J.A. Murphy and D.M.Drew.

TV/26 July 1973

CONTINUED FRACTION SOLUTIONS OF

LINEAR DIFFERENTIAL EQUATIONS by

J.A. Murphy and D.M.Drew.

The mathematics presented in this report was completed whilst I was a Guest Worker in the Division of Numerical Analysis and Computing at the National Physical Laboratory, I would like to thank Mr. Albasiny and his team for making me so welcome, and Professor Crank and the Vice Chancellor for making my visit possible.

D. Drew.

W9260760

CONTINUED FRACTION SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

Page

Introduction

11. Laguerre's Problem.

12. Jacobi Functions.

Error Function and Related Functions.

13. Three Distinct Singularities.

14. M Fraction Solution.

Examples.

15. C.F Solutions of Second Order Differential Equations.

16. J Fraction for the Laplace Transform of f(a+t).

Conclusion

1

3

12 16

18

26 29

34

39

42

1.

INTRODUCTION

We will develop a general theory to unify a number of the results presented in TR/25, and which will enable us to discuss more complex examples. The theory is a refinement of a paper by E. Laguerre

published in 1885. First we construct the theory for J fractions for a certain class of functions, and then we show that the theory readily extends to deal with M fractions. Although numerically we can generate both J fractions and M fractions from suitable series for a function, in general, depending on the number of significant figures carried in the computation, the rounding errors eventually terminate the procedure. The purpose of this paper then is to develop the theory so that these fractions may be obtained directly from more compact descriptions of the functions, in our case differential

equations. We will construct relations which successively generate coefficients of the C.Fs. Further we indicate by examples how

asymptotic formulae for the coefficients in a C.F can be constructed, and how an almost periodic C.F is approximating the positions of the branch points. Two notions that generalise.

The theory as presented is very limited dealing only with second order linear differential equations which are, or can be reduced to,

equations in which one solution is a polynomial. Nevertheless the results can be extended certainly numerically to other differential equations closely related to these. We briefly indicate this aspect, but our work is by no means complete.

The method of constructing our theory is directly applicable to a variety of first order problems. We conclude by deriving the J fraction for the Laplace transform of a function.

This paper is a sequel to TR/25, and relations in that technical report are referenced directly. We recall that in TR/25, we showed that the f i in the linear relations

f 1 = d 1 f + ^C 1 (-1) f 2 = d 2 f 1 + c 2 f

···

f n = d n f n-1 + c n f n-2

where c i d i are polynomials in x,

can be expressed as rational functions by terminating the C.F

,

1 f n f n d c n n

1 ...

d i 1 c i 1 D i

D i A i 1

f i

− − +

+ +

− +

= +

−

where A. = (-1) ^i-1 c 1 c 2 ... c i and D. is a polynomial.

In particular the n ^th approximant to f is

,

D n n c d n n ... c d c 2 2 d 1 c 1

f o/n =

+ +

= + the error being

n . D f n D n C n

f − =

3.

11. Laguerre's Problem

Apart from some minor refinements which we will discuss later, Laguerrre in his paper (1885) set out to determine and study the J fraction.

− +

− +

− + +

d n x c n 2 ...

d x c 2 d 1

x c 1 o (x)

c (11.1)

for a function f that satisfies a first order differential equation ,

U dx 2vf

w df = +

(11.2)

where U, V, W and c 0 (x) denote polynomials in x.

To do this, set

D n f n D n C n

f = + (11.3) and choose the polynomials C n (x) and D n (x) so that their ratio

D n

C n matches the terms in the series for f up to and including the

term _2n x

1 . Then

. 1 ....

x 2n 1 A n D n f n as

terms lower

1 x 2n

2v 2

x 2n 1) w (2n 1

A n

(11.4) D n

f n 2v D n

f n w dx U

D n c n 2v D n

c n dx w d

+ +

= +

+ + + +

+ +

=

∂ +

−

=

−

−

⎭ ⎬

⎫

⎩ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎡

.

Performing the differentiation

^W [ ^D _n ^{C '} _n ^{− C} _n ^{D '} _n ] ^{− 2vC} _n ^D _n ^{− UD 2} _n ^{= A} _{n + 1} ^{D 2} _n ^{{ }}

=A n+1 θ n (11.5)

where θ n (x) is a polynomial, for the left hand side is a polynomial.

Further θ n can be regarded as being of fixed degree µ, the degree of

the highest term in _⎟

⎠

⎜ ⎞

⎝ + ⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

V x

W x 1

1 2

2

,only its coefficients depend on n.

Thus we can now split (11.4) into two equations ,

(11.7) 2 .

D n θ n 1 A n D f n n

2v D f n n

dx W d

(11.6) 2 ,

D n θ n 1 A n n U

D n 2v C D n n

C dx W d

− +

⎥ =

⎥

⎦

⎤

⎢ ⎢

⎣

− ⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

= +

−

− _⎥

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎡

The. latter is particularly important because of the second order differential equations that we will derive from it, and because integrated it gives the error in approximating f by

D n n C .

To construct (11.1) we must derive the recurrence relations between the f n and this we can do in the following manner.

Recurrence Relations for D n and f n .

Replacing n in (11.6) by (n—1) and subtracting the result from (11.6) gives

2 . 1 D n

1 θ n A n 2 n

D θ n 1 A n 1 D n D n

A n 2V

1 D n D n

A n dx

W d

n , n A

1 D C n 1 D n C n As

2 . 1 D n

1 θ n A n 2 n

D θ n 1 A n 1

D n 1 C n D n C n 2V 1

D n 1 C n D n C n dx W d

−

− −

= +

−

−

−

− =

− −

−

− −

= +

−

− −

−

−

− −

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

But A n+1 = c n+1 An, so that

^W [ ^{D '} _n ^D _{n - 1} ^{− C '} _{n - 1} ^D _n ] ^{− 2VD} _n ^D _{n - 1} ^{= − − c} _{n + 1} ^{= θ} _n ^{D 2} _{n + 1} ^{+ θ} _{n + 1} ^D _n ²

5.

which rearranged gives

[WD ^' n ;+VD n + c n+1 θ n D n-1 ] D n-1 = [W D ^' n − 1 + V D n-1 - θ _n-1 D n ] D n (11.8) These are two expressions for the same polynomial. D n and D n-1 are

clearly factors of that polynomial. We can therefore equate both sides to Δ _n D n D n-1 . where Δ _n (x) is a polynomial.

