1
k=n+1
a,
=
L00
k=n+1
a k P(1
+ Gk) /00 L
=. p + Un'
k=n+1
ak
(11) (12)
(The foregoing operations are valid since it will turn out that s, of. s.) Thus jUnl~·
00
L
k=n+1
~.Cgn+1
lakGkl/lan+IJFn+1
00
L
k=O
Iplk(1
+ gn+d
Cg n+ 1
(13)
l-p(l+gn+I)'
which actually shows a bit more, namely, Sn+1 Sn -
S S
=
P
+ o(suplak+1 k>n
pak
_
II).•
(14)
8
I. Sequences and Series
Corollary. Proof
Cfl a :=; Cfl/.
This is true since (15)
limlanl 1 / n slim lan+1/anl . •
Another useful result has to do with the order of growth of partial products. Theorem 2.
If
», = for some t
E Cfl N, Gj
n-l
Il (1 + G),
j=O
n ~ 1,
Vo = 1
i= -1,j ~ 0, then there is an t*
E CflN
(16)
such that (17)
Proof
We have
n-1
Un
= ell (1 + Gj)'
(18)
j=no
(19) but the quantity in square brackets is the Cesaro means of a null sequence, and hence the nth term of a null sequence, say, <5 n • So (20) and this may be extended to all n ~ 0. (s", because of the multiple valuedness of log, is not unique.) • 1.5. Further Properties of Complex Sequences Some unusual convergence properties have recently been demonstrated for complex sequences. These properties are a help in determining whether important sequence transformations are regular or accelerative. Sources for this material are Tucker (1967, 1969). In what follows let s, a E Cfls and be related in the usual way. For all an i= and n ~ 0, define Pn by
°
(1)
and if s is convergent, r, by Tn
Otherwise Pn' r, are undefined.
= (s - sn)/an·
(2)
1.5. Further Properties of Complex Sequences
9
Since we shall in general be concerned only with members of a sequence with large index, the notation "xnR.Yn" (see Notation) will be employed constantly.
Lemma 1. Let
S E C(/C Cn
and an #. O. Let c E
=
C
+ (sn
C(/,
- s),
n
and define
~
(3)
0;
then I - Pn) 1 + C( - an+ 1
C n+ 1 I - Pn ( + -c, - - =. - S-
an
an+ 1
an+ 1
Sn)'
(4)
Proof
Pn) +Cn- - C-n+-1 1-1 +C ( an+ 1 an an+ 1 =.1
+
=. 1 +
c(_I__~) + an+ 1
S -
C
+ Sn
Sn
=. S
an
Sn + 1 _
S -
an+ 1
an
an
-
-
+ Sn+l
S _
C
Sn _
S -
an + 1
an+ 1
an
-
S
Sn
•
(5)
Theorem 1. Let (6)
Then S diverges. Proof Assume s converges. Since (1 - Pn)/an+ 1 is bounded, there is an 8> 0 such that 18(1 - Pn)/a n+ 11 <.l Let C be any complex number satisfying Ic I = 8, so that
Set
Cn
=
C
+ (s,
Re[1
- Re[c(1 - Pn)/a n+ 1]
<·l
(7)
- s). From the previous lemma
+ c(1 -
Pn) an+ 1
+
C
n
an
_
Cn+l] =. Re[1 - Pn (s - Sn)], an+ 1 an + 1
(8)
so (9)
10
1. Sequences and Series
Using (7) and (9) shows
~ + Re c" <. Re C,,+ 1 a"
2
a"+ 1
P,,)] _ ~ <. Re C,,+
Re[C(l a" +1
_
from which it follows that Re c.fa; a,,¢. {z l arg c
-+ 00
+ 3n/4 ~
4
1
a"+1
and so Re c"/a,, >. O. Since C"
argz ~ argc
+ 5n/4}.
(10) -+
c,
(11)
Choosing arg c to be successively 0, n/2, n; 3n/2 shows that a cannot be a complex sequence, a contradiction. • [This beautiful proof is due to Tucker (1967).] We state without proof a similar result for infinite products. Theorem 2.
Let l/a,,+ 1
Then
-
Uo;
= 0(1).
(12)
Il:'=o (1 + a,,) diverges.
Lemma 2.
Let P" be defined ultimately and
Ip,,1 ~.p <
t·
(13)
Then
0<(1 - 2p)/(1 - p)
Proof
~.lr,,/p,,1 ~.
1/(1 - p).
