MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 146
APRIL 1979, PAGES 815-826
The Hankel Power Sum Matrix Inverse and the
Bernoulli Continued Fraction
By J. S. Frame
Abstract.
The m x m Hankel power sum matrix
Vandermonde
matrix)
ing a statistical
has (/, /)-entry
problem
—1
suffices.
As by-products
Hilbert determinant,
nator m
of W
by first factoring
and a convergent
+ (m + 1)
would be less than
W into easily invertible
of the proof,
T
(where V is the m X n
where Sp(n) = 2^—^.
on curve fitting it was required
for n > fim) all eigenvalues
lating W
Sj+j_2(n),
W = VV
to determine
1. It is proved, after calcu-
factors,
that fim) = (13m
close approximations
continued
In solv-
f(m) so that
2
— 5)/8
are given for the
fraction with mth partial denomi-
is found for the divergent
Bernoulli
number
series
XB2k(2x)2k.
1. Introduction. Defined as the product W = VVT of the m x 77Vandermonde
matrix V = (j''1) with its transpose VT, the m x m Hankel power sum matrix W =
Wm has (7, /)-entry Si+J_2(n), where
r-1 I\
(U)
7ZP+1 + 7L
77P + [P/2]
B2k i-^j.
I
P
S,,S,(„) = £" *,_ ¿Lj
£
Jt
\
and where B2k axe the Bernoulli numbers [3], [6] :
(1.2) B2 = 1/6, 54 = -l/30,
B6 = l/42,
Bs = -1/30,
Bxo = 5/66, . . . .
In solving a statistical problem involving the fitting of polynomial curves of degree 777to 77> 777points, for increasing m and n, it was required [4] to find a function
fim), such that, whenever 77> fim), the eigenvalues pk of M = »V-1 would all be less
than 1. We evaluate wm = det Wm in Section 2 as
m
0-3)
wm = det lVm= fcm fi
(» + *-/>.
'./= 1
where hm is the determinant of the Hilbert matrix Hm, and we obtain close estimates
for hm. In Section 3 we factor Wm as a product of easily invertible matrices of which
only diagonal matrices involve 77,and we also explicitly invert Wm and Hm. In Section
4 we estimate the trace of M = W-1 and find that the function
(1.4)
/{m) = (13m2 - 5)/8
suffices for powers oXM to converge when 77> f(m).
As a by-product of this investigation, we find in Section 5 that the divergent
Received October 26, 1977; revised August 4, 1978.
AMS (MOS) subject classifications (1970).
Primary 65F05, 15A09, 30A22.
© 1979 American Mathematical
0025-571 8/79/0000-0078/$04.00
815
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Society
816
J. S. FRAME
asymptotic series B(x) with Bernoulli number coefficients
(1.5)
B(x)=
¿
B2k(2x)2k,
k=l
related to the Laplace transform of x coth x - 1, has the convergent continued fraction
expansion
(1.6)
B(x)
|1 + 1/2 + 11/2+ 1/3 + 11/3+ 1/4 +
In fact, 5(1/12) = it2 - 9.865 is given with error < 2 x 10-12 by the sixth convergent of this continued fraction.
2. The Determinants. Since S0(n) = n and 77(71
+ l)/2 divides Sk(n) for k > 0,
it follows directly that wm = det Wm has the algebraic factor 77m(77+ l)m_1.
For r
< 777,(77- r)m~r is also an algebraic factor of wm, since the matrix Wm(n) has rank r
and nullity 777-7-,when n —r for r = 1, 2,. . . , m - 1. Since the polynomials Sp(ri)
axe generated by the function
(2.1)
G(x, n) = (exn - 1)/(1 - e~x) = ¿
Sp(n)xplp\,
p=0
we see from the identity
G(x, n) + G(-x, -77 - 1) + 1 = 0
that
(2.2)
Sp(n) + (-lfSp(-n
- I) + 8p>0= 0.
Hence, wm has the algebraic factor (77 + 1 + r)m_1_r. We have found tt72 linear functions of 77as factors of the polynomial wm(n) which is of degree t?72in n. The re-
maining factor is the determinant of the leading coefficients l/(/ + / - 1), namely the
determinant hm of the ill-conditioned Hilbert matrix Hm of order m [5], [7]. This
proves Eq. (1.3).