This yields the two relations

W D

^'_n

+ ( V - Δ _n ) D n + c n+1 θ n D n-1 = 0, WD

^'n−1

+ (V +Δ _n ) D n-1 θ n-1 D n = 0 . To eliminate D we step n by one

W D

^'_n

+ (V+ Δ _n+1 )D n - θ n D n+1= 0 . (11.11) (11.9) (11.10)

and subtract (11.11) - (11.9) D n+1 =

θ n 1 (



n+1 + n ) D n -c n+1 D n-1, (11.12)

the required recurrence relation between the denominator polynomials.

To obtain the recurrence relation between the f n , we simply multiply by f n and use the result (11.13) , D n f n-1 ,- D n-1 , f n = A n ,

D n+1 f n =

θ

n

1

 n+1 + n ) D n f n -c n+1 [D n f n-1 ] D n f n+1 =

θ

n

1  n+1 + n ) D n f n -c n+1 [D n f n-1 ]

∴ _{f n+1 =}

θ

n

1 ( n+1 + n ) f n - c n+1 f n-1 (11.13) We can also deduce relations similar to (11.9) and (11.11) for the f n , we simply state them

W f ^' n - ( n + v) f n + c n+1 θ n f n-1 =0 (11.14)

W f ^' n + ( n+1 - V ) f n - θ n f n + 1 = 0 . (11.15)

Let us elaborate on the way (11.13) generates (11.1).

To solve Laguerre's problem, given

dx df

W = 2Vf + U

first we extract the polynomial c o (x) part of the solution f = c 0 (x) + f o

so that = 2vf o − c 1 θ o × dx

df o W

In fact o is V, so rewriting this equation as θ o f 1 =

dx o

W df + (Δ ₁ - v) f 0 = (Δ ₁ + V) f o - c 1 θ o (11.16)

we get f 1 = θ

o

1 ( Δ ₁ + V) f o - c 1

f 2 = θ

o

1 ( Δ ₂ + Δ ₁ )f 1 - c 2 f o

etc.

These linear equations are equivalent to (11.1). All that remains is to deduce a convenient means of generating successively the c n and the polynomials θ n (x) and Δ _n (x).

But first we digress and note that D n (x) and f n (x) are essentially the two solutions of some interesting second order differential equations.

* NOTE if θ 0 ≠ constant U cannot be chosen arbitrarily

θ o c 1 2Vc o dx o W dc

U = − −

7.

Second Order Differential Equations

We consider the two cases θ n =constant and θ n ≠ constant separately.

Wf

^'_n

D n - W f n D

^'_n

- 2V f n D

^'_n

^{= -} A n+1 θ n, (11.17)

this follows from equation (11.7). Differentiating we get (wf

^'_n

- 2Vf n ) D

^'_n

+

dx

d + [Wf

^'_n

- 2Vf n ] D n +

- Wf n D

^'_n

- W'f n D

^'_n

- Wf

^'_n

D

^'_n

A n+1 θ

^'_n

(11.18) For θ

n

= constant, θ

^'_n

= 0

∴ _{W D}

^'_n

+ (W' + 2V) D

^'_n

-

dx d f

n

1 _{[W f- 2V f}

n ] D n = 0

which reduces to

W D

^'_n

+ (W' + 2V) D

^'_n

+ K n D n = 0 , (11.19) where K

n

(x)is necessarily a polynomial of fixed degree. In addition

dx

d [W f' n - 2Vf n ] + K n f n =0 (11.20)

0.

dx y 2V W n e k dx y'

2V W dx We

d

find we , y' We y ' 2Ve

Wf n

y, W dx.

2v n e

f put Now

∫ =

∫ +

+ ∫

= ∫

= ∫

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ as

Hence the second order differential equation

W y”+(W’+2v)y’+k n y=0 (11.21) is satisfied by both D n and 2V W dx f n.

e - ∫

8.

Alternatively we could verify that 2v w dx

e ∫ .D n as well as f n

satisfied (11.20).

For θ n ≠ constant, we use(11.17)to eliminate A n+1 in 11.18),then n 0 n D θ

' n ) θ f n ' 2V f n (W n ] f ' 2V

f n W dx [

d f n ' 1 D n n ) θ

' n W θ 2v ' (W'

D n

W + + − − − − − =

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

W θ n D

^"_n

⁺ [(W' ⁺ 2V ) θ n -W θ n ' ] D

^'_n

⁺ k

^*_n

D n = 0 , (11.22) where K

^*_n

(x) is a polynomial of fixed degree.

The appropriate generalisation of (11.21) is that the equation

w θ n y" + [(w' + 2V)θ n - W θ

^*_n

] y ' + k 0 (11.23)

^*_n

has the complete solution

+ ∫

= dx f n

2V W e B n (x) D A

y (11.24)

Interrelation between θ n Δ _n and K n .

In the preceeding analysis we have introduced three polynomials of fixed degrees θ n ,D n and K n . To obtain the relation between them we now derive (11.22) directly from (11.9) and (11.10). Eliminating D n-1 from (11.10) using (11.9)

-c n+1 θ n W D' n-1 + (V + D n ) W D n ' + [ V ² - D n 2 + c n+1 θ n θ n-1 ] D n = 0 W must be a factorof the polynomial in the square brackets,[ ] = W S n

say,

∴ _{-c n+1 θ n D}

^'_n−1

_{+ (V +} Δ _n ) D

^*_n

+ S n D n = 0 . (11.25) Substituting (11.9)

n 0 n D ' S D n n ) Δ (V θ n

D n n ) Δ ' (V

D n W dx n d

θ + − + + + =

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

9.

and rearranging,we find

w θ n D n " + [( w' + 2v ) θ n - w θ n ' ] D n ' + [( v' - Δ _{n +} s n ) θ n - (v - Δ _n ) θ' n ] D n =0 . (11.26)

With θ n =constant, comparing this equation with (11.19)

(11.28)

1.