(14)
Note that r" is, ultimately, defined and that
(15)
r" =. P" + P"P,,+ 1 + P"P,,+ 1P"+2 + "',
+ 1 =. r,./p" and the above series converges. We have Ir,,1 ~·lp,,1 + Ip"P"+11 + ... ~·lp"I/(1 - p) ~. p/(1 - p) Thus I»J», I s. 1/(1 - p) and 1iJ», I =.11 + r"+11;::::.ll -lr"+lll =.1 -lr"+11
since r,,+ 1
< 1.
;::::.1 - p/(1 - p) = (1 - 2p)/(1 - p) > 0 . .
(16)
(17)
Theorem 3. Let s, s* be two sequences such that a:/a" = 0(1) and Ip,,1 ~.p < t, Ip:1 ~.p* < 1 for some numbers p, p*. Then s* converges more rapidly than s.
Proof An implication of the hypothesis is a" #-. 0, a: #-. O. The previous lemma shows 0<(1 - 2p)/(1 - p)
~·lr,,/p,,1
(18)
1.5. Further Properties of Complex Sequences
and
II
(19)
One concludes that
I Sn S: -
S* S
I=.1 aa:+n+ II ':/P: I 'n/Pn 1 1
~.la:+ll[(1 an + 1
-
p*)(l - 2p)(1 - p)r 1
=
0(1).
•
(20)
Tucker gives an example (1967, p. 358) to show that 1- cannot be replaced by a larger number.
Lemma 3.
Let b, S, s* E rrls with (21)
Then s* converges more rapidly than s to the same limit if and only if s, = 0(1). Proof Either hypothesis implies the convergence of sand S either case, therefore, bn + 1 ~
bn+tI(s - sn)
+ (s -
and from this the lemma follows.
Theorem 4.
(22)
S -
s:)/(s - sn) =. 1,
-
s, #. O. In (23)
•
Let t, s E rrlc and (24)
and suppose t converges more rapidly than s to the same limit. Then u converges more rapidly than s to the same limit if and only if {In ~ (J.n'
Proof From the previous theorem, a n+ l(J.n+ 1 ~ S - Sn = 0(1). Also u converges more rapidly than s to the same limit if and only if a n+ l{Jn+ 1 ~ S - Sn' Since S - Sn #.0, we have an+l(J.n+l #.0, an+1{Jn+l #.0, and so an' (J.n' {In #. O. By transitivity of ~ we conclude a n+ 1 (J.n+ 1 ~ a n+ l{Jn+ 1 or (J.n ~ {In' This step is reversible, so the theorem follows. • Theorem 5.
Let
S
be convergent and (25)
Then the three conditions below are all equivalent: (i) s* converges more rapidly than s to the same limit; (ii) (J.n+ 1 ~ 'n/Pn; (iii) (J.n ~ 1 + Tn'
I2
1. Sequences and Series
Proof
From Lemma 3, s* converges more rapidly than s iff a n + 1(Xn+ 1
~
(Xn+1
~
1.6. Totally Monotone and Totally Oscillatory Sequences Definition.
s is totally monotone (written s E
«i
(-I)kNsn~O,
~TM)
:«
if (I)
s is totally oscillatory (written s E 9fT O ) if {( -I)nsn} E 9fT M • Here n
~
0,
k
~
1
n, k ~ O.
If s E 9fT M , one has (2)
so s converges since it is monotone decreasing and bounded. On the other hand, if s E 9fTQ, S is alternating and so converges to O. Examples. s~1)
= I/(n +
The sequences S(k)
E ~TM'
where
2),
(X
> 0,
(3)
since
S~k) = ft dl/lit) n
(4)
for I/Ik bounded and nondecreasing
1/12 = (I - tin t),
(5)
(see Theorem 3). ~TM is an important regularity space for certain nonlinear transformations in ff(~s, ~s).
Theorem 1. Let s E ~TM' Then the sequences whose nth elements are given below are also E ,'3iP T M • (Empty products are interpreted as 1.)
(so < 1);
13
1.6. Totally Monotone and Totally Oscillatory Sequences
nj:6 (1 -
(ii)
s)
(iii)
A.(-I)k+l~ksn
(iv)
A.J:~lSj
(so:s: 1); (0 < A.:s: 1, k
(0
:s: A.:s:
> 0, a, k fixed);
1).
Proof The proofs are straightforward. We prove here only (i); the reader is referred to Wynn's paper (1972) for the others. Write
1/(1 - sn) = tn·
(6)
Multiplying both sides by 1 - s; and using the difference formula 1.6(40) gives
(-~)ktn = (1 -
°
sn)-I
±(~)(-~)k-jtn+i-~)jSn.