If we take 777= 77in (1.3), the Vandermonde matrix V in W = VVT is square.
Its determinant vm is
(2.3)
vm = det Vm = 1! 2! 3! • • ■(m - l)\ =(m-
1)!!.
Hence by (1.3) and (2.3)
(2.4)
hm=detV2m /n
i.m + t-f) = v*Jv2m.
The ratio of successive hm's is
(2-5)
Klhm+i = (2m+ l)!(2m)!/(m!)4
= (2m+ 1)I ^J .
The following theorem gives a close approximation for hm.
Theorem
2.1. The determinant hm of the Hilbert matrix Hm of order m is
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
HANKEL POWER SUM MATRIX INVERSE
817
given to 10 significant figures for m > 4 by
(2.6)
hm = 4~m(m-1\Til2)m-lm-ll4exv
where the remainder function
(2-7)
Rm,
Rm is defined by
Rm = J7 (e~2t -e~2mt) tanh2(t/2)(4ty1 dt
and is approximated to 9 decimals for m > 5 by
Rm = 0.013081539 - 2-6m-2
(2,8)
+ 2"8m~4 - 2"8sm_6
+2-8m-8-2-7m-10.
Proof.
for 77by
The ratio hm/hm + x in (2.5) is related to the Wallis approximation n m
n Q\
5» _ 1 1 i i. . .
K '
21335
2m
2m-12m
2m _ 4m
+ l
ftm+ i/«m-
If we express In 77in the form
(2,o)
ln„ = j;
£„££
e-^=j;(e-<-e-»>-^
then ln(7Tm/2) and its limit ln(7r/2) are expressible as
(2.11)
ln(7Tm/2)= rV'
J°
- 2e~2t + 2e~3t - 2e-2mf + e-(^
= r(e-f-e-(2m
+ 1)f)(l-e-f)(l
+ i)t)t-i
d{
+e~trlrldt,
J0
(2.12)
ln(rr/2) = j"~ e"f tanh (f/2)f_l di,
(2.13)
ln(7r/7rfc)
= J*~ e-(2fc+1>f tanh (t/2)rl
(2.14)
(ir/2)'"-1 n
dt,
m-l
(irkl*) = 22m<m-1>hmlhl.
fc=i
Summing in (2.13) from fc = 1 to m - 1 yields
„ 1C, toI(ff/2)m-12-2'B<,*,-1>/Am]
(2.15)
= r (e~2t - e-2mt)(e^2
J°
+ e-^2y2r'
dt
= (l/4)ln(2m/2)-/?m
by (2.10), where Äm is defined by (2.7). Equation (2.6) follows from (2.15). To obtain (2.8) we evaluateR4 = .012119610988 from (2.6) setting ä4 = 1/6048000 in
(2.6). Then we compute R„ - Rm from (2.7) by replacing tanh(r/2) by the first five
terms of its series, and set m = 4 to get R„ in (2.8). We check the tenth decimal by
working from h5 instead. This gives expRm and hm accurate to 10 significant figures.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
J. S. FRAME
818
For m = 20 we find R20 = .0130425009 and
h20 = 4.206178954 x 10~226.
(2.16)
The matrices Hm and Wm(n) axe ill conditioned. In fact, IV3(3) has the eigenvalues
\x = 113.4132, X2 = 1.564253, X3 = .02254695 and the conditioningratio X,/X3 =
5030. So the usual computer methods for inverting Wm(n) are unreliable [5], [7].
3. Inversion by Factoring.
To invert the ill-conditioned m x m matrix W =
yVm(ri)with (/', /)-entry Si+i_2(n), we first factor it into easily invertible factors, restricting the variable 77to diagonal matrix factors EP = (diag e¡p¡) and Q = (diag qj),
where
(3.1)
e, = (-ir\
In + i - 1N
n -]
n- m
n - m
Pi
We denote by T = (tu) the lower Pascal triangle matrix with
7-l\
(3.2)
/i-l\
(-iy+/l
'// =
'-/
We note that ETE has entries (."(), so
(3.3)
(TETE)(j= ti']
k=j V - *.
-/
i-j-l
k-j
i-j
h
and T~l = ETE. Next, we define an m x m lower triangular row stochastic matrix
A = (aA that converts the integral powers in V into binomial coefficients by the formula
(AV)ik=Z
(3.4)
airV~l =
r=l
k + i-2
*=1,2,
«'"I
The air are related to Stirling numbers of the first kind, and
(3.5)
0
0
0
0
1
0
0
0
1/2!