1 n θ n θ n 1 c n 2 Δ n V 2 S n

W Where

) . ( n ,

' S Δ n n V'

k

− ≥ + +

−

=

+

−

= 11 27

with θ n ≠ constant, comparing with (11.19) gives

(11.29) Connection between θ _{n and S n}

' n n )θ Δ n (V

n )θ ' s

Δ n

* (V'

k n = − + − −

W S n = V ² - Δ _n ² + c n+1 θ n θ n-1 . ^{W S} n+1 = V ² - Δ ² _n+1 ^{+ C} _n+2 θ n+1 θ n ' subtracting

w(s n - s n+1 ) = Δ ² _n+1 - D ² n + c n+1 θ n θ n-1 -c n+2 θ n+1 θ n . (11.30)

With our definitions θ n (x + d n+1 )= Δ _n+1 + Δ _{n ,}

∴ ^W ( ^S n - ^S n+1) = θn[(x + d n+1 ) ( Δ _n+1 - Δ _n, ) ⁺ C n+1 θ n-1 -c n+2 θ n+1 ], (11.31) and θ n is therefore a factor of w(S n - S n+1 ).

In fact as Laguerre shows the square bracket is W, that is for n ≥ l

(x + d n + 1 ) ( Δ n + 1 − Δ n ) = W + c n + 2 θ n + 1 − c n + 1 θ n − 1 (11.32)

and

θ n = S n − S n + 1 . (11.33)

Thus we have a polynomial relation (11.32) connecting our unknowns.

We shall find that from it, or (11.28), knowing little more than the

form of θ n and ∆ n we can deduce the coefficients in our C.F. Before

we apply our results let us say therefore a little about θ n and n , in particular about

n

.

The Polynomials θ n and n.

First let us recall that θ n is a polynomial of fixed degree µ, the term in x ^µ being given by the first term in the expansion

_.

x 2V x 2 1) W

(2n + + (11.34) Then observe that from (11.9)

(11.35) x n ...

1 d x n 1 θ c n V 2 ...

x 2β n 2 n (α x n n α x n W Δ that so

2 ...

x n β n 1 x n α n x n 1 ....

)x n d n 2 ....

1 d n (d n x D write and

n , D n 1 n D 1 θ C n n V

D ' n W D Δ n

⎥⎦ ⎤

⎢⎣ ⎡

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − + − + + + + − +

=

− +

− + +

≡

− + +

+ + +

= + − + +

=

where, because ∆ n is a polynomial, the expansion on the right must terminate. For most purposes all we will require is the leading term of A and the form of ∆ n . We will simply substitute θ n and

D n in (11.32), or (11.28), equate coefficients of x and hence determine our unknowns and in particular the coefficients of our C.F. The

other polynomials that we have introduced, S n , K n and K* n can then be determined by the appropriate expression (11.27) to (11.29).

To clarifv the preceeding analysis we will construct the recurrence

relations and hence the C.Fs. when W is quadratic in x. There are

11.

essentially two cases. The first is when f has two distinct singularities

these we will take at ± 1 by taking W = x ² -1 , the denominator polynomials

will turn out to be the Jacobi polynomials. Then we will consider the

case when the singularities coincide by taking W = x ² , the resulting C.F

gives useful approximations to the error function and related functions.

12. Jacobi Functions

Let us consider the differential equation (x ² - 1)

dx

df = 2( x + λ )f + U(x) . (12.1)

For this equation

W = x ² -1 , V = (λx + µ) and the weight function

dx (1 x) λ u , (1 x) λ u (12.2)

1 x

u λ 1 x

u exp λ W dx

2V e

w(x) ∫ + = − + + −

+ −

−

∫ = +

= ⎭ ⎬ ⎫

⎩ ⎨

⎧

which is the weight function for the Jacobi polynomials P n (α,ß) (x) if we take α = λ + µ , β = λ - µ .

From (11.34) and (11.35) we readily deduce that θ n = 2n+1 + 2λ

Δ _n = (n+λ) x + δ n

The key information is now obtained by equating coefficients in the identity (11.28) ,

WS n = V ² - Δ

_n²

+ c n+1 θ n θ n-1

(x ² - l)S n =(λx + µ) ² -[(n + λ) x + δ n ] ² + c n+1 (2n +1+ 2λ)(2n-1+2λ).

S n is necessarily a constant,

∴ _{S =} λ ² - (n +λ ² ) = - n(n + 2λ)

λ n n λµ δ = +

1 . n

] 2 1 λ) (n

2 [4 λ) (n

2 ] 2 µ

λ) (n

[ 2 λλ 1 n(n

c n and

− ≥ +

−

− +

= + +

Also from the expression for n ,

13.

Δ _{n+1 +} Δ _n = (n + 1 + λ)x + (n +λ )x +

λ λμ μ

λμ

+ + +

+ n

n 1

where (2n + 1 + 2λ) is θ n Thus from (11.13) the J fraction for the particular integral of (12.1) is generated by the recurrence relation

(12.3) 1 .

n - ] f u 2 λ) 2 (n [ 2 λλ n n(n

λu f 1 x

f n + = ^{⎢ ⎡} + ^{⎥ ⎤} − + + −

] 2 1 λ) (n 4 2 [ λ) λ) (n

(n λ) 1

(n + + + _⎦ + + −

⎣

⎥ ⎦

⎢ ⎤

⎣

⎡

+ +

+ + +

+

= ( n 1 λ) (n λ)

x λu 2λλ 1

(2n

In particular with U (x)= θ 0, λ > - ,we would obtain the J fraction for

(x) dx, w 2 1) (x

1 x 2λ

W(x)

f −

∞ +

= ∫

the first partial numerator c. 1 being unity.

The denominators of these J fractions by (11.12) satisfy the same recurrence relations as the f and are readily shown to be the Jacobi polynomials P n (α,ß)(x) arranged so that the coeffioient of x ⁿ is one.

For by (11.27)

K n = V' - Δ

^'_n

+ S n = -n (n + 2λ + 1),

thus by (11.21) the denominator D n (x) and f n (x) w(x)

1 are both solutions of

(x ² - 1 ) y" + (2x + 2Ax + 2µ) y' - n(n + 2λ + 1) y = 0 (12.4) which is the differential equation satisfied by Jacobi polynomials

P n (α,β) (x); 2λ = ( α, β ) 2µ = α - β .

Particular Cases

In passing we observe that the recurrence relation (12.3) simplifies

in the following three cases:-

1 1 n

f n 2 1

4n n 2 µ 2 f n

1 x f n 0 λ

i) − − ≥

+ − + =

=

(12.5)

which generates the C.P (4.22) for the Associated Legendre functions 1

f n 2 1

λ) 4(n

) 2 λ n n(n

f 1 x f n 0 µ

ii) + − −

+ + + =

= (12.6)

which, with λ = 2

− 1

v generates the C.F (5.6) tor the Laplace transforms of Bessel functions.

