THE RICE INSTITUTE
CONTINUED FRACTION FORM
FOR PERTURBATION SOLUTIONS
ly
ROBERT C. YOUNG
A THESIS
SUBMITTED TO THE FACULTY
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTER OF ARTS
Houston, Texas
May 1957
TABLE OF CONTENTS
Introduction
1
Perturbation methods
2
Goldhammer - Feenberg modification
6
Continued fraction
9
Example I
16
Example II
16
1
Introduction
Approximation methods have long been important in physics and
mathematics) since only a comparatively few problems can be solved exactly.
For example) the motion of the planets presented at an early date the
many body problem, which can only be solved approximately.
In addition,
it is sometimes more convenient to employ approximate methods for prob¬
lems which can be solved exactly, for the evaluation of the exact solu¬
tion may be much too complicated*
Let us consider a situation which may be approximated by an
exactly solvable problem.
The deviation from the exactly soluble prob¬
lem will be referred to as a perturbation.
It is assumed that the
perturbation may be increased gradually from zero to the desired magni¬
tude,
This is expressed analytically by requiring the perturbation to
be a continuous function of a parameter \, which measures the strength
of the perturbation.
The question of convergence is of great interest for any practical
application of perturbation methods.
only a few terms need be taken.
If the series converges rapidly,
If the perturbation is large, the
simile series converges slowly or not at all, and more elaborate ex¬
pressions are required.
For a given perturbation, an improved expansion
provides a certain accuracy with fewer terms; or, conversely, larger
perturbations may be handled with fewer terms than the number required
by a more elementary expression.
The present work considers perturbation methods for the solution
of eigenvalue problems; in particular it is concerned with an accurate
determination of the eigenvalues.
the purpose of comparison.
Several methods will be presented for
The first method to be considered was
2
(9)
introduced by Lord Rayleigh in 1873 , and later generalized by Ritz.
,(11)
The technique was applied to quantum mechanics by Schroedinger in 1926 ,
and is still in wide use.
Another procedure, which obtains a more general
form for a series expansion of the eigenvalue, is the Brillouin-Wigner
method
(1 9 13)
(4)
. This method has been modified by Goldhammer and Feenberg
,
who introduce additional parameters and adjust them to obtain an improved
series for the eigenvalue.
Our principle result is to show that this
resulting improved expression is identical to a continued fraction
expansion equivalent to the original series.
in connection with the problem of convergence.
This result is of interest
The methods described
are illustrated by tiro examples.
Perturbation methods
We shall consider problems defined by an equation of the form
H ijr(x) = E t(x)
(1)
where H is a differential operator, and both the function \jr(x) and its
corresponding eigenvalue E are to be determined, subject to certain
boundary conditions.
For the perturbation calculation, the operator H
is decomposed into a basic part, and a perturbing part, and will be written
H = HQ + \V
,
where the perturbation \V is smaller than HQ .
H u
o
are assumed to be known.
k
(2)
Solutions of the equation
= E u
k
k
For reasonable operators, HQ and V, \]r will be
well behaved and can be expanded in a series of the unperturbed eigen¬
functions u .
If the u, are non-degenerate, as the perturbation is
k
k
3
decreased to zero, a solution will approach a particular unperturbed
function u .
a
Hence one may write
«
t =
u
+
a
xX
»
2
\
+
\
k
X
^
b
k
\
+
which explicitly gives \jr as a power series in X.
allowed values of k.
• • •
(4)
k
Each sum is over all
The prime means that the value k = a is excluded*
Similarly,
E=E+e. +e_ + e„ + ...
a
1
2
(5)
3
The most straightforward procedure at this point is the RayleighRitz method^’"^, according to which one substitutes these expressions
into the eigenvalue equation (l).
The expansions (4) and (5) are sub¬
stituted into this equation and the coefficients of each power of \
equated to zero.
H
o
The first two equations are
u
a
= E„ u
a a
t
X
^\ak“k
+
XVaa
+
= X
a
(6)
U
k
The first equation is seen to be satisfied identically for this nondegenerate case.
In the degenerate case, this equation will contain
more terms, corresponding to other functions belonging to the same
eigenvalue E .
In this event, the first equation can be manipulated
a
(5)
to give the zero order combination appropriate to the perturbation V
.
