upon the padé table derived from a stieltjes series

Upon the Padé Table Derived from a Stieltjes SeriesAuthor(s): Peter WynnSource: SIAM Journal on Numerical Analysis, Vol. 5, No. 4 (Dec., 1968), pp. 805-834Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2949427 .

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SIAM J. NUMER. ANAL. Vol. 5, No. 4, December 1968

Printed in U.S.A.

UPON THE PADt TABLE DERIVED FROM A STIELTJES SERIES*

PETER WYNNt

1. Introduction. This paper is concerned with the Pade table constructed from a series E'=0 (-l)8c,z' whose coefficients are given by

go

c=f u'dt,(u), where 4'(u) is a bounded nondecreasing function in

O < u < oo. It is shown that under certain conditions, when z is real and positive, the Pade quotients along both forward and backward diagonals form monotonic sequences; an optimal property of the quotients lying upon the principal diagonal is proved. Some new convergence results are derived. The Pade quotients are compared with the transformed sums produced by certain linear methods.

2. Notation. We make extensive use of the theory of continued fractions and therefore introduce at the outset a notation similar to that originating with Pringsheim [1]; we write the continued fraction

a1 a2 at

as 00

(2) [ tb, at

& j_ t=1i

Since, in certain cases, we wish to indicate that the structure of the partial quotient a1/b1 differs from that of the remaining partial quotients, we shall also write (2) as

_100 ( b a, at (3) bo - - b1 bt_Jt=2

We write the rth convergent of (1) as

(4) ~~~~~a, a2 a. j abt~ (4) ?+bo +b2 + b boX ttl r =1, 2, b + b+b+ br

L btj t=1

and where necessary use also an extended notation, similar to (3), for the rth convergent

_r

b0, a, at 'b ' b-tj t

Since, in the formulation of certain results, we wish to preserve a uniform * Received by the editors March 22, 1968, and in revised form July 5, 1968. t Mathematisch Centrum, Amsterdam, The Netherlands.

805



806 PETER WYNN

notation, we adopt the convention that

(5) bo=,a o+a [Xbl ' bt] t=2 bi

and

(6) box a, at bo. L b1'bt_J t=2

If, in expansion (1), the value of the coefficient bo is zero, then we write

(7) a..a = F-T b( + b2 + Lbt]t=1

and make use of a similar convention regarding expressions (4)-(6). We are also concerned with power series and their formal manipulation.

We refer to a power series by a symbol such as g(z): thus

(8) g(z) = a a,z8. 8=0

The formal equivalence between two expressions involving power series is indicated by the symbol -; for example, if in conjunction

80 80

(9) g(1)(z) = E a, (1)z8 g(2) (z) = a8. 2z, s=O s3=0

then

(10) q(')(Z) ? g2(Z)}

means that (1) (2) 1 (1) (2) (2) -2 (11) ~~ao (ao ), a, a, (ao)

and so on. We denote by,-- the process of formal expansion: thus

(12) G(z) -g(z)

means that the coefficients {as} (which are independent of z) are uniquely determined by the function G(z); - signifies a directed relationship; no ambiguity results in our investigations from its use.

The sum or formal sum of g(z) is denoted by S{g(z)}. In many cases we shall be dealing with divergent power series, either series which only converge when z lies within a certain circle I z I = R, or series which diverge for all nonzero values of z. However, we remark that in the main body of this paper we are concerned with functions of a quite well-defined class. If f(z) converges for i z i < R, then for such values of z in i z i > R as we



PADE. TABLE 807

consider, 2{f(z)} is unambiguously determined by analytic continuation. If f(z) diverges for all nonzero values of z, it may occur that many functions, of the class which we consider, have the same formal expansion f(z); in these circumstances we mean by '5{f(z )}, depending upon the context, any one of these functions, or the value of any one of these functions.

Since, in many cases, the functions which we consider are not well- defined when z belongs to a segment [a, A] of the real axis, we shall use comp (a, ,3) as a generic term to denote a bounded open domain, in the z-plane, which does not include any point of the real interval [a, A].

3. Corresponding continued fractions. In the following we shall consider certain continued fraction expansions which may be derived from power series. With a view to the applications which we have in mind, the theory of such expansions (see, for example, [2, Chap. III, ?23]) may be introduced in the following way: given a power series

(13) f(z) = >Ei (-1)csCz s=o

it is formally possible to derive the continued fraction

(14)

by imposing the condition that the rth convergent (which is, of course, a rational function of z) of (14) possesses a formal series expansion in powers of z which agrees with the series (13) as far as the first r terms: specifically,

(15) co vt z E ) sa zs + E ( - 1) " dr,s z8

-1 1 t2 8=0 =

where in general dr, F cs , s = r, r + 1, *- . (The coefficients {vt} are functions of t; they do not depend upon r.) The continued fraction (14) is said to correspond to the series (13); its convergents provide a sequence of rational functions approximating 5{f(z )}.

Naturally the delayed power series

( 16 ) f ( z ) ( 1 )8C.+.Z8 8=0

also possesses a corresponding continued fraction which we write as

[Cm Vt (m)z 1 o

(17) L 1 1t2

so that the coefficients {vt(?)1 of (17) are equal to the {vt} of (14). The conditions which determine the successive coefficients {vt(M)} are similar to (15).



808 PETER WYNN

The successive convergents of the corresponding expansion (17) offer a sequence of approximations to {fm(z)}. Since the relationships between fm(z) and the series f(z) of (13) is

m-1 (18) f(Z) - E (-1)sC8Zs + (-z)mfm(Z),

s=O

we see that the convergents of the continued fraction [r-1 (1l)mCmZm Vj(M) zlr

(19) [ ( 1)sz 1 Ls=ot 1 ' t=2

also provide a sequence of rational functions approximating (25{f(z )}. It is a consequence of equations of the form (15) that

(-1) ItCncm Vt Z]

(20) m+r-1 00

(-l)scszS + Z (-1) r (m)zs s=O s=m+r

where now, in general, drms) 5 c's s = m + r, m + r + 1, *vv

The coefficients {vt((m)} may be derived from those {cs} of the series f(z) by a number of simple recursive processes. Explicit formulas may be given by adopting the notation

Cm Cmn+l ... Cm+k-1

Hk() [c]= Cm+l Cm+2 ... Cm+k

(21 ) Cm+k-1 Cm+k . . . Cm+2k-2

m = 0, 1,**., k-1, 2, ..*

Ho (m)[c] = 1, m = 0,1, **,

for Hankel [3] determinants; we then have [2, Chap. II, ?23]

(m) Hr(ml) [cr]Hr_ [cr]

(22)~~~~~~~~~~m Hr l[cr,]Hrm+ )[c,] (22) (m) _ 1m) r]ir(i+i) r cI

V2r+i Hr(m)[Cv]HIr(m+l)[cr]v

m-=0,l,,r = 1,2,

4. The Pade table. The introductory theory of the Pad6 table [4] may be presented for our purposes as follows: given a power series

00

E ( 1 )'c ,z5 8=0



PADE TABLE 809

it is formally possible to construct a certain double sequence Pi,j, i = 0, 1, *s, j = 0, 1, *., of rational functions of z. The numerator polynomial of Pi,j is of the jth degree, the denominator polynominal of degree i; this quotient is characterized by the property that its series expansion in ascending powers of z agrees with the series '=0 (-1 )8c,z' as far as the term in z' . Specifically,

x i i+j 00

(23) Pi 5 -sZ (1)s (-1) di,j,5zS, Zs= Pi,jsZ s=0 s=i+j+

where in general di,j,,8 cX = i + j + 1, i + j + 2, The functions Pi,j may be set in a two-dimensional array in which the

first suffix i indicates a row number and the second suffix j a column number. This is the Pade table:

Po,o Po,l ... Po,j_1 P?,i ...