(12.7) 1

f n λ) 2 (n

2λλ n(n 4 n 1 λ) f (n λ) 1 (n

2 λ 1 1 x

f n 2 µ 1

iii) + −

− + +

+ + +

+ =

=

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

Written as a Laplace transform the function f becomes

(12.8) u

λ 1

a b 2 ( 1 λ ) where

dt 2t)

; b

; 1b (a 1 F 1) t e (x

o = + −

+ + =

∫ ∞ −

and consequently is related to the Kummer and Whittaker functions.

Error Analysis for Jacobi functions.

With U(x) = - e , our J fraction for f is

1 f n f n d n

x c n d 2

x c 2 λ

1 x u f 1

− −

− +

−

− −

− + + +

=

where c n+1 and d n+1 are given by (12.3).

Now the error in terminating this expression after n terms is D f n n

D n n f − C =

and this expression satisfies,by (11.7),the differential equation

2 n . D W

θ n 1 A n D f n n

W 2V D f n n

dx

d − _⎥ = − +

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎡

Hence the error can be written as an integral

(u) du 2 n D w(u) W

θ n 1 A n w(x) x

D n

fn = ∫ ∞ +

and the asymptotic value of this integral for large n we have deduced is

15.

(12.9) n

0 1 i 1

e Nς 1

2e Nς 1) β

(x 1) r(β 1) β r(α 2 α

1) α (x 1) β r(α π

n 0 1 2 1

i 2 ( Nς tanh 1

) π 1 2 i (α β e 1) (x 1) r(β 1) β r(α 2 α

1) α (x 1) β r(α D f n n

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎟

⎠

⎜ ⎞

⎝

⎛

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎟

⎠

⎜ ⎞

⎝

⎥ ⎛

⎦

⎢ ⎤

⎣

⎡

+ + + −

+ − + + + + + −

=

+

− + −

− +

+ + + + + −

=

φ

φ π

where x = cosh ς , N = (2n + 1 + α + β ) and ) . 2 (α 1 π φ = +

This expression for the error ia of a similar form to the error

estimates obtained in TR.25 and in fact generalises a number of these

results, in particular (5.28) for the Laplace transforms of Bessel

functions.

Error Function and Related Functions

For our second example we consider the equation

dx 2 df

x = 2(λx + µ)f + Ux) (12.10) where

W = x ² , V = λx + µ

An almost identical analysis with that for the Jacobi functions produces the recurrence relation ,

f n 1

2 1]

λ) 2 4(n

λ) (n

2 λλ 2 n(n

n u λ) f ) (n

λ 1 (n x λu 1

f n + + − −

+ + + −

+ +

+ = ^⎢ _⎣ ^{⎡ ⎥} _⎦ ^⎤

[ , (12.11)

as the relation which generates the J fraction for the particular integral of (12.10).

Now the weight function for this set of

functions (12.12)

x e 2u 2 λ x 2 dx

x 2u x exp 2λ W dx

2V e

w(x) = = ∫ + = −

⎭ ⎬

⎫

⎩ ⎨

∫ ⎧

is clearly closely related to that for the generalised Laguerre functions. However the C.F generated by (12.11) is not to be

confused with (2.17); it is matching a different series. It provides a powerful sequence of approximations to the error and related

functions.

When U(x) = -2λx, the solution of (12.10) is

2u x dx ,

1 e x 2λ x 2λ 2u x 2 λ e

x

f(x) − +

− ∞

= ∫

we take µ = ½ . As an example we put µ = 4

1 so that

2 dx 1 3/2 e x

2 x 1 1 2 2 e x 1 f(x)

−

∞

= − ∫

17.

and thus

(12.13) 2 .

45 z 1 2 25.21 z4

24 z 2

3 1 2

2 3

z 2 1 t2 dt z e o z 2 z 1 e E 2

f + − − − −

+ + +

−

− =

= ∫

⎟ ⎠

⎜ ⎞

⎝

⎛

Luke [ 6.4] derives an estimate for the error in terminating the C.F for

x dx 1 1 e x 2λ x 2λ 1 2 2λ e x

f(x) − ∫ ∞ +

= (12.14)

after n terms. Making the zero convergent one, he estimates the error behaves as

. 1 2 1) n 2λ (2n 1 r (2x) 2n 2 2λ

) 1 (2 λ 2 r r 1 1 2 1 e 1) n (

~ n (x)

E + + +

− +

− + ⎟

⎠

⎜ ⎞

⎝

⎛

(12.15) Luke's figures for these formulae are impressive, so we will reproduce them.

Putting z = 1 in (12.13) f(l) = 0.5380 7951 n f 0 /n (1) f - f 0/n E n

0 1.0 - -

1 0.5238 0952 0.14(-1) 0.15(-l) 2 0.5382 4561 -0.17(-3) -0.17(-3) 3 0.5330 7854 0.97(-6) 0.98(-6) 4 0.5380 7951 0 0

For our next example we increase the degree of W and hence increase the number of singularities in f. We will find that no longer can we derive the c and d explicitely, but that we must be content with

producing a set of equations which successively generate the c n and d n .

13. Three Distinct Singularities

We take W = x - 1 so that the three singularities are symmetrically placed about the origin. Consider

=

− dx x 1 ) dy

( ³ _{3 (αx} ² + ßx + y)y + U(x) (13.1)

For this equation W = x ³ - 1, 2V = 3(αx ² + ßx + γ) (13.2) and the weight function

⎭ ⎬

⎫

⎩ ⎨

⎧

⎪⎩

⎪ ⎨

⎧

+ − + −

= −

− +

≡ +

= ∫ ∫ ∫ _dx

w 2 x

C w

x B 1 x exp A

) dx 3 1 (x

y 2 βx

3( α(

exp W dx

2V e W(x)

giving

w(x) = (x - l) ^α+ β • +γ (x - w) ^α+ß 2

• w ^+γw (x - W 2 ) ^α+ßw+γw2 (13.3) From (11.34) and (11.35) we can readily establish the form of θ n and

n

θ = (2n + 1 + 3α)x + φ _n (13.4) D n = α)

2

(d + 3 x ² + ß n x + y n (13.5) where θ n, ß n and y are to be determined.