Multiplying the second equation of (6) by u * and integrating
s
over the range of variables gives
X as Es +
kju
s
*
v
a
dx = X E& ag + e,| 6
sa
(7)
4
which yields
e
1
=
y ‘a*
T u
a
4,1
= <a|XV|a^>
l>,= <s/XT|a> (Ea - Eo)'1
(8)
s 5= a
This notation will usually be further condensed by setting
<UalXVlUb> =
T
ab
(9)
*
In a similar manner one obtains
2
2
X
=z
’Tak
E
k
T
a -
ka
A
V
V
aa sa
' Vsk V
bs = x
Y (Ea-Ek)(Ea-Es)
=
^ " ta
km *ma
(VEkXW
(7)
An alternate method'
(Ea-Es)2
_ y
ak
ka
aa
k (Ea - %)2
' makes use of the Green’s function
uk(x) uk*(x0)
G
E
(x,x0) = 2
' '
T
(11)
E - Ek
which corresponds to the operator
conditions.
(10)
HQ
- E and the appropriate boundary
Then, since
(H -E) >!/■ = - X V
(x,xQ) V(x0) \V(xo) dxQ
= X
k
<uk|V|\V> uk
E
-
\
If we require again that Sjr be normalized so that the coefficient of
u& is one,
(12)
5
<uklVjfr> %
t = ua +
(13)
E - %
Upon multiplying the differential equation for \Jr (l) by uatt and integrating,
one obtains
<ualHo+XVl^ =
= ^alE^> =
E
(U)
E = Ea + <ua|\V/^ >
These expressions are exact, but depend on the unknown function
^1/
may be determined explicitly by an iterative procedure
*(n) = u a + z
E
-Ek
k
(15)
Ur
Then if ^°^ = u& , several iterations yield
T!T
(n)
- ..
"
+
Ua +
V
f
E^
Vkm V,ma
^
ka
k +
+ ^
M (E^JCE-Ea)
^km ^mp * * ^za
u,_
+
k
^
kj^.z (E-Ek)(E-Em)..(E-Ez) ^
(16)
*
The last sum has n products in the numeratorj the prime on the summation
symbols again signifies that none of the indices takes on the value a.
The indices take on all other integral values independently.
The corre¬
sponding expression for the eigenvalue is
E
(n)
= E
a
+ V
aa
+ £o + £q +
<
J
(17)
in which
•jr' W
2
k
®- %
_ 5- ' Vak Tfa, Vma
3
etc.
ta
(S-EicKS-V
The last term in (17) is
(18)
6
(1,13)
series for the eigenvalue
The more accurate Brillouin-Wigner
may be obtained by inserting the same approximate solution (16) into the
variational integral
E
_ ^>lfl Hi
The form of the result is identical to the previous equation (17), but
extends to order 2n+1 rather than to order n+1.
E in the energy denomi¬
nators is to be identified with the approximate value of the eigenvalue
given by the variational integral.
is an implicit function of E.
Thus the Brillouin-Wigner expression
The Rayleigh-Ritz formula may be obtained
from this by writing out the E in the denominators to a sufficiently
high order and then expanding these denominators in powers of
X.
The evaluation of the eigenvalue from (17) has perhaps become
more difficult, but convergence can generally be expected for larger
values of
X.
This is an advantage, even for small values of
fewer terms are needed for a given accuracy.
X,
because
If the sums are to be
performed by a computing machine, the computation is, however, actually
simpler than before, since there are fewer types of sums.
Goldhammer-Feenberg modification
(4)
Recently Goldhammer and Feenberg
have described a refinement
of the Bri11ouin-Wigner scheme that further improves the accuracy and
rapidity of convergence of the series.
The approximate wave function
is replaced by
(20)
7
The expression for the eigenvalue becomes
* = \ *
T
aa
+
<2 *1 "
A 2)e
1
2
+
2 A2 *
(A 2 +
1
A1
2
A )c
2
3
+
+ (2 A3 - A22 + 2 A1 A2 - 2 A1 A3)e^ +
+ (A22 + 2 A^ + 2 A1 Ag - 2 A1 A4 - 2 A2 A3)e5 + . .
• •
+
^-1 “ fn2)e2n
+
e
2n+1
Under the substitution A. = 1 + K this becomes
i
i
2n+1 n
E = Ea + Vaa + 2- + 2 2 %
i=2 i=1
e
i-hi+1
+
S(KI,K2,
in uhich S is a homogeneous quadratic function of the K's.