Pi * P. ... *1j- pi

.

P0_,0o Pi-l, Pi-lj-i Pi-o,j Pi,O Pi,1 . .. Pi s_1 Pil, ...

We refer to the positions occupied by the sequence of quotients Pr,m+r,

m < 0, r = m, m +1, * ; m > 0, r = 0, 1, *., for a fixed finite integer m, as a forward diagonal of the Pade table; the forward diagonal containing the sequence Pr,r X r = 0, 1, * * *, is called the principal diagonal of the Pade table; the quotients P,n_r,r , r = 0, 1,*.., m, with m > 0 finite and fixed, lie upon a backward diagonal.

It is clear that (15) and (23) are effectively formulations of the same property: the Pad6 quotients are convergents of certain corresponding continued fractions. With regard to the quotients lying on and above the principal diagonal, we have quite simply

m-1 ~~(-1) mcm zm 7. (m) z ]22r P,mr [7 ( - )'a' zs );m X = Pr,m+r-1,

(24) LsO1 Jt2

M 0, 1, ly.. r = 1, 2, .

and

m-1 (-l)/\m m

Ctm) 1-2r+1

E Yc- -z )crnzVt=Z Pr,m+r, (25) L[( l)c ljt=2

m, r = 0, 1,

Thus the forward diagonal sequence Pr,m+r, r = 0, 1, ... , with 0 < m < oo can be exhibited either as a sequence of odd order convergents of a



810 PETER WYNN

certain corresponding continued fraction ((25)) or, if m > 0, as a certain sequence of even order convergents ((24) with m replaced by m - 1).

In order to discuss the Pade quotients lying below the principal diagonal we must first introduce the reciprocal series

00

(26) f(Z) = E 1)5Z 8=0

obtained from f(z) by means of the formal equation

(27) 1 ( - 1)Yc Z + z E+ =0 cs

If the various corresponding continued fractions obtained from f(z), analogous to expansions (17), are denoted by

Fm-1 I 1\mamz t(m) j0

(28) ZE (- ) sz, ()CmZ Vt Z 8=O 1 1 t=l

then we have

jynm-1 m m t (m) 12r-1 -1

Pm+r,r-, = co{ 1 + z [E (-1)8z8a 1 X Z

m 0,I,1 , r= 1,2,

If the Pade quotients derived from the series f(z) are denoted by Pi,j, i, j = 0, 1, , then equations of the form (29) may be rewritten as

(30) Pi j = l - i = 1, 2,* .. , j = 0, 1, ., -1.

In later sections we exploit (30) in the following way: assuming that f(z) is of a certain class we obtain results concerning the behavior of the Pade quotients {Pis,A lying above the principal diagonal; we then show that J(z) is of the same class as f(z), and in consequence derive further results relating to the quotients {Ph,j lying above the principal diagonal of the Pade table derived from f(z); equation (30) is then used to present these latter results in terms of the quotients {Pi,j} lying below the principal diagonal.

There is a simple relationship, of which we make subsequent use, between the coefficients of expansions (19) and (28), with the superscript m = 0 in both cases; it is

(31) O = V2) t=2,3,*.

Up to this point we have been concerned with purely formal results: if the various quantities occurring in a given formula can be determined, then they satisfy the stated relationships. In this context we remark that



PADEI TABLE 811

considerable theoretical and practical interest attaches to the so-called normal Pad6 table, which can be characterized by the property that no factors can be canceled from the numerator and denominator of any Pad6 quotient; each of the rational functions is distinct. A simple necessary and sufficient condition for the Pade table derived from the power series f(z) to be normal is that the Hankel determinants Hk(m)[c,], Hk(m) [cv], m, k - 0,1, ... , defined by (21) should be nonzero. This condition is also sufficient to ensure the existence of the coefficients {vt(m)} and {Io(m)I in expressions (15) and (28), respectively.

We conclude with a point of practical interest: if we do not have at our disposal all the coefficients of the series f(z), but only those of the finite subsequence c8, s = 0, 1, ***, r, then we can only construct a limited section of the Pad6 table, namely, the triangle bounded by the first row, the first column, and the backward diagonal stretching from Po,r to Pr,,o

5. Stieltjes series and continued fractions. In order to facilitate the derivation of the quantitative results which are the prime concern of this paper, we now assume that the coefficients {cj} can be expressed as

(32) Cs= u d4,b(u), = 0, l

where s6(u) is a bounded nondecreasing function in 0 < u < oo;f(z) is then called a Stieltjes series.

If the function 4t(u) is given, and all the integrals (32) exist, then the quantities { c8} are uniquely determined; the converse problem of construct- ing the function +(u), the sequence {c8} being given, is called the Stieltjes moment problem. First of all the members of the sequence { c8} must satisfy certain necessary conditions which are also sufficient for a solution to exist; these conditions may be formulated [5] as follows: either

Ht(0) [C] > O, t = O, 1, ... t,

I33)HI[c'] = 0, t = t' + 1, t' + 2,*-,

WI() [c'] > 0, t = O, 1, I ... It'I-1,

Ht(l)[cY] = O, t = t', t'+ 1, *

or

Ht(1)[cj] > 0, t = 0, 1, t HTT(o) rC' = rw t =t + 1 t + 27..



812 PETER WYNN

or

(35) Ht(I)[c"] > O, HtI(1)[c'] > O, t = 0,1, .

If all solutions to the moment problem are normalized by a condition such as 41(0) = 0, it is possible that many normalized functions 4V(u) satisfying equations can be constructed, or that only one such function exists; in the first case we speak of an indeterminate Stieltjes moment problem, and in the second say that the moment problem is determinate. A number of criteria have been given which are sufficient to ensure that the Stieltjes moment problem, if soluble, is determinate; perhaps the sim- plest is that the series E., C-(21) should diverge [6].

Clearly we may speak of the moment problem associated with the function Af(u); if this problem is determinate, 4I'(u) is its solution; if the problem is indeterminate, 4t(u) is one of many solutions.

By formal expansion, we have

(36) ii d( E) E __ )'cz 1 + zu 8=0

We remark that if VI(u) is prescribed, then the function

.l; d~ (u) (37) F(z =fd() 1 +zu

is uniquely determined and may be shown [2, Chap. IV, ?32] to be regular in comp (- o0, 0). Conversely, if we are given a function F(z) and know that it can be represented in the form (37), then the corresponding function i1(u) is uniquely determined [5].