(7)
j= 1 )
Now s, - Sn+ 1 ?: 0, so :S: s, < 1 for all n, and thus Eq. (7) provides an immediate induction argument on k. •
Theorem 2. Proof
Let
S,
Obvious.
t
Bf T M • Then {sntn}, {asn + bt n} E BfT M , a, b > 0.
E
•
Theorem 3. S is totally monotone if and only if there is a function t/J(t) bounded and nondecreasing on [0, 1] that satisfies s;
=
f
n t dt/J(t),
(8)
n ?: 0.
Proof
Write
<=:
( -1)k~ksn = For all k and
=:
°:S: n :S: k,
f
(1 - t)ktn dljJ(t) ?: 0.
(-It-n~k-nsn
or
?: 0,
k-"(k L - n) (-I)'sn+r = r-.
r=O
where r _
sn -
E Bf~.
r
(10)
°
(11)
:S: n :S: k,
It is easily seen that this system of equations has the solution
(k - n) r; -_ L m(m L m=n m - n m=n k(k k
(9)
k
1)··· (m - n + 1) L m, 1)··· (k - n + 1)
°
:S: n :S: k,
(12)
14
I. Sequences and Series
where (13) This can be written
°::; n s k,
(14)
where cDk n(t) .
t(t - llk)(t - 21k) ... [t - (n - 1)lk) 1(1 - llk)(1 - 2Ik)··· [1 - (n - l)jk]
=
-,---:-:---,---;c:-,---:-:---=-:-:-,.------=cc----,-----~_=_
= t" + O(k- 1)
(15)
uniformly in t, and l/Jk(t) is the step function defined by
°
t ::; 0,
< t ::; 11k, 11k < t s 21k, Lo + L 1 + Lo + L 1 +
+ Lk + Lk ,
(k - 1)lk < t < 1, 1 ::; t.
1,
Because of (12) with n = 0, So
(16)
= l/Jk(1) :2: l/Jk(t) :2: l/Jk(O) = 0,
°s t ::; 1.
(17)
But any sequence of bounded nondecreasing functions on [0, 1] contains a subsequence converging to a bounded nondecreasing function [see Wall (1948, p. 246)]. It is easy to justify taking the limit over this subsequence inside the integral sign (Wall, 1948, p. 245), so for some bounded nondecreasing l/J(t),
s, = For
SE
fr
dl/J(t),
n :2: 0.
•
(18)
fJil s , the determinants sn
H~k)(s)
= Sn+ 1 Sn+k-l
Sn+ 1 Sn+ 2 Sn+k
are called Hankel determinants.
+k- 1 Sn+k
Sn
Sn+ 2k - 2
n, k :2: 0,
(19)
1.7. Birkhoff-Poincare Logarithmic Scales
Theorem 4. If s E 3l TM ,
H~k)(S) 2:
O. If s
E ~TO, (_l)nkH~k)(s)
15
2: O.
Let
Proof Q~k) =
L sn+i+A¢j =
k-1
I1 0
i.j~O
tn(~o
+ ~lt + ... + ¢k_ltk-1)Z dt/J(t)
2: O.
(20)
But this means by a known result on quadratic forms (Bellman, 1970, p. 75) that the determinant of the coefficients of Qin) must be nonnegative, which gives the first part of the theorem. The second is similar. •
Theorem 5 (Brezinski). Let f(x) = Lk~O CkXk be a power series with nonnegative coefficients and radius of convergence p > O. Let s E 3l TM and So < p. Then {f(sn)} E 3l T M • Proof
Obvious, since the sequence S(k) S~k) =
Co
E ~TM
when
+ C 1Sn + '" + CkS~
by Theorem 2, and any limit of totally monotone sequences is totally monotone. [This also provides a proof of Theorem l(i).] •
Theorem 6. If S E ~TM' then H~k)(~2rs)
If S E
~TO'
2: 0
and
then
( - 1)kn H~k)(LlZrs) 2: 0
Proof
Obvious.