•
• KJJ)
(22)
Now since
E obeys a variational principle, E may be improved by obtaining an ex¬
tremum with respect to each of the new parameters.
derivative of E with respect to each of the
Setting the partial
equal to zero gives the
set of equations
W1 ■ -
2
•
i
= 1.
2, .
.
(23)
n
which has the explicit solution
% =
K
° =
K
-j
z v~1
Vj
-
= 0
= -1
>
i = 1, 2| • •
3
=
% 2, •
•
n
•
(24)
Here A is the determinant
e
2
e
3
* * *
£
n+1
e
3
A
(25)
e
n+1
e
n+2
•
•
•
e2n
/
and
F
■ Li
is obtained, from A by the substitution of
for the kth column of A.
leaves the rsluit^,
a
• ••
W
e.2n+1
This is essentially the form in which Goldhammer
A superscript on A will be used here, where ambi¬
guity may arise, to designate explicitly the order of the determinant,
,/ ¥
< The denominator of the
can be written conveniiently as a single
determinant
■2
£
„(“) _
/.
£
0
•
3
£
n+2
(n) . £ >)
i=1
(26)
i
/f
n+1 ’ * 2n+1
1
1
..
1
Thq expression for the eigenvalue can be simplified by noting that
n
n
/
2
$1
Ki Ei+n+1
K, II- = - 2 S
" " gi "i 3%
(27)
/ by Euler's theorem on homogeneous functions. Thus if
E
=
\ *
v
aaa
"
2n+1 n
E<" S
)=
+1 E (n)
(28)
h +
Collecting over a common denominator, E
E
(n)
=
(n)
can be expressed in the form
N<n>
^T
is the quantity defined in (26), and the numerator may be written
as the determinant
(29)
9
e
"h+2
2
(30)
N
e
* *
0
e2 (e2+e3) . •
n+1
e
2n+1
n+1
For most problems requiring this additional accuracy, a computer would
be used to evaluate the e^, and the subsequent evaluation of these deter¬
minants would require little additional labor.
Continued fraction
It will now be shown
that the expression (29), with (26) and
(30), is identical to a continued fraction expansion of the BrillouinWigner series(17).
few definitions.
For this purpose, it is convenient to introduce a
In place of the numerator and denominator used above,
let
= N(n)/A<D)
B* = D(n)/A(n)
(31)
The nth approximant to the eigenvalue is then
E<n)
=
(32)
Next one introduces the terms
A^l = H^VA'^
8 = DW)AW)
where
(33)
10
s
e
01+2
£n+3 • * ’ £2n 2|
+
n+2
3
N
n+3
M)
^2
(£2+e3) *
£
*
•
e
3
D
6
4
4
n+3
(n+i)
e
n+2 en+3
1
. . .
1
e
* * * £2n+2
e
3
1
^+2
4
(n+i)
, A
01+2
en+3
(£) ,
s
= 1 . (34)
• * • e2n+1
This permits an additional series of formal approximations to the eigen¬
value
e(D+H
= AWBa*i
<35 >
which will presently be identified as approximations intermediate between
the Ev(n)' of successive order. Finally one introduces
A(n-1 )A(„+»)
20
A(n)
A(n-§)
Afc+1) A(n4)
2n-1
A (n)
.(n+M
’ *1
= E
2 ’
These definitions may now be utilized to investigate the recursion
(36)
11
relations obeyed by the A's and B's.
For this purpose, it is convenient
to first investigate the properties of determinants of the types which
make up the A's and B's.
Let
be defined as
• • •
£2
^+2
e
3
•
•
(37)
£
n+1
a
2
where the
CL
6
_
n*2
• • •
cc^ • • •
£
2n+1
°n+2
One may write the product, A
are arbitrary.
(n-4-)„(n)
X
, as
a single determinant
e
3
e
4 * * ^+1
0
• •
0
0
• •
0
* *
E
e
* *
E
0
• •
0
0
•
. . 0
e
• •
s
0
. . 0
e
• •
e
a
a
e
4
n+2
•
•
(
a
n-i)x(n)
=
n+1
a
3
a
4 * *
2n-1
n+1
(38)
2
n+1
2
♦ •
n+2
2n+1
°n+2
In this determinant, add row n to row 1, row n+1 to row 2 , . . . row
2n-2 to row n-1.
result is
This, of course, leaves the value unchanged.