If condition (33) holds, 4/(u) is a simple step function with a saltus at the origin of magnitude /to, and saltuses at points Xi of magnitude p, i = 1, 2, * *, -1; F(z) is a rational function of the form

t'-1

(38) F(z) = Ao + E a M= 1 + zXi

If condition (34) holds, i1(u) is a simple step function with saltuses at the points XS' of magnitude ui1 i 1, 2, * * , t'; F(z) is a rational function of the form

t' (39) F(z) = Z +A

i=M + zXi'

If condition (35) holds, then F(z) is not a rational function. In the first two cases the moment problem associated with 4/(u) is determinate; in the third case this may or may not be true.

If the moment problem associated with u(u) is determxinate, S{f(z)j



PADEI TABLE 813

refers to one function; if the moment problem is indeterminate, {(f(z)} refers to many functions each of which is obtained by replacing Jt(u) in (37) by another solution of its associated moment problem. In the indeterminate case we have many functions connected with one series, with one assemblage of corresponding continued fractions, and with one Pade table.

For real positive z, f(z) now becomes a series of real terms which alter- nate in sign. The function F(z) is real and positive. The successive partial sums of f(z),

(40) Sm = E (-1)8c8z8 = ] -(u) dat'(u), m = 1,2,* *, _ 1 + uz

form an oscillating sequence. In particular we have

(41) S2m-> f d() = 1, 2

If the moment problem associated with aJ(u) is indeterminate, then clearly ,6(u) may be replaced by any other solution in inequality (41); the term on the left-hand side remains unchanged, that on the right-hand side is altered, but the inequality retains its validity; in all cases the right-hand side of inequality (41) may be replaced by (2{f(z )}, and more generally we have

(42) S2m-1 > ({2f(z)} > S2m, m = 1, 2, *..

Concerning the corresponding continued fraction (14) derived from a Stieltjes series (we then speak of a Stieltjes fraction), either [5]

(43) vt > 0 t = 2, 3, ..)t,t vt = t t + t + 2) ..

where t 1 is a finite positive integer, or generally

(44) Vt > 0) t = 2, 3,

Condition (43) is equivalent to (33) when t" - 2t' - 1 is odd, and to condition (34) whent = 2t'is even; conditions (35) and (44) are equivalent.

Conversely we remark that if the coefficients of a corresponding continued fraction obey a system of inequalities such as (43) or (44), then a representation of the form (32) can be found for the coefficients of the series from which the continued fraction originates. Now it is clear that if conditions (43) or (44) hold, then the coefficients {Ivt of the continued fraction

cn to te st=27

corresponding to the series Y(z) of (27), also satisfy similar conditions



814 PETER WYNN

(see (31)). Thus the coefficients {c8} may be given a representation of the form

(45) U8= j U8 d*(u), s=O 1,*-*.

Furthermore, if we write

(46) P(z) = d;(U) 1 +zu'

then (31) leads to

(47) F(z) =1 +CzZ)O

If the moment problem associated with the sequence {cj} is determinate, only one function F(z) of the form given by that of (37) exists; only one function F_(z) can be obtained from (47); only one function 6(u) can be constructed. If the moment problem is indeterminate, then to the many solutions t'(u) there correspond many functions {(u).

In the case deriving from condition (43), the corresponding continued fraction (14) terminates and does no more than reconstruct the rational function (38) or (39), whichever is appropriate. With regard to condition (44), if the moment problem associated with At(u) is determinate, the corresponding continued fraction

[CO VtZt=

converges uniformly ini comp (-oo, 0) to F(z) [5]. If the moment problem is indeterminate, the continued fraction diverges by oscillation: the sequence of even order convergents converges uniformly in comp (- 0, 0)

to a function f dco(u)/(l + zu), the sequence of odd order convergents

to a function f dih(u)/(l + zu); /o(u) and APl(u) are the extremal

solutions to the moment problem (32); of all possible solutions, 'o(u) is that which is constant throughout the largest possible interval [0, 5];

Al(u) is that which has the greatest saltus at the origin [5]. In addition to deriving the inequalities (43) and (44), Stieltjes [5] also

gave a characterization of the convergents of the corresponding continued fraction derived from a series whose coefficients are represented by (32). Stieltjes' result is that when z is real and positive,

l (u-) [CO vt z12r (48) . dxtKu) .

I inf 1 + L { + E X(l + zu),f. r = 1, 2, Xil.. Xn? + zu 8=



PADE TABLE 815

the left-hand side of the above equation is visibly the remainder term associated in this case with the 2rth convergent of the corresponding continued fraction (14). For the odd order convergents:

rCO vt z-2r+1 d4l d(u)

(49) - K zu1$+ X(1 + zu)8} r = 0,1,

XI ... 1 + zu __

If we compare expression (48) with a similar expression in which r is replaced by r + 1, we see that

|O d4l(u) CO covtz > 2r

d*,(u) rCO vt Z2r+2

(50) f 1p+ zu [ t=2 fo1+ zu ']t=2 '

r = 1) 2, ..

we may also derive an inequality concerning the odd order convergents of expansion (14) upon the basis of formula (49). In summary, assuming that the convergents referred to exist, in the sense that the continued fraction has not terminated with a lower order convergent, we have

F 1 2r?2 CO ?t '- < CO 't Z < SIAzZ )1}

(51)~~~~~~= {f} 1 I 1 t=2 ' ]=

r = 0,1,

The even order coilvergents of expansion (14) coinstitute a bounded monotonically increasing sequence; the odd order convergents form a bounded monotonically decreasing sequence.

6. The Pade table derived from a Stieltjes series. If condition (43) holds, the Pad6 table degenerates into two semi-infinite bands of quotients, one horizontal, the other vertical; when t = 2t' -1 is odd, the quotients belonging to and lying between the horizontal sequences {Po,r}, {PV'i,rj, r = 0, 1, * , exist, as do also those belonging to and lying between the vertical sequences {Pr,0}, {Pr,t'-i}, r = 0, 1, * ; when t" = 2t' is even, the bounding sequences are {PO,r}, {Pt',r} and {Pr,0}, {Pr,,-l}, r = 0, 1, ..., respectively.

If condition (44) prevails, we encounter an example of a normal Pade table.

Although in the following we shall assume that we are concerned with a normal Pade table, all the results derived are true for a terminating Pade table, insofar as the sequences referred to actually exist.



816 PETER WYNN

If the {c8} are given by an expression of the form (32), then for a fixed m the { cm+8} are given by a similar expression (merely by replacing d{I(t) by tm dc(t)): inequalities (44) may be extended by adding a superscript m to each of the variables; they may also be extended by the addition of a tilde (see (31)), and indeed by both a superscript and a tilde.

An alternative derivation of the same conclusions was given by Van Vleck [7]. Firstly he used a well-known theorem of Sylvester [8] to show that conditions (35) actually entail that

(52) Hr(m)[cP] > 02 m, r = 0,1, ...

This, in conjunction with (22), proves that vim) > 0, m= 0, 1, ...

t = 2, 3, ... ; secondly he used a result of Hadamard [9] which expresses the determinants Hk(m)[a,] in terms of the Hk(m)[c,], and formulas (22) again, to show that Vt(m) > 0, m , 1, * , t= 2, 3, *

From inequalities (52), it follows that if condition (32) holds, then the resulting Pade table is normal. In addition, it can also be shown in this case, that if the points of increase of At(u) are restricted to the interval [0, b], then for values of z in comp (- oo, -b-1), no two neighboring Pad6 quotients have equal values; since we restrict our attention in this section to values of z which are real and positive, the Pad6 tables with which we are concerned also have this property.