The interrelation of these quantities with the coefficients c n and d n

in the continued fraction can be obtained directly by equating coefficients in (11.32)

(x + d n+1 )(∆ n+1 - ∆ n ) = W +c n + 2 θ n-1

i.e. for n≥1

(x+d n+1 ) [x ² + (β n+1 – β n ) x + (y n+1 – y n )] =

x ³ -1 + c n+2 [(2n+3+3α)x + φ _n+1 ]-c n+1 [(2n-1+3α)x + φ _n-1 ]

(13.6) 1

Φ n 1 c n 1 Φ n 2 c n 1 n )

1 y (y n 1 d n

) 3α 1 1 (2n c n 3αα 3 2 (2n c n y n 1 y n n ) 1 β (β n 1 d n

n 0 1 β β n 1 d n

⎪ ⎭

⎪ ⎬

⎫

−

− + + + +

−

= + −

+

+ + −

− + + +

= + −

+ + −

+

= + −

+ +

19.

In addition we have that θ n ( x + d n+1 ) = ∆ n+1 +∆

i.e.

[(2n + 1 + 3α)x + φ _n ](x + d n+1 )-(2 n + 1 +3α))x ² +(ß n+1 + ß n ) x + (γ n+1 + γ n ) so that

(13.7) n .

1 γ γ n 1 d n n

β n 1 β n 1 n

d n ) 3α 1 (2n

⎪⎭

⎪ ⎬

⎫ + +

+ =

+ +

= + +

+ +

ι

φ

A simple rearrangement of these five equations gives the following scheme for successively computing d n+1, ß n+1 , γ n+1 , c n+2 , Φ n+1 :-

(13.8) n )

1 γ (γ n 1 d n 1 1

n 1 c n 1 n 2 c n

2 1 d n n ) 1 γ (γ n 1 c n ) 3α 1 2 (2n c n ) 3α n 3

(2

1 d n n γ n

1

γ n n 1 β n d n 1 β

n 2β n 1 d n ) 3α 2 (2n

⎪ ⎪

⎪ ⎪

⎭

⎪⎪

⎪ ⎪

⎬

⎫

+ − + +

− +

= + + +

− + + −

+ + +

− + =

+ +

+ = + =

− + + =

− + =

+ +

φ φ

φ φ

where the last two equations can be used with n = 0 if the terms containing c n+1 are dropped.

The initial conditions are partly determined by U(x). We can eliminate any polynomial part to y by taking U(x) = - θ 0 and to keep the number of parameters to three, let us put U(x) = - [(1 + 3α)x + 3β].

Since 2 V 3

Δ 0 = = (ax ² + ßx + γ ), we then have the initial conditions

γ

2 0 3 γ , 2 β 0 3 β , 3 β , 0

1 1

c = φ = = = (13.9)

From (13.8) we then obtain

y α 1 c 2

1 1 α , 1 2 y

0 , 2 y 1 3

y , 2 β 1 3 β 1 ,

d = = − +

− +

=

−

=

=

= 0 φ

etc.

3a , 4 1 3β

+

−

d 2 = φ

c and d are rational functions of α, ß, γ.

In general the above technique applied to the linear first order differential equation (11.2) will yield a set of interrelations like (13.8) from which the coefficients of c n and d n of (11.1) can be derived. The process is superior to simply determining a series solution of the equation and then converting it into a C.F by the method indicated in (1.17) of TR.25 in that it is numerically more stable.

The error in truncating the C.F. after n terms is still given by the integral of (11.7)

W(t) dt 2 (t)

D n W(t)

n (t) 1 θ A n w(x) x

D f n n +

∫ ∞

=

dt ,

3 1) (t 2 (t) D n W(t)

)t n 3α 1 x (2n 1 W(x) A n

− + +

∫ ∞ +

= + φ (13.10)

where w(x) = (x-1) ^A (x-w) ^B (x-w ² ) ^C . A n+1 we can if necessary compute, although it is worth noting c n c n+1 c n+2 →

16 1 as we shall observe later, A. very crude estimate for D n (x) would be x ⁿ , for a rather better estimate we could use the differential equation satisfied by D n for x large and n large

x ² y' → (2+3α )xy' - n ² y = 0 .

What we have tried to do and what ideally we would like to do is to obtain asymptoticc estimates depending on n for c and d n (or simple combinations of the cs and/or ds). For unless we can do this there seems little possibility of deriving a suitable error estimate that depends on n in the form we obtained when W was linear or quadratic;

further such asymptotic estimates might guide us in our handling of a

much wider class of problems.

21.

We observe that there are certain combinations of our unknowns which remain constant as n increases. Equating coefficients in the identity (11.28)

WS n = V ² - D n 2

+ c n+1 θ n θ n-1

we find

(13.11) )

y 4 (2β 3αα 9 n(n n ]

) 3α 1 1 (2n

) n 3α 1 (2n 1 [ c n y n 2β n

y) 2 2α 4 (β 1] 9 ) 2 3α (2n 1 [ c n y n ) 3α 2 (2n

β n

β) 2 2α 4 (y 1 9 n n 1 c n β n ) 3α 2 (2n

y n

⎪ ⎪

⎪

⎭

⎪ ⎪

⎪

⎬

⎫

= + + +

+

− + +

+ +

−

+

=

− + +

− +

+

+

− =

− + +

+

φ φ

φ φ

This set of equations can be used instead of the equations (13.6) and should be regarded as a first integral of that set of finite difference equations.

Quasi-Periodicity

To gain insight into these approximations we have computed the values of c n d n φ _n ß n γ n using the formulae (13.8) for various values of

α ,ß ,γ . These formulae seem to be numerically remarkably stable, the results tabulated in Tables A were obtained carrying ten figures.

Table A for a = 2ß= 2 γ =

3 1 clearly indicates the almost periodic nature of the partial numerator c n and of the partial denominator x + d n of the C.F. This three term periodicity results from the distributing of the poles and zeros of the convergents along the three branch cuts. The distribution along the three branch cuts is disturbed by the introduction of one extra pole and zero, is improved by a further pole and zero and finally regains a position similar to the first when the third pole and zero are added. For plot3 of poles and zeros see graphs on page V7. As more and more poles and zeros are introduced, as n is increased, so these poles and zeros etch out the branch cuts of the function y which satisfies the differential equation (13.1).

For other values of α ,ß ,γ.. the C.F settles to being periodic, but often only after an initial disturbance caused by the initial conditions has died away. Table A for

3

α = 1 , β = 0,

3

y = 2 shows a large leap in the value of c n to - 1009, before c n and d n slowly settle towards their periodic values. Usually the effect of the initial conditions is less dramatic taking longer to be absorbed. In all cases computed, the three term periodicity eventually dominated.