The
12
e
•3
•
* *
e
e
* *
e
e
* *
e
e
* ’
e
0
•
•
. .
o
e
• * *13+2
0
. .
0
e
. .
a^i
a
n+1
a
3
2
n+1
2n-1
n
2
n+1
* *
2
n+2
2n
(39)
e
2n+1
• • °n+2
Now make a Laplace expansion using rows 1 thru row n-1 and row 2n for
one set of determinants.
It will be seen that, because of either null
or identical columns, only two terms enter.
A(n-4) x(n)
These are of the form
s x(n-4) A(n) _ x(n-1) ^(n+i)
(40)
with
n+2
X (n-W _
(41)
n+1
e
2n
°h+2
Consider now the A coefficients,
A
2n-1
+ a
2n
A
2n-2
N(n-4)
A(n^)A(n-^)
^
N(n-1)
^(n-i")
A(n)
*
A(n-1)
= (A(n4)A(n))"1
A(n-4)
j^A(nVn^}
-A^N(n"1)]
i-1
If we identify the a. in (37) with 21 e. , the bracketed term in (42)
1
(n-JO (n) k=2
is, from (40), just Av "^'N' . In a precisely similar way, setting
(42)
13
cc^ = 1, we get the result for the B's.
A
A
B
= B
2n “
2n
2n-1
2n-1
+ a
2n
A
+ a
2n
B
2n-2
(43)
2n-2
(n+1) (n*rf<j|')
We also need the product AX, which may be written as
the determinant
fc
u
» • £ ,o
0
. . 0
0
. .
n+3
(n+1)
A
X
0
n+2
'3 * *
•
•
(n-40
“n+2
e
* *
2n+2
• • 0
. . 0
a.
0
£*3
(44)
n+2
n+1
'2n
a
h+2
n+3 3
Subtract column 1 from column n+2, column 2 from column n+3, • • *
column n from column 2n+l to obtain
* *
e
n+2
~n+3
A
A
(n+1)
"e2
-£,
•
•
—6
n+1
. -£n+2
(n-J-)
X
. £
n+2
0
.
. o
.
. 0
2n+2
a
n+3
—£
'2n+1
-e
n+2
n+2
e
n+1 ‘ *
0
. .
£
2n
0
Next, make a Laplace expansion using columns 1 through n+1 for one set
(45)
14
of determinants.
As before, only two terms enter so that
e
3 * *
£
n+2
. e.n+3
A(n+1 )x(n-£)
. A(n)X(n+i)
_ A(n+|)
(46)
. e.2n+1
-
n+1
°n+3
This is not quite the desired result, however, since the last row in the
(n)
determinant written out in (46) is not correct for X
the two special cases we desire, namely
= 1, and
GL
in (37). For
i-1
= 21 ej »
result nevertheless follows. It is obviously so for a* - 1, but for
i-1
a. = 21 e4 one need only subtract row 1 from row n+1 to obtain the
J
2
desired result
k(n)x(n+|-) =
^(n-i-i-)x(n) _ ^(n+1 )x(n-£)
(47)
Then, since for the intermediate case
A
2n
+ a
A
2n+1
N (n)
A(n)
2n-1
A(n+1 )A(n-4) N(n-j~)
A(n)
A(n+f)
^(n-4r)
L(n+tVn). A
(n+1
=
the bracketed term is just A'
y v
N
vn^J
(43)
, and the desired recursion relation
is found to be
A
2n+1 ~
B
2n+1
k
2a
= B
2n
+
a2n+1
+ a
2n+1
A
2n-1
B
2n-1
(49)
15
where the relation for the B's again follows in a completely similar
manner.
Relations (43) and (49) with (36) are just the relations ob¬
tained for a continued fraction expansion of the Brillouin-Wigner series
(17), and suffice to prove^ that the quantities A /B = E^ and
(n+&) *n/ 2,1
A
/B
= E
are successive approximants to the continued fraction
2n+1' 2n+1
E
(k/2)
s=
ai
1 + a2
1 + a„
1
+ .
1* + ak
(50)
where the a's are defined in (36).
This result is of great interest in connection with the problem
of convergence.
Not only does the continued fraction generally converge
more rapidly, but in many cases the continued fraction expansion will
converge where the formally equivalent series may not.
serves as an illustration of this behavior.
The gamma function
The gamma function may be
represented by
log T(z) = - z + (z - •§■) log z + log </2rr + J(z)
(51)
The function J(z) may be represented by a formal power series expansion
, x
P(z), or by a continued fraction.