We now investigate the bounding sequences of the Pade table. Inequal- ities (42) may simply be reformulated in terms of the quotients {Po,rI. Concerning the quotients {Pr,O} we have in the notation of (30),

CO (53) Pr, 0 r = 1, 2, *..

1+ ZP0,r-1' 12

furthermore

(54) P0,2r > P(Z), r = 0, 1,

(55) PO,2r+l < F(z), r= 0,,2

Inequality (54), together with (53) and (47), may be used without diffi- ficulty; when using inequality (55) however, it has to be borne in mind that it is possible that the function 1 + ZPO,2r+?i2 occurring in (53) as a possible denominator on the right-hand side, may be negative. In conclusion we formulate the following theorem.

THEOREM 1. In a normal Pade table derived from a Stieltjes series f(z) with positive real argument z,

(56) PO,2r > 5{f(z)}, r = 0, 1,

(57) PO,2r+l < ({f(Z)}, r = 0,1,



PADE' TABLE 817

if, for a certain value of r, P2r,O > 0, then also

(58) P2r,0 > 5{f(ZA)}; finally,

(59) 0 < P2r+1,0 < S{J(z)}, r = 0, 1,

We may investigate the forward diagonal sequences lying above the principal diagonal of the Pad6 table, upon the basis of inequalities (51). To investigate the forward diagonal sequences lying below the principal diagonal we again resort to (30), and in addition write

(60) P*_l = F(z) + n*_j,

so that we now have

(61) P*j = cot 1 + zP(z) + ZXi1,1}1

Let us suppose that as we proceed along a prescribed route in the P*, array, the values of the quantities ji- j, as given by (60), are w('), s = 0, 1, * . If {&8)} is a monotonic sequence of positive numbers, then clearly the corresponding sequence Pi,j produced from (61) is also a monotonic sequence, and we may also deduce that for quotients lying upon the route in question,

(62) Pi,j < el{f( WI

If {W(s)} is a monotonic sequence of negative numbers, the behavior of the corresponding sequence of quotients Pi,j is a little more complicated; let us assume that the numbers i(?), M), ... decrease in magnitude and that w(O) at least is sufficiently large for the corresponding quotient Pi,j to be negative, whilst there exist members of the sequence, iIM, ii('+1), say, such that the corresponding quotients Pjj are positive. The quotients Pi,j lying upon the route corresponding to the sequence { then behave progressively in the following way: they form a sequence of negative numbers increasing monotonically in magnitude; possibly there exists a quotient whose value is infinite; thereafter they form a decreasing sequence of positive numbers and, for these members of the sequence,

(63) Pi,, > e{f(z)}.

In conclusion we formulate this theorem. THEOREM 2. In a normal Pade table derived from a Stieltjes series f(z)

with real positive argument z: (a) the sequence { Pr,2m+r}, where m is zero or a fixed finite positive integer,

decreases monotonically; also

(64) Pr,2m+r > S{f(z)}, r = 0, 1,



818 PETER WYNN

(b ) the sequence {P2.rn?r,r where m is a fixed finite positive integer, is in general composed of three consecutive constituents: initially the sequence consists of negative numbers which increase in magnitude; there exists an infinite member of the sequence; the remaining members form a monotonically decreasing sequence of positive numbers, and if these are P2m+r,,r, r =1,

l+1, *, then

(65) P2m+r,r > (2{f(z)}2 r = 1,1 + 1,

(in any given example it is possible that any one, or any adjacent two, of these constituents are missing);

(c) the sequence {Pr,2m+r+l} where m is zero or a fixed finite positive or negative integer, increases monotonically, and

(66) Pr,2m+r+l < I{f(z)}, m> O r= -0,-1, < 0, r = -2m - 1, -2m, .

We now consider the backward diagonals of the Pade table: in the first instance we compare Pr,2m+r with Pr-1,2m?+r+ . We have in conjunction

Go u2m d [,6(u) C V (2m)z

2r+1 2md4u _ cX

v

(67) 1 + ZU t=2 (67)~ ~ ~~~~ ~G = J

u Z2m+1 diP(u r { 2

mxi d + Z X8(1 + zu)8}

and

() , r 1 (ZU)mdl,(u) F C2mZ2m Vt(2m)Zl2r+? (68) Pr, 2m+r = l(u + L 'J =

thus - f d4P(u) Pr, 2m+r Jo- + zu

(69) = min p (zu)m+dk(U){1 + ZX8(1 + zu)j}

Xl r 1'X + zu = We may also derive a similar relationship for Pr-1,2m+r+l X and transferring a factor of zu inside the braces, have

00 d4l(u)

Pr-1, 2m+r+l - d + zu

min= m I ()2m?l ) zu + E zuXJ(1 + zu)S}. Xl.. X.r_1 1+ zu 1

The minimum on the right-hand side of (69) is taken over all poly- nomials of degree r whose square, when u is put equal to - z', is unity.



PADEi TABLE 819

The expression in braces on the right-hand side of (70) is such a polynomial; thus the magnitude of the expression on the right-hand side of (69) is certainly not greater than that occurring in (70). Since, for the values of z under consideration, no two neighboring Pad6 quotients have equal values, we see that the value of expression (69) is strictly less than that of expression (70). We have, in summary,

dVl(u) (71) Pr-1,2m+r+l > Pr,2m+r >f +zu, r =1,2,1 , m = O,1,* .

The quotients PO,2m , Pi,2m--li P2,2m-2 ... * Pr,2m-r ... which are encountered by commencing at the entry PO,2m of the first row and con- tinuing along a backward diagonal of the Pad6 table, decrease monotonically until we meet the entry Pm,m of the principal diagonal.

Similarly we may show that the neighboring sequence Po,2m+1 X Pi,2m .**

increases monotonically until the entry Pm,.+, is encountered. The preceding analysis may also be extended so as to concern that part

of the Pade table lying below the principal diagonal. Our conclusions concerning the backward diagonals of the Pad6 table

are the following. THEOREM 3. In a normal Padg table derived from a Stieltjes series f(z)

with real positive argument z, the sequence {Pr,2m.-},X with a fixed finite positive integer m, consists of positive numbers which decrease monotonically until the entry Pm,m lying upon the principal diagonal is encountered; thereafter, in general, this sequence increases monotonically until a certain quotient P212m-1 <_ ? (where 1 > m) is encountered; then the quotients constitute a sequence of negative numbers decreasing monotonically in magnitude, which terminates with the entry P2m,O ; we have

(72) Pr,2m-r > {ff (z)} r = 0, 1, * , 1, 1 > m

(in any given case the infinite and negative members of the above general sequence may not exist; we then take 1 = 2m in inequality (72)); the sequence {Pr,2m_r+1}, again with a fixed finite positive integer m, increases monotonically until the entry Pm,m+i is encountered; after passing through the value Pm+i,m the sequence decreases monotonically, terminating with P2m+1,0; we have

(73) Pr,2m7r+i < e {f(z) } r = O, 1, *,2m + 1.