and
•
1.7. Birkhoff-Poincare Logarithmic Scales Let p E J+, Ill' Ilz, ... , II p, ebe complex constants, Ilo an integral multiple (positive or negative) of lip. Define Q(w) = lloW In w + 1l1W + IlZW(p-1)/P + ... + IlpW1/ p, WE 3l+. (1) Consider the sequence of functions t/J;jw) = eQ(W)w6 -
j/P(ln
wy,
i,j = 0, 1,2, ... ,
co
E ~+,
(2)
and let F = {t/Ji,j}' It is easily verified that F is strictly (nonreflexively) well ordered under the operation written"
16
1. Sequences and Series
bounded, i = 0, 1, 2, ... , p, then one may define a unique asymptotic scale on F, say, {
I/Ii", k"'
(3)
This scale is called the Birkhoff-Poincare logarithm scale (B-P log scale); p is called the index of the scale. If p = 0, it is called simply the BirkhoffPoincare scale. The ~pecial case J1; = () = 0, p = 1 is called the Poincare scale. Any function satisfying a fairly general difference equation or differential equation is known to possess an asymptotic expansion in a B-P log scale, or, more precisely, the function can be written as a linear combination of such expansions, once it is decided how to interpret sums of asymptotic expansions. [In fact, this can easily be done; see Wimp (1974b).] For difference equations, this is called the Birkhoff-Trjitzinsky theory (1930, 1932),and for differential equations, the theory of subnormal forms. [See Wasow (1965) and the references given there.]
Theorem 1 (Birkhoff-Trjitzinsky). Ao(w)y(w)
+ Al(w)y(w +
1)
Consider the difference equation
+ '" +
Am(w)y(w
+ m) =
0,
(4)
where Ai is defined for w E ~o and Am(w) i:- 0. Let Ai have an asymptotic expansion with respect to some B-P scale F with Q = J1j = () = in ~+. Then there is a B-P log scale G and a basis of solutions Yl' Y2'" ., Ym of the equation such that Yj has an asymptotic expansion gi with respect to G in ~ +. The general form of these solutions is
°
+ aOlw- l/ p + ...) + (a l O + allw- l/ P + .. ·)lnw + '" + (a mo + amlw- l/ P + .. -)(In co)"], (5)
y(w) = eQ(W)w 8[(aoo
Proof The proof is the subject of two papers. The first (Birkhoff, 1930) treats the formal (constructive) theory of the question; the second (Birkhoff and Trjitzinsky, 1932) treats the analytic theory. •
While the theorem is simple to state, in the construction ofthe asymptotic expansions there are many complexities. For instance, p for G and F need not be the same. Further, once certain expansions gi are obtained others may be found from these by formal manipulations, thus vastly simplifying the work involved. For example, the difference equation () yw
+
(w (w (w
+
+ b + c + 1) + A] yw+ ( 1) (w + b)(w + c) + 1)(w + 2) + b)(w + c) y(w + 2) = 0, b, c > 1)[(2w
°
(6)
17
1.7. Birkhoff-Poincare Logarithmic Scales
has solutions
h1(w) =
r(b
r(w hz(w) = f(w
+ w)r(C + w) r(w + 1) 'P(b + w, b + + b) + 1)
1 - c; A), (7)
1 - c; A)
(see Wimp, 1974a). There is a formal basis of solutions
gj(w) = exp[( -IY+ lAl/2Wl/2]W9
LE 00
k=O
k
(
_ly k w - k / 2,
() =
-!(b
+ c)
-
i (8)
and
h1(w) '" fiA(e-b)/2-1/4e~/2g1(W)
in
(9)
~+.
Here the A i are rational; p = 1 for F; for G, P = 2. Let us see how the construction proceeds for the important case m = 1. For the formal computations, assume all series are convergent for w > R, say. Then Eq. (4) can be written yew + 1) = w/i/P(aoe/i/P + a1w- 1/ P + .. -)y(w), f.1 E J, ao # O. (10) Let
yew)
Then
z(w
+
1) = b(w)z(w),
Finally, let u(w)
u(w
+
= w/iw/Pa~z(w).
=
b(w) = 1
In z(w). Then u satisfies
(11)
+ b1w- 1 / P + .... w- 2 / P
1) - u(w) = In b(w) = b1w- 1/P + (2b 2 - bi) 2
=
C1W- 1/ P
In this equation write
+ C2W-2/p + ....