How can we take advantage of this three term quasi-periodicity in the coefficients c n and d n ? For a suitable large n

R *

2 d n x c n 2 1

d n x c n 1 d n

x c n

R +

+ + + + −

+ + + −

= − (13.12)

where R* is almost the same function of x as R. Treating (13.12) as a C.F and writing its convergents

1 ) c n 2 d n 2 o n d n 2 d n 1 d n (d n

2 )x c n 1 c n 2 d n 1 d n 2 d n d n 1 d n (d n )x 2 2 d n 1 d n (d n x 3

2 )]

c n 2 d n 1 (d n 2 )x d n 1 (d n [x 2 c n D c 3 3

1 ) c n 1 d n (d n 1 )x d n 2 (dn x

1 ) d n n (x c D 2 2

, c d n

x c n D c 1 1

+

− +

− + + + +

+ + + −

+ − + +

+ + + +

+ + + +

+

− + + + +

+ + + +

= −

− + + +

+ + +

+ +

= − +

= −

we can then write .

D 2 R * D 3

C 2 R * C 3

R +

= + (13.13)

If we replace R* lay R, R satisfies the quadratic equation

D 2 R ² + (D 3 - C 2 ) R - C 3 = 0 (13.14) from which we can determine R. When the problem is simple one of

calculating y that satisfies the differential equation (13.1), this R

can be used to plug our C.F ,

1 R(x) d n

x c n 1 d 3

x c 3 d 2

x c 2 x

1 +

+ − −

−

−

− − +

−

− + (13.15)

and will go a long way towards inserting the singularities of y(x).

The branch points of R are contained in the discriminant of (13.14), a simple manipulation gives the discriminant.

∆ = [C 2 + D 3 ] ² - 4c n c n+1 c n+2

= [x ³ + σx ² + Tx ² +υ] - 4c n c n+1 c n+2 (13.16) where σ = d n + d n+1 + d n+2

T = d n d n+1 + d n d n+2 + d n+1 d n+2 - c n -c n+1 - c n+2

ν = d n d n+1 d n+2 -c n d n+1 -c n+1 d n+2 -c n+2 d n.

To indicate how closely the branch points of y(x) are being approximated by those of R(x), we have calculated the discriminant (13.16) using our numerically results for o and d in Tables A.

6 y 1 5 , β 1 3 , α 1

For = = = and taking n = 40

σ = ·00010 = [x ³ + .00010x ² + .00007x - '50001] ² - 0.24997

T = · 00007 branch points x ³ + ·0001 Ox ² + .00007x - .50001 = ± 0.49997 υ = -.50001 hence x ³ + .00010 ^{X 2} + .00007x - 0.99998 = 0

4c n c n+1 c n+2 = 0.24997 or x ³ + .00010x ² + .00007x - 0.00004 = 0 3

y 2 0, β 3 , α 1

For = = = and taking n = 54

σ = 0.00003 ∆ = [x ³ + 0.00003x ² + .00007x - .49996] ² - 0.24994 T = 0.00007 branch points x ³ + .00003x ² + .00007x - .49996 = ± 0.49994 ν = -.49996 hence x ³ + .00003x ² + .00007x -. 99990 = 0

4c n c n+1 c n+2 = 0.24994 or x ³ + .00003x ² + .00007x - .00002 = 0

23.

For α = 3, β = 2, .γ = 1 and taking n = 28

σ = 0.0153  = [x ³ + 0.015x ² + 0.0066x - 0.4849] ² - 0.2348

^T = 0.0066 branch points x ³ + 0.015 ^{X 2} + 0.0066x - 0. 4849 = ± 0-4845 ν = -0.48489 hence x ³ + 0.015x ² + 0.0066x - 0.969 = 0

4c n c n+1 c n+2 = 0.23480 or x ³ + 0.015x ² + 0.0066x - 0.0004 = 0 In each case increasing n by multiples of three we find the branch points of R(x) tend to those of y(x) as we would expect. In this particular problem the branch points of y(x) are given directly by the polynomial W = x ³ - 1 of the differential equation. The 'artificial' triple

branch point at the origin arises because we are forcing the branch cuts towards the origin by our approximations at infinity. Our numerical considerations suggest that we write

[x 3 + σx 2 + rx + v − 2 c n c n + 1 c n + 2 ] → x 3 − 1 , (13.17) as n increases, this implies

σ → 0 d n +d n+1 +d n+2 → 0

T → 0 d n d n+1 +d n d n+1 + d n+1 dn+2- c n - c n+1 - c n+2 → 0 .

In addition, the forcing of the branch cuts towards the origin suggests V + 2 c n c n + 1 c n + 2 ^→ 0

∴ ν → ½ d n d n+1 d n+2 - c n d n+1 - c n+1 d n+2 - c n+2 d n → -

2 1

4c n c n+1 c n+2 →

4 1 .

Now it is not a simple matter to take advantage of these conditions

as n increases in the non-linear generating relations (13.8). Instead

they more naturally line up with our linear method of generating the

C.F outlined in (1.17) of TR25, and strongly suggest fitting the series

six terms at a time once the disturbance due to the initial conditions

has died away.

25.

Our analysis in section 11 can be extended in various ways. First we will look at another solution to the first order differential

equation (11 .2). Then we extend our considerations to functions

defined by second order linear differential equations. Finally we

will derive a general continued fraction matching the series in 1/s

for the Laplace transform of a function.

14. M Fraction Solution

The particular function f satisfying the differential equation (11.2) usually possesses a series for both x large and x small, so that besides our J fraction for f we can also construct an M fraction for f. A set of approximations derived from an M fraction will tend to give good approximations near the origin, at infinity and also along particular lines in the complex plane joining them. For a discussion of Murphy's M fraction see McCabe (1971).