As a curiosity we mention that if we start from any quotient of the form Pr,2m+r , with 0 < m < oo, and move in a diagonal direction away from the upper edge of the Pad6 table without crossing the principal diagonal, then the quantities encountered will constitute a monotonically decreasing sequence. We are led to the following theorem.



820 PETER WYNN

THEOREM 4. In a normal Padg table derived from a Stieltjes series with real positive argument, the entry Pr,2m+T, where m is a fixed finite positive integer, constitutes an upper bound to all the Padg quotients lying in the semi- infinite band bordered by the entries Pr,2m+r, Pr+1,2m+r) , * Pm+r,m+r ,

the forward diagonal Pr+s,2m+r+s, s = 0, 1, * * * , and the principal diagonal. Similarly one can show that Pr,2m+r+l, again with m a fixed finite posi-

tive integer, constitutes a lower bound for a certain iilfinite set of Pad6 quotients; furthermore one can extend these considerations so as to concern that half of the Pade table lying below the principal diagonal.

We now derive an optimal property of the principal diagonal. The convergent

(74) CO '

is obtained from the coefficients occurring in the partial sum 2r-1

E (-1)'c8Z8. s=0

Stieltjes, who introduced a motif of which we have already made use, showed that the error in evaluating (37) by use of the convergent (74) is less than that produced by the use of the partial sum _ (-1 )8c8z8. In order to prove this, he wrote

f d+/ (u) 2r-1 (1)sc zs d() (Z)2r 1 + zu s=O 1 + ZU

(uz)r is a polynomial of the requisite type occurring in braces in (48), and his result follows for reasons with which we are already familiar. He also compared the convergent of order 2r + 1 with the partial sum

_ 8o (-1) 8c,z and came to a similar conclusion. What we have proved, although our methods are no more than a straight-

forward adaptation of Stieltjes' reasoning, is considerably more extensive: we have in fact derived the following theorem.

THEOREM 5. In a normal Pade table derived from a Stieltjes series with positive real argument z, whose coefficients are given by the representation (32), of all the quotients which may be derived from the coefficients occurring in the partial sum 1_o= (-1 )c3z8, 0 < r < oo, that entry lying upon the principal diagonal, namely Pr,r, provides the most accurate approximation to the value of the integral (37); furthermore, of all the Pade quotients which may be derived from the coefficients occurring in the sum -3=0 (-1)8c8z, either Pr,r?+ or Pr+11, (the entries neighbor to the principal diagonal) provides the optimal value.

We wish to emphasize that the validity of our analysis does not depend



PADE TABLE 821

upon the convergence or divergence of the original series f(z) and is also independent of the convergence of the corresponding continued fraction derived from this series.

7. Questions of convergence. We are able to use the preceding theory to obtain a variety of convergence results; our analysis is presented in terms of two cases: in the first of these the function +1(u) is presumed to be constant in [b, oo], where 0 < b < oo, and to have a point of increase at u = b, i.e., 41(b -0) 0 41(b); in the second we shall assume that f(u) has points of increase for arbitrarily large values of u.

If b

(76) C's Us d4k(u), < b < , s=0 ,1,*,

where ,6(u) is bounded and nondecreasing in 0 < u < b, then it may be shown that the Stieltjes moment problem associated with the sequence cj} is determinate; ({f(z)} refers to one function and indeed

(77) i{f(Z)} = ff U)

is analytic in comp (-oo , -b-) (see [2, Chap. IV, ?32]). With regard to the power series f(z), we may assert that it converges for

I z I < b-', since 2{f(z)} is analytic in I z i < b-'. On the other hand, we have for arbitrarily small 6 > 0,

b

(78) Cs > us d4(u) > (b - 3)8{14(b) - 1(b - 6)- ;

thus f(z) diverges for I z > b-1. To examine the case in which z =b-, we put

(79) u bu', (bu')- = (u'

when co 00

(80) > (-c),b-S b= (-Z )"I s=O s-0

where

(81) 7 = f u's df'(u') s = 0,1,*

It is easy to show that { md is a monotonically decreasing sequence with- limit zero; thus f(z) converges when z = b-1.

We now investigate the series (z) and the function #(u) of (45). From



822 PETER WYNN

(27), we have

(82) S{f(Z)} [ f - -z ] z

from this equation it appears that the only possible singularities of e5{J(z)} in comp (- 00 -- b') are at the point z = 0 and at points at which S{f(z )} is zero. However we first verify that

(83) ({Jf(o)} = _; CO

thus there is no singularity at the origin. Furthermore, writing z x + iy, we have

(84) ({f(z) = (1+ du i ,t(u) b Ud4u (84) 25 jb xu) I2 f y(u

(1+ xu)2 + y u2 (1 + xu)2 + y2u2

Clearly Im ['{f(z)}] # 0 when Im (z) # 0; and when z is real and lies in the bounded interval -b-1 < z M < oo,

(85) Re [(B{f(z)}] I 41(b) - (0) > O. 1 +Mb

Thus I{f(z) } is never zero in comp( - oo, -b-'), and in conclusion 5{f(z) } has no singularities in comp(- oo,-b-1). We know f(z) to be a Stieltjes series; thus there exists, in this case, a rep-

resentation b

(86) = fU8 d(u) , = 0,1, ...

for the coefficients of f(z). The function i(u) is constant at least in the interval [b, oo ]; for if b were to be replaced by a higher lim-it of integration in (86), and the values of the coefficients {c } thereby sensibly increased, then the series J(z ) would diverge at some point in I z < bV1 (see (78 ) ), which is impossible since 5{f(z )} is analytic in comp (- oo, -l0). For the sake of completeness we remark that the function +(u) may be constant in some interval 0 _ a < u < b; as an example, we have only to consider the series whose coefficients are given by

(87) c8=b8, s=0,1, *1

thus the radius of convergence of f(z) may be greater than that of f(z). In the notation of (76) and (86), we have

(88) {~( (1-)8Cm+sZS} = fd4l n(u) d4m(u) = um di(u), 0 < u < b;



PADE TABLE 823

(9 { (1-)8 m?s Z8} = ,Z d, ;m(U) = U d (u)

0 < u ? b.

We now derive a general result concerning the convergence of the forward diagonal sequences of the Pade table derived from f(z). Our result is based upon an early theorem of Markov [10] which for the present purposes may be formuLlated as follows: if

(90) d, =j tu8 dG(u), 0 < < oo, s= 0, 1* *.*.

where 0(u) is a bounded nondecreasing function in 0 _ u < 3, then the corresponding continued fraction derived from the series ??o (-1 )" d8z8

b

converges uniformly in comp (- oo, -b-1) to f d(u)/ (1 + zu). The

next theorem follows immediately by inspection of (88) and (89). THEOREM 6. In a normal Pade table derived from a Stieltjes series whose

coefficients are given by (76), with 1(b- 0) 5 VI(b), all the forward diagonal sequences {Pr,m+r}, r = 0, 1, *., m < so fixed, converge uniformly

in comp (-oo, -b1 ) to the function f d/(u)/(1 + zu).