+ d2_pWl-2/P + ... + d_1W 1 / P + d1w- 1/ p + d 2w- 2/P + "',
(12)
+ ... (13)
u(w) = dl_pWl-l/P
+ de In w
and it is immediately found that d 1 with
P'
d2 -
P"
' "
(14)
do are uniquely determined,
(15)
18
I. Sequences and Series
by comparison of terms w- I / P, w-z/p, ... ,w- I • On comparison of terms w- k / p , (k > p), one gets equations of the form
+ I,
k :::: p
(16)
in which IY.k is a known polynomial in d ,_ p, dz-p, ... ,dk-I-P' Thus all the d, are determined in succession and uniquely. Then, writing (17) and exponentiating the series for u(w) gives the desired formal series. This construction, of course, shows that a unique formal asymptotic series always exists, and also that, for the first-order case, p for F = P for G and the index of G is zero. The next result shows how the partial sums of a series grow when the general term of the series has an asymptotic expansion of the form eQs.
Theorem 2.
Let (18)
with
S E
Cfic . Let an'" eQ(W)wlJp(w),
where
~
co = n
+(
J+
(19)
+ IY.IW- I/ P + IY.zW- z/ P + ....
(20)
In
is arbitrary and complex and where pew) = IY. 0
Then
s, where ()*,
(;(6 are as
Case I.
S '"
eQ (W)w 8' p*((I))
(21)
follows:
Q"¥= O. Denote the first nonzero fl j in the seq uence
by u., Then
+ flo, - () + (r -
()* _ {()
r=O I)/p,
S r
Case II.
Q == O. ()*
=
8
+
I,
IY.6 =':1.0/(8
p;
r=O = I;
T
s
r
+
I).
2
s s
(22)
(23)
p.
(24)
1.7. Birkhoff-Poincare Logarithmic Scales
19
Proof. A straightforward application of Theorem 1; see Wimp (1974b) (whose r = 0 values are incorrect). •
Also of interest is a related result for the partial sums of divergent series.
Theorem 3. Let s
~
Then
rcc and let an' p, to be as in (19), for some constant c. r:
in where lJ*,
(25)
are as in (22) and (23) except in the following cases:
O(~
llo;6 O. Then O(~ =
Case I.
lJ* = lJ.
0(0'
Q == 0 and p(w) contains a term w -
Case I I.
S,,-
Corollary.
C
+ din w
I.
Then for some c, d,
-- W6+ lp*(W)
(26)
Let 0(0;6
(27)
O.
Then for some constants /3j' 'Ij, h j ,
An+ Ino
(
/31
/32
(
n+n
)
- - IXo+-+-+"', 2 n
(A-I)
Sn -
S --
and
l
6
1
-n + (8 + 1)
OCo +
'II
n
'12
2 o In n + lJdn + lJ 2/n
< I (28)
)
A = 1, Rc lJ < - 1,
+ ... ,
2
Sn -- iX
1..1.1
+ ... ,
A = I, lJ = -I.
(29)
As examples of the use of these formulas, consider the computation of
e' and (s) from their defining series. Let Sn
(u) = s~ - r~,
For
L k'
k=O n
=
L
k=1
•
1 k'"
(30)
Re a > I.
Sn'
p
= 1,
and for p
s~
=
" xk
= t,
IIp = -I,
III
= In ex,
= III = 0,
() =
-u,
0=
0(0
= I/J"~, (31)
s~,
JLo
0(0
= l,
IXj = 0,
.i >
0,
(32)
20
I. Sequences and Series
and (for both may be taken to be O. We have (33) and the series on the right can be rewritten (34) Higher coefficients are easily determined by formal series manipulations [see Smith (1978)]:
Pl =
PI = 1,
P3 = Xl -
X,
2x,
(35)
For the Riemann zeta function, ((a) -
~ L.
k= I
a1 '" n I - " ((Xo + -(Xl + 2(Xl + ... ) , n
k
n
(36)
where (Xo
(37)
= 1/(a - 1),
and (Xlk+ I
(Xlk
= 0,
k '2. 1,
(ahk-l = (2k)!
k '2. 1,
Blk>
(38)
the higher coefficients being obtained by the formula in Wimp (1974b, 2.42) or from the integral representation for ((a). Equation (36) is known to hold for all complex a =I- 1 [see Olver (1974, p. 292)]. In what follows let (39) be a formal asymptotic series. In future sections we shall need to know the effect of certain difference operators on this series.
Lemma 1
L~Ju(n)v(n) = Proof
±
r=O
(j)l.l.ru(n)!.l.j-rv(n r
See Milne-Thomson (1960, p. 35).
•
+ r).
(40)
1.7. Birkhoff-Poincare Logarithmic Scales
21
Lemma 2
An(A- 1)PnO( IJ( O+ Pdn + "'), kid, APy(n)= { (-OM-l)Pno-P(lJ(o+Ydn+ ... ), A=l,
0#0,1,2, ... ,p-1, (41)
for some Pi' Yi' Proof
Left to the reader.