1/2!
0
0
2/3!
3/3!
1/3!
0
6/4!
11/4!
6/4!
1/4!
= («*/)•
From (3.5) and (3.4) we obtain
(3.6)
(^
_('
+ *-*
■¿r;
k- 1
/-i
To factor W we now write
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Jt-l
-i.-v.>
HANKEL POWER SUM MATRIX INVERSE
,
-r,
819
A fk+i-2\lk-l\
(3.7)
i+/-2\
» lk + i-2
7-1
/ k=j V/ + /-2
Summing over k yields
-i
r(AwcrlA)\
i ' + / - 2\
(n + i- I
\
\; + I - 1
i- 1 /
(3.8)
f/ + / - 2\ ¡m + i-l
,-. ;v»+i-iF*
Theorem*3.1.
(3.9)
77/(2inverse matrix M = W~l has the factorization
M = W_1 = ATEBEA, B = TTQTTTTP~1,
where E, P, Q, T, A are defined in (3.1), (3.2) and (3.5).
Proof Entries of TTT and TT7"?/are
(3..0)
,~-.t
//- l\ /fe - l\
li + k-2
(n^£X C-J
,-,)- (,-,
^^■¿rap-ctMHc:;:.1
Summing over fc yields
(3.12)
T~~ = / ' + / ~ 2\
iTTrT)
i-I
/"Í +1-1
I V/ + 7-1
Combining (3.8) and (3.12), we have
(3.13)
AWAT =PTTTTQ~1TT.
Since the diagonal sign matrix E commutes with P and Q but transforms T and TT into their inverses,
(3.14)
(AWAT)~i =ETTQTTTTP-lE
= EBE
and (3.9) is proved.
Equation (3.12) provides a simple method for inverting the Hilbert matrix H.
Theorem 3.2. The inverse of the m x m Hilbert matrix H = (h¡A with (i, /')entry h¡¡ = l¡(i + j - 1) is given by
(3.15a)
(/r1),.,. = d^h^d),
l-m - l\
,.,
(m-I
)'
di = m\j-x
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
820
J. S. FRAME
Proof Factoring (3.12) yields
„
. . Im + i - l\
(3.16) iETT-n^i-l)-^
i_i
m
/m - l\
j-S-^J-rf^
Since i'TT^r is involutory, £>'/¥£)= (D'/YD)-1. tVIso,
/m
(3.17)
/7m=det/Ym=±l/n
»=i
rf,4
Although the matrix /J in (3.9) is symmetric, its symmetry is not obvious from
formula (3.9).
Theorem 33. 77?esymmetric matrix B = (èy) in (3.9) has entries expressible
in terms of descending factorials (jc)r = x(x - I) • • • ix - r + I) as follows:
(3m
h
Y
(3-18)
'''J?»
where (,?"/.)
(m+i-l)r(r»+J-Vr(
r-1
(n-m + r)S.
\
[j-ij-J'
is the trinomial coefficient
r-1
\ li+j-2
7+7-2/
\
/— 1
Proof. To transform ß in (3.9) we evaluate
(rw^i
(;:;)(;;^)(r;:;2
(3.19)
=^
\
+ S~2\y
A + s-2\/77-r
i-1
\i + s-2)\n-m
/;£}
i + s - 2\ /« + s - 1
i-l
"> /í + s-2\
»*-£.(«-.
/ i
In +s-
m- i
l\/s-
1
)(-*)(,-!
(3,0) -('!':*)£
('f:1) )r>i+j-iz (\r-i-j s"'+,)("++'V
\ 7-1
/ s=j \ ¡¡-I
+ 1/ \m +;1 -r
r>f+/-i
// + J-2W
*=/ \
r-1
r-1
\/n+/-l
/\i-l,j-l/\n-m
+ r/
Summing over s and dividing by p., we have
""*-^.f+rOU,Jf,T,)/f":i
Writing (x)r = r!Ç), Eq. (3.21) becomes (3.18).
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
HANKEL POWER SUM MATRIX INVERSE
821
To conserve space in displaying the symmetric matrices Mm(ri) = Wm (n) we
show the upper half of M3 and the lower half of M4.