Suppose then that

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

−

−

−

− +

+ +

−

−

−

− +

+ +

= x 3 for x large

b 3 x 2 b 2 x 1 b

(14.1) small

x 2 for

2 x a 1 x o a a f

and that f satisfies the linear differential equation (11.2) 2Vf U

dx

W df = + (14.2)

We begin by writing Q n n f Q n n

f = p + (14.3)

and choosing the polynomials P n (x), Q n (x) so that their ratio Q n n

P matches n terms in each of the series (14.1). We can write this ratio as the M fraction

x . q n 1

n x p 2 x

q 1

2 x p 1 x

q 1 p 1 Q P n n

+ +

−

−

− + −

+ +

= + (14.4)

Proceeding as we did in section 11, we obtain

n U Q p n n 2V

Q p n dx W d Q n n

2v f Q f n n

dx

W d − = − _⎥ + +

⎦

⎢ ⎤

⎣

⎥ ⎡

⎦

⎢ ⎤

⎣

⎡

27.

which on differentiating gives

2

UQ n Q n

2VP n n ]

n Q ' P P n [Q n n W

Q f n n 2V

Q f n dx w d Q f n n dx 2V

w d 2 n

Q − − = − − + +

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡

⎩ ⎨

⎧

= a polynomial, (14.5) as W(x), V(x) and U(x) are polynomials. Further as

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎛ + + + +

−

− + + + +

= x n 2 x large

0 1 1 x n

1 1 q n ) 2 q n q 2

(q 1

1 A n

smal x

1 ) 0(x n x n

1 A n Q n n

f (14.6)

from the L.H.S we deduce that the degree of the terms in this polynomial lie between n-1 and n-1+υ where the term of largest degree in

x w +2V is v, v is fixed.

Thus the corresponding result to the key relation (11.7) is

2

Q n

n (x) 1 x n 1 A n Q n n 2V f Q f n n dx

W d ₊ ⁻ φ

=

⎥ −

⎦

⎢ ⎤

⎣

⎡ (14.7)

where φ _n (x) is of fixed degree υ, and A n+1 = (- 1) ⁿ P 1 P 2 - - - - P n+1 . The factor x ^n-1 is the only difference between this result and (11.7), it removes some of the elegance of the subsequent analysis of section 11 , but nevertheless it does not prevent parallel results being derived.

We will not repeat the analysis but simple indicate our results.

(11.9) and (11.10) become

Δ n Q n P n 1 x n Q n 1 0 2

1) W (n ' n Vx

Q

Wx + _⎢⎣ ^⎡ − − + _⎥⎦ ^⎤ + + φ − = (14.8)

Δ n Q n 1 n 1 Q n 0

2 1) W (n ' Vx

1 - Q n

Wx + _⎢⎣ ^⎡ − − + _⎥⎦ ^⎤ − + φ − = (14.9)

and so eliminating Q' n , the recurrence relation for the Qs (and fs) is 1

Q n 1 x P n Q n Δ n 1 2 n

W n 1 1

Q n + = φ _⎢⎣ ^⎡ + Δ + + _⎥⎦ ^⎤ + + − . (14.10)

Eliminating Q n-1 from (14.9) we obtain the differential equation for Q n,

Wx Q n H n (x) Q n 0 n

' n W 1) (n 2Vx x

' n W' Q

Wx + + − − − + =

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

φ

φ (14.11)

(14.12) Δ n

2 1) W (n n Vx

' n

S n x n ' n Δ Δ x w'

W 2

1) x (n

V' n (x) H where

⎥⎦ ⎤

⎢⎣ ⎡

⎥ ⎦

⎢ ⎤

⎣

⎡ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛

+

−

−

−

+ +

− +

− − +

=

φ φ

and the crucial relation corresponding to (11,28) is

2 Δ 2 n P n 1 x n n 1 . 2

1) W (n n Vx

S

Wx = _⎢⎣ ^⎡ − − _⎥⎦ ^⎤ − − + φ φ − (14.13)

where again S n and n are polynomials whose degree does not depend on n.

To determine the M fraction (14.4) we therefore must first find the polynomials φ and n . Now we are approximating f for both x small and x large and both of these considerations will yield information on the coefficients of these polynomials φ _n and n .

φ _n is defined by (14.7)

,

Q f n n n 2V

Q n f dx W d 2 n Q n (x) 1 x n 1 A n

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ⎥

⎦

⎢ ⎤

⎣

⎡ −

− =

+ φ

substituting (14.6) we find

⎪

⎩

⎪ ⎨

⎧

⎭ ⎬

⎫

⎩ ⎨

⎧ + + +

− +

+

−

= W 2V other terms x large (14.15)

x 1) (n 1 Q n

1

(14.14) small

x terms

other 2Vx)

(nW n (x)

φ

where the dominant terms are contained in these expressions.

Prom the relation (14.8), the polynomial n can be written

Q n n 1 n Q 1 x P n Q n

' n Wx Q 2 Vx

1) W n (n

Δ −

− +

−

−

−

= φ

and again we cons ider x both small and large.

29.

For x small, Q n = ¹ + α n x + - - - --- where α n = q 1 + (p 2 +q 2 ) + - - + (p n +q n ),

+ + − −

−

−

− + + +

− − + +

−

−

− +

−

−

=

∴ 1 α n x

1 x α n n 1 1 x P n n x

α 1 α n Wx 2 Vx

1) W n (n

Δ φ

2 1) W (n −

= -Vx - Wx α n - P n+1 x φ _n + higher terms . (14.16) For x large, Q n = (q 1 q 2 (x ⁿ + w n x ^n-1 + - - )

,

q n 1 q n

n ) 1 p (q n q 2

q 1 2 ) 1 p (q q 1 n 1 w Where

−

− +

−

− + +

+

=

∴ ∆ n

2 1) W (n −

= -Vx - W

q n n 1 ) p

x n

(n − w − + φ _n + lower terms. (14.17)

These conditions,together with the relation φ n

Δ n 1 Δ n

2 + + + =

w ( 1+q n+1 x ) , (14.18) will be sufficient to determine Φ n and ∆ n in the two simple examples which we will now use to illustrate the theory.

Dawson's Integral

One of the neatest M fractions belongs to the function which satisfies the differential equation

2x f' + (1+x) f = 1 f(0) = 1 (14.19) With W = 2x, 2V = - 1+x), we find

_⎪ { }

⎩

⎪ ⎨

⎧

−

−

− + +

=

−

−

− + +

− + +

−

−

−

− + +

=

−

−

− + + +

= x x large

1 q n x) 1

(1 1) 2(n 1 q n

1

small x

1)x (2n x)x

(1 2nx φ n

consequently φ n = (2n + 1 )x and

1 2n 1 1

q n + = + .