•
Theorem 4
L dijA)AP+ry(n) = i
r=O
An(A - 1)P( -l)i( - O)ino- i(lJ(o
+ lJ('dn + ."), (42)
A # I. 0 # 0, 1,2,,,. ,j - 1,
where IJ('J'
IJ(~, .•••
depend on j and p and di.rCA)
Proof
= C)A-i(1 - A)i- r.
The first part of the previous lemma gives A-nAPy(n) = (A - l) PnO(lJ(o + Pdn +
(43)
" -).
(44)
Then letting v(n) = A-n and u(n) = APy(n)and using the two previous lemmas gives the theorem. • The paper by Wimp (1974b) includes a number of applications of the previous results, particularly to the problem of finding asymptotic formulas for the remainder terms in expansions in orthogonal polynomials. Brezinski has shown that something like Theorems 2 and 3 is true for series of arbitrary real terms provided the terms are ultimately positive and that their differences are ultimately of one sign. First, note that Un ~ V n is equivalent to the statement that u; #.0, Vn #.0, and (45) Lemma 3.
Let an >.0, s diverge to
+ 00, and b; =
n
L akbk = o(sn)'
k=O
0(1). Then (46)
Proof n
L
k=O
f.lnkbk = 0(1),
(47)
22
l. Sequences and Series
where (48) by the Toeplitz limit theorem, Theorem 2.1(3). Note b may be a complex sequence. •
Theorem 5.
Case I.
Let
S E ~s,
an >.0 and h; = an/l1an with I1h n = 0(1).
Sa; <. O. Then s converges and (49)
Case I I.
l1a n >. O. Then s diverges and (50)
Proof
Case I. h; - ho
n-I
= k~O I1hk =
0
(n-I ) k~O = o(n), 1
(51)
by the lemma (with s = {n}). This means
1)r 1 = 0(1),
[n(an+ dan -
(52)
or, since an+ dan <. 1, the sequence {n(an+ dan - I)} is definitely divergent to - 00. By Raabe's test s is convergent. Thus 00
00
L ak = L hkl1ak =
k=n
k=n
00
-hna n -
L
k=n+1
akl1hk-t·
(53)
Now
f akl1hkIk=n+1
l !
S sup Il1h kl(s - Sn) s.sup/l1h kl(s - Sn-I)'
(54)
L
(55)
k~n
k~n
Thus
00
k=n+1 where
~ E ~N'
so (53) may be written S -
or, since
S -
akl1hk_1 =(s - Sn-I)¢n,
Sn =1=.0,
Sn-I
= -hnan - (s -
-hnan/(s - Sn-I) and letting n ->
CJJ
gives the result.
=.
Sn-I)~n,
1 + ~n
(56) (57)
23
1.7. Blrkhoff-Poincare Logarithmic Scales
Case II.
Sincean+1 >.an'Sn--' +00. Also, n
L hk~ak =
Sn =
hnan+ I
k=O
-
hoao -
n
L ak~hk-I'
(58)
- ho a o .
(59)
~n)
(60)
k=1
or ao
+
n
L ak(l
k=1
+ ~hk - I) = h; an+ I
But, by the lemma, n
L ak(l + ~hk-I) =
k=1
where; E
fYtN .
sn(l
+
Since s; =1-.0, hnan+ Js,
=. 1 +
and letting n --. 00 gives the result.
ao(l + ho)/sn + ~n,
(61)
•
Sometimes the variable appearing in a Poincare series is n + f3 rather than n. This is immaterial, however, as the following result indicates. Theorem 6.
Let
L crCn + a;o-" 00
Un '"
Then, for any f3
E
,=0
e, «e «.
(62)
'fl,
L c~(n + f3)O-', 00
Un '"
(63)
,=0
Proof
k k ( a _ f3)O-' L c,(n + at-' =. L c,(n + f3)O-' 1 + --f3 ,=0 ,=0 n+ k
=. ,=0 LcrC n +
0 00 (f3 - ay-'(r - (})s-, f3) s=, L (s-r )'( . n + f3y
=. i>,(n + f3t ,=0
±
se r
(f3 - aY-,'(r - (}):-, (s - r).(n + f3)
k
=. L c:(n + f3)o-S + O(nO- k- I ), s=o
and from this the theorem follows immediately.
•
+
O(nO- k- l )
(64)