(3.22)
16«3 + 24n' + 56« + 24
9(n2 + n) + 6
-18(2« + 1)
(»),
(»)a
(")4
-120(«2 + «)- 100
1200(«4 + 4.5n3 + 7n2 + 5« + 11/6)
(24n + 12X8«+ 1)
-180(« + 1)
(« + 3),
(n + 2),
(n + 2)5
("),
120(2« + 1)
-140
280(6n2 + 15« +11)
(n + 3),
(«),
4. Estimation of trM
180
360(2« + 1X9" + 13)
-300(n + 1X3« + 2)(3n + 5)
(« + 3),
(«)4
30
(«)3
(« + 2)s
(« + 2)5
-4200(n + 1)
2800
(« + 3),
(« + 3),
Since W and M are positive definite for n > m, all eigen-
values pk of M will satisfy jufe< 1 if tr M < 1, m > 1. For .4 in (3.5) and e¡ = (-1)1-1,
the first and last diagonal entries of M = Mm are bx x and bmmj((m - l)!)2. Numerical computation shows that the maximum 77for which det(Wm(77)-1) = 0 axe given
for m = 1, 2, 3, 4 by
(4.1)
(m, 77)= (1, 1), (2, 5.82090), (3, 13.3776),(4, 24.24453).
The parabola through the first three points is
(4-2)
77= gim) = 1.5679m2 - .0828m - .4851,
and we find g(4) = 24.270 > 24.24453. A slightlyhigher value than (4.2) will be required for tr M < 1. We first estimate the dominant diagonal entry b x, of M.
77\
m)
m
U
/m\ 2 /n\
r=i
\r)
/n-m
\m//\
+ r\
r
/
22,
Im
r=i
\r
n
(4.3)
77+ m\
m I
n + m
l+bxx
(4.4)
(4.5a) !„(! +bxx)=
=
m
In
\m
n\
"L
2n+ I +(2k-
m)~
k= i
2n + l-(2k-l)
»>
1 + (2k - l)/(2n + 1)
£ Ink=l
(2k - l)l(2n + 1)
=Z
1)
6(m, r)
r=i (2r - l)(2n + 1)\2r-1'
where
m
(4.5b)
0(m, 7")= ¿
2i2k-l)2r~l<
f2mx2r-1 dx m i2m)2r\2r.
k=i
We now assume the inequalities 77> /(m) in (1.4).
Theorem 4.1. 77i»?matrix M = W~liri) has trace < 1 if
(4.6)
77> 1.625m2 - .625
and
m>5.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
822
J. S. FRAME
Proof
If (4.6) is satisfied for m = 5, then 77> 40, and
(4.7a)
bx 1 < (45)s/(40)5 - 1 = 62639/73112 = .856754.
If (4.6) is satisfied for m > 6, then (2t? + l)/2m2 > 1.6215 and
2m2
ln(l+èn)<
°L
2m
\2»-2,'
——
Z
277+1
,= , \ 277+ 1
r(2r - 1)
(4.8)
<
(4.7b)
1
/
£
1.6215 "1
ft1,<.8734
1
\2r-2
izTZz)
\9.729
/H2r-l)<.
62777,
for m > 6,77 > (13m2 - 5)/8.
The rest of tr M is given by
m
m
txM-bxx=Y, z «*(-i)V-i)V
(4.9)
k=2
i,j=k
We replace i, j, r by 1 + 1,7 + I, r + 2 and write
(410) b.
=V
yfi) v« -
(w + ,W"+/)r+i
(7j-m
k=l
(r + l
+ 7-+ 2),+ 2(r + 2)! \ 1, j
Then
2m-l
(4.11)
trJf-*n-jriV
r+1
£
*MII(r), *mn(r) =
r=l
£
c^ftyvft,
i'+/=2
where the entries of the (m — l)x(m-l)
matrix C = (cy) are
m-l
(4.12a)
cij _ ( 1)
2_< ai+ 1,kaj+ 1,k ~ cjh
k=l
(4.12b) C =
720
-360
240
-180
144
-360
360
-300
255
-222
240
-300
280
720 -180
255
-255
242.5
-228.5
144
-222
233
-228.5
220.1
-255
233
The dominant term y[^ satisfies
(4.13)
/i)<
11
(« + 0,(m + Da/3 < &£
(13m2/8-m
+ 19/8)3
(38)3
200
< .094832,
2109
since the rational function decreases for 5 < m. The function <pmn(l) is 1, but for r
> 1, then ymn(r) in (4.11) are bounded by rational functions which increase for
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
HANKEL POWER SUM MATRIX INVERSE
823
m > 5, and which we replace by their limits as m —►°°.