We now wish to derive a general result concerning the horizontal and vertical sequences of the Pad6 table. We know already that the sequences IPo,r) r = 0, 1, ... , and {Pr,o}, r = 0, 1, * * *, converge for real values of z lying in the interval 0 < z ? 1b'. In particular we easily show that for such values of z,

PO,2r+l > 0, r = 0, I, ...

(91 ) P2r(O > 0 r = 0, 1,***.

Our results concerning the monotonic nature of the backward diagonal sequences of the Pade table tell us immediately that when 0 ? z < b-1 the Pad6 table consists entirely of positive numbers, and furthermore that for such values of z all the row and column sequences i.Pmr} { Prrm1 m ? 0 fixed and finite in both cases, converge.

To prove a further result concerniing these sequences, we remark that [2, Chap. IV, ?35] the even order convergents of the corresponding continued fraction derived from the series f(z), whose coefficients are given by (76), have a partial fraction decomposition of the form

- 2r r (92) C Vt Z r,s

where

(932} II- ur >() n,> Xr, >-hb r = 1) 21 ... I s = 11 2 ..s



824 PETER WYNN

and, in addition, r

(94) Z: /r,s = co 8=1

Thus we have m-1 ( r (m) '

(95) P =,m+r E (-1)c8 Z' + (-Z)m E r, s30 ts=0 + z)

where again

(96) >0, 0 > X(m) > -b, r = 1,2,**, s= 1,2,* **,r

and r

(97) (,= Cm

We now consider the behavior, when I z I < b-' and as m tends to infinity, of the expression on the right-hand side of (95). Clearly the first constituent tends to a limit, since f(z) converges uniformly for I z I < b-'; we also remark that

(98) lim cm zm = 0, z i < b-1. M--OO

i\Moreover, when I z I < b-1, we can find a nonzero positive real quantity 6, such that

(99) 1 + zxr4s I > ,

where 8 is independent of z, m, r and s. Thus r (m) r

(100) E jm) < 6- E r,s s90 1 + (irn)

Z g0

and r (m)

(101) lim (z)iZ +rT,os = 0.

Thus we have shown that

(102) lim Pr,m+r = (2{f(z)}, oo > r > 0,

uniformly for I z I < bV1. We derive a siniilar result for the vertical sequences.

In conclusion we formulate the following theorem. THEOREM 7. In a normal Pade table derived from a Stieltjes series whose

coefficients are given by (76), with 4'(b - 0) = $ (b):



PADE' TABLE 825

(a) when z is real and lies in the interval 0 ? z < b-1, all the quotients IPi, j are positive real numbers;

(b) the horizontal sequences {Pm,rJ, r= 0, 1,*.., and the vertical sequences {Pr,m} r = 0,1, *... , m _ 0 fixed and finite in both cases, converge uniformly for z lying in the circle I z I < b-1, and also at the point z = b-1; indeed, for such values of z the sequence of Pade quotients lying along any route which is consistently directed away from one or both of the bounding sequences also converges uniformly.

If we make an additional assumption concerning the structure of the function At(u), we can derive still further convergence results. Our investigations are based upon a theorem of de Montessus de Ballore [11] which may be stated as follows: The Pad6 quotients derived from the series g(z) are denoted by P$ j; it is assumed that this series converges in the neighborhood of the origin to a function 2{g(z)} which is regular in the circle I z I- R except at v (< oc) poles (counted according to their mul- tiplicity) lying within this circle. De Montessus de Ballore's result is that

(103) lim P,, 2{g(z)}

uniformly in any open domain in z j < R which does not include a pole of 25{g(z)}. It follows from this result that if 2{g(z)} has no poles within the circle j z j = Ro, then of course the sequence IPO,,) converges to 25{g(z)} for I z I < Ro ; if 25g(z)} has onepole at z- )1, within the circle I z I = R, , then {Pi,T} converges to 25{g(z)} for I z i < R, except at z = ',; if g(z) has two poles at z = ,71, z = 772, respectively, within the circle I z I R2, then {P2,r} converges to g(z) for I z I < R2 except at z = z = 2; and so on.

One is, at first, tempted to read more into de Montessus de Ballore's result than it, in fact, says; the result, as stated, says nothing concerning the further sequences {P7 v, } = 0, 1, * ; if e {g(z)} is analytic in an annulus R < I z 1 < R' with the exception of a further v' ( < oo ) poles, then of course we can say something concerning the convergence of these further sequences, as can also be said if 25{ g(z)} is a logarithmic or alge- braic function; but these are special cases. In order to dispel the belief that the further horizontal sequences must necessarily converge, Perron [2, Chap. V, ?45] gives a simple example of a Pade table in which the sequence {PO,r}, r = 0, 1, . , converges for all finite values of z, but already the sequence {P,,A, r = 0, 1, ... , diverges for values of z which are everywhere dense in any finite part of the z-plane. We are, however, about to show that in certain circumstances the convergence of the further sequences may be proved.

Our additional assumption concerning A/'(u) is that in the range bk < U

< b, where bk > 0, the function Vt(u) is constant with the exception of



826 PETER WYNN

saltuses at the points u bk-1, bk-2, ***, bo = b. Thus, in this case, we have

(104) 25f(z)} ) b 4U + 1 zE ( 104 ) 17 ~~~1 + zu =o 1 + zbi'

where we stipulate that b = bo > b, > ... > bk > 0, and

(105) .1i = VI(bi + O) - V(b - 0), i = 0, 1, .. 2 k-1

We may immediately apply the results of the preceding paragraph to deduce the convergence of the sequence {Pv,r} r = 0, 1 ... v < k, to

g{f(z)} , except at the points z = -bo-, z = -bj-, * , z = With regard to the vertical sequences of the Pade table, we remark first

that in the case under consideration, (5{f (z )} is analytic in the circle I z - bk- except at the simple poles z= -bo0', - b1, *..., -bki1. Thus in the notation of (27), Si{f(z) } is analytic in the circle z I = bk-l except at those points at which 25{f(z)} is zero. Such points certainly exist. For in the interval [-bi-' + 6, -bi-' + 8 + 8'], 0 < i < k - 1, where a

and 6' are arbitrarily small nonzero positive real quantities, 25ff(z)} is real and positive, and in the interval [-b, - 8 - 6', -bb7 l- 8], 1 < i < k, (2{f(z)} is real and negative. However (2{f(z)} has only one zero in the interval [-bi-', - b-i-1], for

d fbk ud4v(u) k-

bA (106) d E2{f(Z)} = LJo(1 + Zu)2 i 1 + zb)2l

and the right-hand side of this expression is certainly never zero when z lies in the real interval [-b1, -b7t1]. Thus 5{f(z)} has a representation of the form

(107) d{u(z)} = f d+ + E A + 1 + zu i==0 1 + biz

where

(108) bi+, < bi< bi, i= 0,1,***,k-1, 0<bk< bk

We may now apply de Montessus de Ballore's result to the convergence of the vertical sequences of the Pade table.