P(z)
J(z)
The first few terms of each are
1
1
12z
(12)
*-
360Z^ 1260Z
1/12—
z + 1/20
z4- 53/210 '
z + . .
The power' series does not converge for any z, although for large z,
(52)
16
a good approximation may be obtained by ending the series after a few
terms.
This continued fraction form converges for all z such that the
real part of z is greater than zero.
For z = g-, the power series yields
1.801 for r, while the continued fraction form for J(z) yields 1.778.
Comparing these values to the correct value of 1.772, one sees that
changing from the power series form to the continued fraction form
decreases the error from 1.7 percent to 0.3 percent.
Example I
To illustrate the theory on a perturbation problem, consider the
Mathieu equation
5 + 4 cos
(
x ) Sjr = E \Jr
(53)
dxr
whose lowest eigenvalue is E = 1.54436 . . •
Successive approximations
to this value, obtained using the continued fraction expansion, are
(4)
shown in Table I,
The modified Brillouin-VJigner method
, obtained
by Goldhammer and Feenberg, provides only the odd numbered approximants
and thus gives a set of monotonically decreasing elements.
However,
successive approximants of the continued fraction are seen to oscillate
about the true value.
Table II shows approximations carried to e or
4
equivalent by means of the unmodified Brillouin-Wigner scheme, a secular
(2 3 10)
determinant method
*
*
, and the continued fraction.
It should be
pointed out that the perturbation here is quite large, and the convergence
is quite poor in the basic Brillouin-Wigner series? the barely de¬
creasing in magnitude.
However, the continued fraction result is quite
accurate and requires little additional labor.
Example II
As a second perturbation example, which will be solved in some
TABLE I
SUCCESSIVE APPROXIMATIONS TO EQ = 1.54486...
Order
E
o
Error
1
2
3
4
5
2.00000
1.26795
1.55505
1.54429
1.54487
+.455
-.2769
+.0102
-.00057
+.00001
TABLE II.
APPROXIMATION TO 4th ORDER BX THREE METHODS
Method
E
o
Error
Brillouin-Wigner
1.1541
0.3908
secular determinant
1.5412
0.0037
continued fraction
1.5443
0.0006
17
detail, consider a square-well of infinite height, with a raised portion
in the center.
This may be described by the expressions
a2
H = -^+V
V =b
-7r/4
=
0
=
0
where the range of x is from
vanish at these points.
-TT/Z
to
< x < 7T/4
-ir/2 < x ^-7r/4
TT/2,
TT/4
< x <7r/2 (54)
and the eigenfunctions must
The unperturbed eigenvalue equation and the
corresponding solutions are
u = n2 u
n n
= JZ/TT
COS
n — 1, 2, ...
n odd
n x
n even
= */Z/TT sin n x
(55)
A fev; of the matrix elements then are
r1
1 / \ (n"1 )/2
V„an
M-1 )
n = b {I
2 + tor
^mn
=
]
n odd
(-12)/4 m,n odd; (m-n)/2 odd
Vmn = ^ (-l)^*11
\
[m-hi]
m,n odd; (m-n)/2 even
\
\ \ = 0
m - n odd
(56)
For the lowest\state, n = 1, and the matrix elements become
V
1l'^
b [1/2 +
i/ir]
)A
).^.
T11m i\.afei
,
l
7r
m - 1
v
m
i ;
\ / . \(m-1 )/4
A 2b H)
IT
m + 1
m = 4j~1 > 1
m = 43+1 > 1
(57)
18
The first sum in the series for the eigenvalue (17) is
i v V
1m ml
= Z
m
E -m
=
2
2
7T
1
1
_&£
2
(_m=44-1 (m -E)(m-l)
m=4j+l
)
(58)
2
(m -E)(m+1 )2J
These sums can be evaluated by expanding in partial fractions giving
'2
^ “
647
'16
*
E = 1
(59)
If E ^ 1, let E = k2 ,
e
2
=
4 v2
1
t—\
2
‘ *2 ” / 8k (k-1) j=i ( j-(k+1)/4 j+(k+1)/4j
1
S'
2
8k(k+1 )
f1
(60)
IT2
\
1
ig ~
jTl ( j-(k-1 )/4 " J+(k-1 )/4 j " (k -1 )
2
2
" pTT
The remaining sums may also be evaluated from tabulated functions by
noting that^
oo
Y (z) = -
Y+
2L (J "
n=1
¥ (z) = ¥ (z-1) + \
(61)
where ¥ (z) is the logarithmic derivative of the gamma function T(z+1),
and Y is Euler’s constant, 0.5772 .