*m„(2)= Cv(,V-y(iV)ly(i\)= 3(1»- 2)(m- 6)/(13m2- 8m + 27)
(4.14a)
< 3/13 = .23077,
^mn(3)- (ÄV -VW + 2>'(,33)/3
+y(232)l2)ly\\)
(4.14b)
< 17(m2 - 6m + 32)(m - 2)(m - 2.4)/(120)(13m2/8 - m + 35/8)2
<(17/120)(8/13)2 = .05365.
Similar calculations yield
(4.14c)
^mn(4) < (l/32)(8/13)3 = .00728.
Since the coefficients of (8/13)r_1 in <pmnir) decrease as r increases, the remaining
sum of <pmnir) is < 2.6<pm„(4). Hence, (4.11) implies
trJtf < .8734 + .095(1.23077 + .05365 + 3.6(.00728))
(4.15)
< .8734 + .095(1.3107)< .998 < 1.
This proves Theorem 4.1. We check directly for m - 2, 3, 4 that
trM2(6) = 97/105, trM3(14) = .95 + 1/7280,
trM4(25) = .87755 + .09359 + .0073 + .0000005< .9719.
This proves the parabolic bound 77>/(m) = (13m2 - 5)/8 to be sufficient for trM
< 1. Although some bound between this and 77> g{m) in (4.2) might also suffice for
all 77,the tight inequality (4.15) indicates that it would be difficult to prove.
5. The Bernoulli Continued Fraction.
The entries Si+i_2in)/n of the matrix
Wm(n)/n have as constant terms the Bernoulli numbers Bi+¡_2 given in (1.2). The
limit as 77—►0 of the leading principal minor of Wm(n)/n is the determinant bm_x
of order m - I expressible as
(5.1)
**_, = det(Bi+j) = lim (nbx x)(n-mwm(n)).
«-►0
Recalling bxx from (4.3), wm(n) from (1.3), vm from (2.3) and 7im from (2.4), we
have
(5.2)
(m\ l( -1 v
lim nbxx = I )m/(
I = (-\f-lm,
n=o
(5.3)
\m/
lim«-'"wm(77)
I \m - 1/
= «m,m(-ir<m-1>/2,
«= 0
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
824
J. S. FRAME
(5.4)
¿m_1=(-l)(m-1)<'"-2)/2mu>2m,
(5.5)
b*Jbm_x = (-l)m-'(m
- l)!(m!)4(m + l)!/(2m)!(2m + 1)!.
Since Bi+i = 0 for odd i + /, we can rearrange rows and columns of the matrix
(t3i+ ■)so the odd numbered ones precede the even numbered ones, and thus factor
bm_x as the product dm_xdm_2 of two determinants, where
B2
BA
B 2k
B*
75,
B2fc+ 2
B2k
"2k+2
B.4fc-2
B*
B.
B.2k + 2
B6
BS
B2k + '
"2k+2
B2k + 4
73,
4k
*2fc-l
(5.6)
d2k =
(5.7)
dmldm-2 = ô»Vèm-l>
-dm-3dJdm_xdm_2
= (m - l)m4(m + l)/(2m - l)(2m)2(2m + 1)
(5.8)
= (l/4)((m - l)m/(2m - l))(m(m + l)/(2m + 1)).
Theorem 5.1. 77z<?
divergent asymptotic alternating series
(5.9)
B(x) = X
B2k(2x)2k = 4*2/6 - 16x4/30 + 64*6/42 • • •
k=l
has the convergent continued fraction expansion (1.6).
Proof. By the general theory of continued fractions [2], [9], if a formal power
series (5.9) with arbitrary coefficients B2k is expanded into continued fractions of the
form
(5.10)
«i(2*)2l
a2(2x)2
x2/c0\
11+
11+
\cx +
x2 I
\c2 +
and if the dks axe defined by (5.6), then
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
x2 I
\c3 +
HANKEL POWER SUM MATRIX INVERSE
(5.11)
825
am = l/4cm_xcm = ~dm_3djdm_2dm_x,
m>l.