In summary we formulate the following theorem. THEOREM 8. In a normal Pade table derived from a Stieltjes series f(z)

whose coefficients are given by (76) for which A'(u), a bounded nondecreasing function in 0 < u _ b, is constant in the interval 0 < bk ? u ? b except for isolated saltuses at the points u = bi , i = 0, 1, , k - 1:

(a) the horizontal sequences {P,,r} r = 0, 1, ... , < k converge uni-

formly in any open domain lying within the circle I z = b7 which does not include any point z = -bi1, i = 0, 12 ... X V - 1 at which 25{f(z)}



PADIr TABLE 827

has a simple pole; the vertical sequences {P,j, r = 0, 1, v * !, v <k, converge uniformly in any open domain lying within a circle z i b-', which does not include any point z = -b', i = 0, 1, *., v-1, at which e5{f(z) } has a sinmple zero;

(b) when z is real, positive, and < bi-', i < k, both the horizontal sequences {Pm,r}, r = O, 1, *., and the vertical sequences {Pr,m}, r = 0, 1,...

m > i fixed and finite in both cases, converge; indeed, for such values of z the sequence of Pade quotients lying along any route which is consistently directed away from one or both of the bounding sequences and does not encounter more than a finite number of quotients belonging to the sequences Pm,r}, {Pr,m}, r = 0, 1, *, m = 0, 1, 1, i-1, also converges.

We now turn our attention to series f(z) whose coefficients {1cj are

given by integral expressions of the form (32) for which the function

t'(u) has points of increase extending throughout the entire positive real

axis.

From inequality (78), we see that f(z) diverges for all nonzero values of

z. Any possible function 25{f(z)} is not analytic in the neighborhood of

the origin; clearly the same is true of any possible function 2{f(z)}. Thus

in the case under consideration, both the bounding sequences {Po,r} and

IPr,01, r = 0, 1, * , diverge. The convergence behavior of the forward diagonal sequences of the

Pade table derived from a Stieltjes series was analyzed completely in the

doctoral thesis of H. S. Wall [11]. If the moment problem (32) is inde-

terminate, then all the forward diagonal sequences converge for z in

comp (- o0, 0), no two neighboring limits being equal. If the moment

problem (32) is determinate, then it may occur that all the forward diag-

onal sequences converge to the same limit (this is the case of chief practical

interest), and it may occur that the band of forward diagonal sequences

commencing with the entries Pi,o, P1_j,o, ... , POO, * * , Po,k- X Po,k

all converge to the same limit, but that outside this band oscillatory diver-

gence of the kind already described takes place.

8. Numerical results. Table 1 displays that part of the Pade table which

can be derived from the first thirteen terms of the series =o (-1 )Is !z when z = 1.0. In this case we have, in the notation of (32),

(109) A (u) = 1-e , 0 _ u < co

Since the Stieltjes moment problem associated with the sequence Is! } is

determinate, we may with complete propriety write

(110) lim { E ( -)si zs4 0.5964.

The results of Theorems 1-5 are clearly illustrated in this table.

Table 2 gives similar results relating to the first seven terms of the



t tt

+- - C

I + o Co 0 0 cO X

Co C0 +

cO cl CD l

_ _ I + +C +o +0

00 Co Co t- o Co oo + 00 '- + - to 00 0 Co Co Co

Co Co Co Co 0. o'- Co Co 0" 00 Co

C Co Co 0D

C 00 Co 0- 00 0 + + + + +

Co Co km 00 cO Co Co 0 C'D C 0

Co o 00 0 00

w~~~~~~~~~~~~k k

cCCo Co C 0 Co Co

+ +o I I I

Co CO OD Co 00 CD

o 0 00 Co Co ' -

Co 00. Col CD Co Co C

+ + I + +

E- 2~~~~~~S

+ + I I I I

_ _ C?~~~~~~~~~~" ^0 0 00 004 00 00

O C km C0o 00 Co 00 0 Co

oq CD 00

LO km

00

km-

Co

CD

oQ o ~0 0 o Co Co Co C

m 5 _0000000 0_

? + +++++++I

Co C0 (

0- Co 00

o 00 C

09 Co ,_ Co O Co Co 00 CO

O 00 o CD Co 00 C

_4 1- 1- CO4 Uti _ I_++ __

+ I II o Co oo ~ o o' 00 Co 00 o Co 0

O CO 00 00 0- _ 00 Co C CD o C Co 00 C!

C' C 0 C900

0

d4 00 ? 00 00 00 00 00 Co

o .'- ~. .o CD Co Co . C. Co C

_ + I++ + + + + +

__ II_+_III

C4 ? ? >1 MO8 X N t ?

I+ + + + + + I + +

Co o Co C 0D00 - 00 C C oo C

+ + + + + + + + I + 8

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~14 -4 _ -4 C Oo tC

_ o o o o o o o8o8



PADE TABLE 829

TABLE 2

\j 0 1 2 3 4 5 6

0 1.0 0.5 0.833333 0.583333 0.783333 0.616667 0.759524

1 0.666667 0.7 0.690476 0.694444 0.692424 0.693590

2 0.705882 0.692308 0.693333 0.693089 0.693169

3 0.685714 0.693396 0.693122 0.693152

4 0.698351 0.693046 0.693154

5 0.689325 0.693197

6 0.696173

series 1jS=o (-1 )s(s + 1 )-1z, again when z = 1.0. We are dealing with a case adumbrated by (76); we have indeed

1

(111) (s + )-1= u du, s =0 1

Furthermore 00

(112) E (-1 )-(s + 1 )-1 = 0.693147. s=O

Again the results of Theorems 1-5 are illustrated; although Theorems 6 and 7 concern the whole Pade table, Table 2 serves nevertheless partially to confirm the results of these theorems.

9. Comparison with certain linear transformations. As an alternative to the continued fraction transformation of the series f(z), we may consider linear transformations based upon the formation of weighted means of the partial sums, having the form

r (113) Sm(r) = E Wr ,,m,sSm?+s

s=O

where rn-i

(114) Sm = E ( -1)8csz8. s8=O

If the transformation (113) is to be regular (i.e., limit preserving), then we must have

ri (115) E Wr',m,s = 1.

s0=



830 PETER WYNN

We find, after use of (32), that

(116) "O d't(u) _ sm(r) = I d(u) {E Wrt,m,s(UZ)4

1 + zu 1 + ZU 1s=

If r' is even (equal, say, to 2r) and the coefficients { W2r, ,,} are such thlat the polynomial EZ=o W2V,m,8(-uz)S may be expressed as the square of a polynomial of the rth degree in - uz, coefficients Yo, Yi, ***, Yr can be found, that is to say, such that

2r fr A2

(117) E W2r,m,a(-UZ) { E Y8(-uz)s};

then we have

(118) f 1+d(u) Sm(2r) = 1() (uz)m{Z y8( - uz)s}2 1+ ZUO1 + ZU ls=o

We still have, of course,

(119) Z Y8 = 1. s=0

Comparing the right-hand side of (118) with (48), we see that "O dl d(u) _Sn(2r) >

0 d4 d7(u)

| o1+ zu | o1+ zu (120) (-

1)'CnZmVt Cm)z 2r

[m1 ~J1 1 t=2

We formulate the following theorem. THEOREM 9. The partial sums Sm.X S.+l ,, * Sm+2r , m, r < o, of the

Stieltjes series whose coefficients {c,} are given by expressions (32) and whose argument is real and positive, may be used in, among others, two ways to approximate the value of the function f(z) of (37); first, by means of a continued fraction transformation the Pade quotient Pr,m+r may be derived; second, a limit preserving weighted mean of the form (113), subject to conditions (115) and (117), may be used. The error resulting from the use of the Pade quotient is less than that resulting from the use of the weighted mean.