Thus (60) becomes
k-1
k-1
k+1
ffi) - f(- r(£?) - v(- s?)
8k(k-1)2 8k(kt1)2
(62)
This particular problem may also be solved exactly.
even solutions we have
For the
19
(^"2
dx
+
k^-V)i|r=0
k2 = E
\}r = N sin k(5 - x)
*
= N A cos
a
>!
X < 2
' 4
x
a2 = k2 - b
k < 4
(63)
The continuity condition requires that
^ dx
im
dx
(64)
22+
4
which yields
k cot k~ = a tan
4
4
k < 4
(65)
If K ^ 4» these expressions are modified
x >2
= N sin k(£ - x)
4
= N A cosh {3x
*<!
p2 = b - k2
k > 4
- k cot k- = p tanh p4
4
(66)
A value for E may now be obtained to second order by each of the
methods previously discussed.
Substitution of (57) and (59) into (5)
gives the Rayleigh-Ritz formula
E = E1 +
+ e2(D
= 1 + 0.81831 b - 0.01349 b2
The Brillouin-Wigner series becomes
(67)
20
(68)
E = 1 + 0.81831 b - e2(E)
where (62) is to be used to determine Eg*
To obtain the maximum benefit,
the continued fraction expression will be written in the form
11/a1
(69)
1 + Vn/E-, - E2(E)AI1
in which Eg
again to be obtained from (62).
A value for E from the
implicit expressions (68) and (69) may be obtained either by iteration,
or graphically after solving for Eg.
The values of E obtained for a wide range of values for b, the
height of the central barrier, are shown in figure 1.
of b, all three methods give accurate values.
For small values
However, as the pertur¬
bation is made larger, the curves draw away from one another.
For very
large values of b, the second order Rayleigh-Ritz series diverges as
-0,01 b2, which is easily seen from (67).
The implicit Brillouin-Wigner
series is observed to follow the exact curve over a wider range, and a
careful examination of (68) and (58) shows a slower divergence for very
large bj as -0.03 b. The continued fraction curve is seen to be the last
to depart from the exact curve, and remains finite for all b.
Thus it
is of the right order of magnitude even as the perturbation becomes
infinite.
Here we see again the improved accuracy and convergence ob¬
tained by expressing the expansion for the eigenvalue in the form of a
continued fraction
LLI
FIG.
REFERENCES
(1) L. Brillouin, J. Phys. £, 1 (1933).
(2) E. Feenberg, Fhys. Rev. 24, 206, 644 (1948).
(3) H. Feshbach, Phys. Rev. 24, 1548 (1948).
(4) P. Goldhammer and E. Feenberg, Phys. Rev. 101. 1233 (1956).
(5) W, V. Houston, Principles of Quantum Mechanics.
(McGraw-Hill Book
Company, New York, 1951); p. 90.
(6) E. Jahnke and F. Etade, Tables of Functions. 4th ed.
(Dover Publica¬
tions, New York, 1945); p. 18.
(7) P. M. Morse and H. Feshbach, Methods of Theoretical Physics.
(McGraw-
Hill Book Company, New York, 1953); Chap. IX.
(8) Oskar Perron, Die Lehre von den Kettenbruchen.
(B. G, Teubner, Leipzig,
1929); pp. 5, 304.
(9) Lord Rayleigh, Theory of Sound 2nd ed. (Dover Publications, New York,
1945); pp. 113-118.
(10) P. I. Richards, Phys. Rev. 24, 835(1948).
(11) E. Schroedinger, Ann. D. Phys. 80, 437 (1926).
(12) H. S. Mall, Analytic Theory of Continued Fractions.
,
(D. 1 Van Nostrand
Company, New York, 1948); p. 365.
(13) E* P. Wigner, Math, u. naturw. Anz. ungar. Akad. Miss. £2, 475 (1935).
(14) R. C. Young, L. C. Biedenham, and E. Feenberg, Phys. Rev.
(in press).
ACKNOWLEDGEMENT
The author wishes to express his appreciation to Professor
L. C, Biedenharn for suggesting this problem, and for his
continuing council throughout the work.