For the Bernoulli series Eqs. (5.5) and (5.11) imply
(5.12)
cm =(m(m + l)/(2m + 1))_1 = 1/m + l/(m+
1), m > 1,
while the condition l/c0c, = 45, = 2/3 implies c0 = 1. Since ¿Zcmis divergent, the
continued fraction (1.6) converges, and Theorem 5.1 is proved.
We can apply this continued fraction to approximate 7r2. It would require about
a billion terms of the series 2"(l/rC2) to approximate tt2/6 to nine decimals. But the
Euler-Maclaurin summation formula gives the remainder after 5 terms by the expression
(5.13)
r
x-2 dx + 1/2 • 62 + ¿
J6
52fc(l/6)2fc+1.
k=i
This alternating series diverges, with minimum remainder of about 10_1 s after the
19th term. Using the convergent continued fraction instead, we have
Tr2= 6(1 + 1/4 4- 1/9 + 1/16 + 1/25 + 1/6 + 1/72) + 5(1/12)
(5.14)
= 9865 + _I2_l_J
|1 + 1/2 +
12-2 1 _12-2
|
11/2+ 1/3 + 11/3+ 1/4 +'
rr2 = 9.865
(5.15)
1/12 |
|12 + 6 +
1
I
|6 + 4 +
1
1
|4 + 3 +
1 I I
|3 + 2.4 +
1
|2.4 + 2 +
1
1 1
\2 + r '
where the sixth convergent with r = 12/7 has an error about 10_1 , and the tenth
convergent (which changes this r to 1.9976) has an error less than 10_1 s, giving 7r2 =
9.869604401089359.
The function s~iB(s~ï) is the Laplace transform of x cothx - 1.
Continued fractions for the Laplace transforms of tanh x, sech x and x csch x can
also be obtained by similar methods, but have already been derived by Stieltjes [8] and
others, and are listed by Wall [9, p. 369]. The author has not found the continued fraction (1.6) in the literature, nor the determinantal formula (5.4) which evaluates the first
principal m x m minor bm = |B/+.|, i, j = 1,...,
(omitting B0 andBx) of the determi-
nant \Bi+j_2\of order m + 1 called Am(5) by Al-Salam and Carlitz [1, p. 93,(3.1)] which
in the notation of (2.3) becomes
(5.16)
Am(5) = (-ir(m
+ 1)/2(m!!)6/(2m+ 1)!!.
Comparing (5.16) with (5.4) for order m, we have
(5.17)
\Bt+i\m =(-iy"(m+
l)|B,+/_2lm + 1.
Department
of Mathematics
Michigan State University
East Lansing, Michigan 48824
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
J. S. FRAME
826
1. W. A. AL-SALAM & L. CARLITZ, "Some determinants
numbers," Portugal Math., v. 18, 1959, pp. 91-99.
2. J. S. FRAME, "The solution of equations by continued
of Bernoulli, Euler, and related
fractions,"
Amer. Math. Month-
ly, v. 60, 19S3, pp. 293-305.
3. J. S. FRAME, "Bernoulli numbers modulo 21000," Amer. Math. Monthly, v. 68, 1961,
pp. 87-95.
4. D. C. GILLILAND & JAMES HANNAN, Detection of Singularities in the Countable General Linear Model, Department
Aug. 1971.
of Statistics,
Michigan State University, RM-217, DCG-8, JH-10,
5. E. ISAACSON & H. B. KELLER, Analysis of Numerical Methods, Wiley, New York, 1966,
pp. 196, 217-218.
6. N. E. NÖRLUND, Vorlesung über Differenzenrechnung,
Springer, Berlin, 1924, p. 18.
7. G. M. PHILLIPS & P. J. TAYLOR, Theory and Application of Numerical Analysis, Academic Press, New York, 1973, pp. 91, 246.
8. T. J. STIELTJES, "Sur quelques intégrales définies et leur développement
en fractions
continues,"
Oeuvres Complètes, vol. 2, P. Noordhoff, Groningen, 1918, pp. 378—391.
9. H. S. WALL, Analytic Theory of Continued Fractions, D. Van Nostrand, Princeton,
N. J., 1948, pp. 369-376.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use