Perhaps the best known linear transformation of the type being dis- cussed is the generalized Euler transformation (see, for example, [13, Chap. XI, ?11.11]): this, it will be recalled, functions in the following way: given the series E?=o u8 to be transformed, it is assumed that its terms behave like those of a geometric progression with ratio a, so that we may write

(121) u3=,S, =0,1



PADE TABLE 831

where the quantities {I } are approximately constant. The new series produced by the generalized Euler transformation is then

m-1 m oo s (122) E n S + 1_ E T)

S=o 1 - a- S=0 1- - a

It is to be expected that the successive differences Arm , s = 0, 1, *** will decrease sharply in magnitude, and that the infinite series (122) will converge numerically with greater rapidity than the original series. A short calculation reveals that if we write

(123) ZUs = Sm, s=O

then, in conjunction with (121), m-1 m r'-1 '/t \8

(124) E s o-s + E r A(

S=o 1 ? io(1

-aO 8=0 S 1-

If r' = 2r, then we are clearly dealing with a case adumbrated by (115) and (117); we now have

(125) YS= (s) (1) s =0, 1,* * , r.

It is not possible to establish a similarly favorable comparison between continued fraction transformation and the formation of generalized Euler means of odd order: to see why this is so we have only to consider the linear combination of two partial sums

(126) WlSm + W2Sm+l.

Suppose that we have evaluated two consecutive partial sums S', and S', + given by

(127) St f 1 (-zu)t d4(u), t = im, + 1,

of a series whose terms are given by OD

(128) u' (_z)sf us d'(u)

and whose sum or formal sum we know to be

(129) s = [ 1 + zu

We wish to devise a linear transformation of the form (126) which will be



832 PETER WYNN

limit preserving for all series, i.e.,

(130) Wl + W2 1,

and will work spectacularly well for the given series, i.e.,

(131) wlSIM + W2SMr`+1 S'.

We need only take

(132) W'= S'S' - S?'' W2=St S W m'+l m' m'+ Sm

for we then find that both (130) and (131) are satisfied. In terms of the generalized Euler transformation the choice of coefficients (132) corre- sponds to taking

(133) S - 2SSm + + SM, (133) 0-

~ ~S' - Sm"+1

and

(134) (S =-sus s=m',m' +1,

where us' is given by (128). It is clear that the parameter a- and the coefficients Wl, W2 are well-determined quantities; it may also be shown that

(135) w 1 = a W2 =

in agreement with (124) for r' = 1. In summary, we see that if we estimate S' by means of the odd order

transformed sum m'-1 m 1 8

(136) us, + t Em S=o 1 o-a 8=0 1 - a/

where us', (' and a are given by (128), (134) and (133), respectively, then the error of approximation is zero, and therefore no worse than that resulting from continued fraction transformation.

A comparison between continued fraction transformation and the use of limit preserving weighted means of odd order, given by (113), can be based upon the use of (49). The conclusion is that if

(137) W2r+1,m,O = 0

and coefficients Y,, Y2, * * , Y,, can be found such that

2r+1 r Y

A2

(138) E W2r+l {mS(-uz)S= E Ys(-uz) s=1 s=O



PADE TABLE 833

then we have

( Sm(2r+l) - f u

d4u)

_139[ -1) C+ zu; ' ]2-I)mZm C

S=O t =2 ~~ + zu

10. Series which are not of Stieltjes type. If the members of the sequence {c8} are not represented by integral expressions of the form (32), then the determinantal inequalities (33), (35) no longer hold; but the validity of these inequalities constitutes a necessary and sufficient condition for minimum values of (48) and (49) to exist; thus we see that our proof of the optimal property of the principal diagonal breaks down if the Pade table is constructed from a series which is not of Stieltjes type. We can indeed furnish a large class of examples for which the optimal property of the principal diagonal does not exist.

It is true of many divergent asymptotic series of positive terms that they should behave in the following manner: if the function of which the series is an asymptotic representation is denoted by S(z), and the successive partial sums of the series by Sr(z), r = 0, 1, , then for a certain value of z we have

(140) So(z) < S1(Z) < ... < Sn(z) < S(z) < Sn+l(Z) < ***

As the value of z is slightly decreased, then the individual terms in inequality (140) decrease in magnitude and, in addition, the relative position of the value of S(z) in the interval [Sn(z), Sn+l(z)] shifts towards the upper limit. As z decreases further, a critical value z = z' is reached at which

(141) S(z') = Sn+l(Z );

as z decreases thereafter, the position of S(z) in the sequence of inequalities (140) must be moved one position to the right. In the case of such a series there is a denumerably infinite number of values of z for which one of the entries in the first row of the Pade table is completely accurate: the entries in the principal diagonal are no better.

11. Acknowledgment. The above work was carried out when the au- thor was a member of the Mathematics Research Center of the Univer- sity of Wisconsin; it is a pleasure to thank Dr. G. W. Petznick for his help in the computation of Tables 1 and 2.

REFERENCES

[1] A. PRINGSHEIM, Uber die Convergenz unendlicher Kettenbrache, Sitzungsber. der math.-phys. Klasse der Kgl. Bayer. Akad. Wiss., 28 (1898), pp. 295- 324.



834 PETER WYNN

[2] 0. PERRON, Die Lehre von den Kettenbruichen, vol. 2, Teubner, Stuttgart, 1957. [3] G. HANKEL, Uber eine besondere Klasse der symmettrischen Determinanten, Leip-

zig dissertation, University of Gottingen, 1861.

[4] H. PADP3, Sur la representation approchee d'une fonction par des fractions ration- nelles, Ann. Sci. Ecole Norm. Sup., 9 (1892), pp. 1-93 (supp).

[51 T. J. STIELTJES, Recherches sur les fractions continues, Ann. Fac. Sci. Univ. Toulouse, 8 (1894), pp. 1-122.

[6] T. CARLEMAN, Les fonctions quasi-analytiques, Gauthier-Villars, Paris, 1923. [7] E. B. VAN VLECK, On an extension of the 1894 memoir of Stieltjes, Trans. Amer.

Math. Soc., 4 (1903), pp. 297-332.

[8] J. J. SYLVESTER, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, Philos. Mag., ser. 4, 4 (1852), pp. 140-141.

[9] J. HADAMARD, Essai sur l'etude des fonctions donnees par leur developpement de Taylor, J. Math. Pures Appl., 8 (1892), pp. 101-186.

[10] A. MARKOV, Deux demonstrations de la convergence de certaines fractions continues, Acta. Math., 19 (1895), pp. 93-104.

[11] R. DE MONTESSUS DE BALLORE, Sur les fractions continues alg6briques, Bull. Soc. Math. France, 30 (1902), pp. 28-36.

[12] H. S. WALL, On the Pade approximants associated with the continued fraction and series of Stieltjes, Trans. Amer. Math. Soc., 31 (1929), pp. 91-115.

13] D. R. HARTREE, Numerical Analysis, Oxford at the Clarendon Press, 1952



upon the padé table derived from a stieltjes series

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