For x small = − + + _⎟ − − + + − − −

⎠

⎜ ⎞

⎝

⎛ x 2x 2 α n P n 1 x n 2

x 1)x 1

n (n

Δ φ

= − )x + 2

(n 1 higher terms

( 4 ² n ¹ ) ^x

1 P n 2 2n

n 1 2 x 2

φ n q n n 1 P x n n W 2x 2 x

x 1)x 1

n (n Δ large x

For

⎥⎦ ⎤

⎢⎣ ⎡

⎥ ⎦

⎢ ⎤

⎣

⎟ ⎡

⎠

⎜ ⎞

⎝

⎛

+ − +

−

− +

=

−

−

−

− +

−

− + −

+

−

=

But the coefficient of x is (n - ½), therefore

.

2 1 4n 1 2n P n and x

2 ) (n 1 2 x 2 Δ n

− − + =

− +

=

Hence the M fraction solution of (14.19) is

− − − −

− +

− +

− +

= +

7 x 1 1

35 x 6 5 x

1 1 15 x

4 3 x

1 1 3 x 2 x 1 f(x) 1

which can be written

. (14.20)

x 1) (2n

2nx x

7 6x x

5 4x x

3 2x x

1

f(x) 1 − − −

− +

− +

−

− −

− +

− +

− +

= +

Further from (14.13) we can readily deduce that S = nx ≡ H n (x)

and that the denominator polynomials Q n (x) satisfy the differential equation

2x Q n " - [x + (2n-l)] Q’ n + n Q n = 0 . The integral

z e t 2 dt z f(2z 2 ) 0

z 2

e − ∫ =

is known as Dawson's Integral. The accuracy of the approximations obtained by simply truncating the C.F after a given number of terms is indicated below. The approximations are of course good for z <1 and z>4.

Z 1 2 3 4

Accuracy 8 terms 5D 2D 3D 5D

Accuracy 12 terms 9D 4D 4D 8D

31.

f = cot ^-1 x

A simple function having the correct behaviour at infinity as well as a Taylor expansion at the origin is f = cot ^-1 x.

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

−

−

− +

− +

−

− =

−

−

− +

− +

−

− =

= −

. large 7 x

7x 5 1 5x

1 3x 3

1 x 1 x 1 1 tan

small 5 x

x 5 3 x 3 2 x

x π tan 1 π 2

f

This function satisfies the differential equation

1 f(0) π 2

dx ) df x 2

(1 + = − = (14.21)

⎪ ⎪

⎪

⎩

⎪⎪ ⎪

⎨

⎧

−

−

− + +

+ +

−

+

− +

− =

−

− + +

=

x 2 1) 1) (x 1 (n q n

1

1 x.

q n n 1 n n

iving 2 )

x n(1 n

Now φ

φ

g

For x snail ∆ n = 2

1 (n-l) (l+x ² ) - (1+x ² ) xα n - p n+1 x Φ n + - - - -

= 2

1 (n-1) - (α +n p n+1 ) x + higher terms.

For x large = − + − + − − + n + − − − q n

1 p n x n )

(n w 2 1) (x 2 1)

(x 1) 2 (n n 1

Δ φ

x lower terms .

1 q n q n P n 1 1)

n (n 2 W

1)x 2 (n

1 +

+ + +

+ + +

= _⎟ ^⎟

⎠

⎞

⎜ ⎜

⎝

⎛

(n 1)x 2 .

2 x 1 δ n 1) 2 (n n 1 Δ

Hence = + + − +

Further φ _n and D are related by (14.18),

2 W ⁺ Δ _n ⁺ Δ _n-1 =n-1 φ _n-1 (1+q n x), and from the coefficient of x we deduce δ n ⁺ δ n-1 = (n-1)q n -

q n

n . (14.22)

This formula and our two expressions for δ n ,

δ n = - (α n + np n+1 ) , δ n = w n + (n+1) , 1 q n q n P n 1

+ +

are clearly sufficient to enable us to successively calculate

δ n , p n+1 and q n+1 .;in fact we could eliminate δ n . For definitions of α n and w n see (14.16) and (14.17). However this is by no means the end of the story. From (14.13) besides showing that

S n = - (n-1) (δ n +np n+1 ) - nx (14.23) we also find that

,

q n n 2 1 q n

2 1 1 n P n n 2 2 n

δ _⎟ ^⎟

⎠

⎞

⎜ ⎜

⎝

⎛ +

+ + −

+

= (14.24)

and

1 . P n 1) 1 (n

q n q n P n 1 1) n (n

2δ − − +

+ + −

= (14.25)

Immediately we see that w n = α n + p n+1 ., and hence deduce

q n

q n 1 q n P n q n

1 1

P n −

+ −

+ = (14.26)

and

q n 1 n P n n ) n q (p 2) 1 (n

q n q n P n 1 1)

(n = − + − + −

+

+ + (14.27)

a simple pair of formulae which successively generate the coefficients P n+1 q n+1 , in the C.F. In practice we find that p n tends steadily to

2 ,

− 1 while a tends steadily to 1 as n → ∞ (which agrees with one

solution of letting p n → p and q n → q simultaneously in these formulae).

A close examination of the computed values of p n and q n , see Table B ,

33.

suggested that n 3 .

0 1 n ~ q 1 2 and

n 0 1 2 ~ n 1

P _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛ −

+

Putting _⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + + + − − −

−

= n 3

τ n 2

ρ 2 n 1

P

and = + + − − − n 2

1 σ q n

in the formulae (14,26), (14.25) and (14.24) successively; we find α = - 2p

= − + + − − − n 2

σ 2 n 1

δ p =

8 1 and r = p.

Hence the asymptotic forms of p n , q n and δ n are

⎪ ⎪

⎪ ⎪

⎭

⎪⎪

⎪ ⎪

⎬

⎫

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎛

−

−

− +

−

−

=

−

−

− +

−

=

−

−

− + + +

−

=

2 , 4n

1 2 n 1

δ

4n 3 1 1 q n

8n 3 2 1 8n

1 2 n 1

P

(14.28)

higher terms could be found.

As p n and q n tend steadily to limiting values we can of course plug this C.F after n terms with R such that

R n ,

1 x q n 1 P n 1

R n +

+ + +

−

=

or simply with R

R x 1

2 x 1 R = − + +

R ² + (1+x) R + 1/2x = 0

i.e. .

2 x 2 1 x)

R = − (1 + + +

The two M fractions that we have considered have both been treated

numerically by J. McCabe [p54, p143] so we will not pursue then

further.