Some infinite products with interesting continued fraction expansions

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C. G. P

INNER

A. J. V

AN DER

P

OORTEN

N. S

ARADHA

Some infinite products with interesting continued

fraction expansions

Journal de Théorie des Nombres de Bordeaux, tome

5, n

o

_{1 (1993),}

p. 187-216

<http://www.numdam.org/item?id=JTNB_1993__5_1_187_0>

© Université Bordeaux 1, 1993, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Some

infinite

_products

with

_interesting

continued fraction

_expansions

by

C. G.

_PINNER1,

A. J. VAN DER

POORTEN2

AND N.

SARADHA3

RÉSUMÉ. We _display several infinite _products with interesting continued fraction expansions. Specifically, for various small values of k -

necessarily excluding k = ₃ since that case _{cannot occur,} we _display infinite products

in the field of formal _power series whose truncations _yield their _{every k-th} convergent.

1. Introduction

Let ... be a sequence of rational

integers,

and let

F(x)

be the infinite

_product

with continued fraction

_expansion

_{~ao(x~ , ai(z) , ... ] .}

In

_[1]

it is shown that for n =

_{0, 1, 2,}

... each truncation

is a

_convergent

of

F( x ~

_if,

and

only if,

1980 Mathematics Subject Classification (1985 Revision). 11A55. Mots-c]6s : continued _fraction, infinite _product.

Manuscrit _{reçu le} 10 _juin 1993.

1

This work was done in _part at the Mathematical Sciences Research _Institute, Berkeley. 2 _Work

supported in _{part by grants} from the Australian Research Council and _by a

re-search agreement with _{Digital Equipment Corporation.}

3

Honorary research fellow attached to the ceNTRe for Number Theory Research,

Moreover,

then the

_degree

of the

_{subsequent partial}

_quotient

is

_given

by

Furthermore,

one sees that the truncations

provide

precisely

_every sec-ond

_convergent

_if,

and

_{only if,}

for

_{n = 0, 1, 2, ....}

In

_fact,

_given

_(5),

the

_partial

_quotients

of

_F(x)

can

readily

be listed

explicitly

and are of

rapidly increasing degree.

Hence-forth,

we

only

consider

products

for which

(3)

is satisfied. So their _every

truncation

_yields

a

_convergent.

Here we will be concerned with classes of infinite

products

whose contin-ued fraction

_expansions

have the

_property

that those

_convergents

_arising

from the truncations of the

_product

are

_{again separated by}

some few

con-vergents.

In this context we _say that two consecutive truncations

and of our

_product

are

k -apart if,

as

suggested by

the

nota-tion,

there are

exactly k -

1 intermediate

_convergents.

Our aim is to use the above mentioned

_2-apart

_examples

to construct additional

_products

with truncations a fixed bounded distance

_apart.

We shall

explicitly give

examples

with

_regular

_gaps of size

_{4 , 5 , 6}

and 8. We also illustrate how

the matrix

_{interpretation}

of continued fractions can be

nicely employed

to

cleanly

generate

the

_partial

_quotients

of the

_products

we

_study.

2.

_Terminology

We use the

terminology

of

[1]

and

[2].

Namely, given

a field

K,

we let L =

lI~((~-1 ))

denote the field of formal Laurent series in

x-1

over K. Then each F E L is of

shape

and we _say that the

_degree

of F is

_deg

F = d. The

integral

_part of

_F,

in

_K[x].

A series F E L has a

unique

continued fraction

expansion

denoted

by

where the

_partial

_quotients

ch are

polynomials

in _{x ,} and have

_degree

at

least 1 for h &#x3E; 1. From the

_{general theory}

of continued fractions we have the fundamental

_{correspondence}

_whereby

for h =

_{o, l, 2, ...}

if,

and

_only

_if,

defining

the

_convergents

_ph/qh

_{by products}

of certain matrices.

_Further,

on

_setting

we observe that for c E

7~ ,

B /

Then we _may write

formally

that,

, - , , /

whence if ~2013~ denotes the

correspondence

between matrix

products

and

continued fractions we have

The

_{correspondence}

between continued fractions and the so called R-L sequences is

particularly

convenient in

studying

the

multiplication

or divi-sion of a continued fraction

by

a rational

_quantity.

This is illustrated in

detail in

_[1].

We assume the relevant

_terminology

here.

We also recall from

_[2]

that a rational function

p( x) / q( x)

is a

_convergent

Phlqh

to F

_if,

and

_{only if,}

and that the

_degree

of the next

_{partial quotient}

is then

_{given by}

3. The

_Examples

Revealed

As mentioned

_above,

we

already

know a class of infinite

products

such that the truncations are

_always

_2-apart;

namely

where A =

_{(Ao, al, ... ~}

satisfies the conditions

for all n &#x3E; 0. It

_transpires

that we can use such known cases to construct

a slue of new

examples

with the

_property

we want. In

_particular

if

then the truncations of G

_(although

in

_general

no

longer

2-apart)

will still be of bounded distance

_apart:

THEOREM 1. Let

where

Then for a~I n =

0, 1, 2,

... , the successive truncations of

G(x)

are

_convergents

of G at most

_(2r

+

₂₎

_-apart.

Proof.

_By

criterion

₍₃₎

it follows that the stated truncations

_certainly

are

Moreover the condition

_{2Ài.1 I}

_Ài+1

leads _{to a,} so to

_{speak, gratuitous}

inter-mediate

_convergent

with

as _may be

readily

checked with an

_appeal

to

_(6).

Between them these two

partial quotients

soak _{up most} of the available

_degree

and

_(as

the

_remaining

are of

degree

at least

one)

we obtain

Note: we see at once the the truncations are

2-apart

if and

only

if

,6;-- v

and that otherwise

_(since

_by

_{degree considerations j #}

1 1~ -

_{1 )}

we must have

_separation

k &#x3E; 4. We shall see later that there are in fact

no

_3-apart

examples.

Since one can

envisage

cases were all the

remaining

quotients

are indeed linear the bound 2r + 2 seems

reasonably

precise;

we shall see cases where this bound is

sharp

when r =

_1,

2 _or 3.

Not

_{surprisingly,}

the

_phenomenon

of one

product

inheriting

the

bound-edness

_property

from another is rather more

_general:

THEOREM 2. If

_F(x)

and

_G(x)

are infinite

products

(whose

truncations

are all

convergents)

which differ

_{only by}

a factor

consisting

of a rational function

, , 1"/ ’B

then the truncations of

_G(x)

are of bounded distance

_apart

if,

and

only if,

the truncations of

_F(x)

are of bounded distance

_apart.

LEMMA 1.

_Suppose

that

_pilqi

and are

_convergents

to a Laurent series

_F(x)

such that

are

_convergents

to

Then

Proof.

The

_{proof employs}

a similar

_{degree counting}

_argument:

We _suppose that there are M + 1

_convergents

₍₀

_j

_M)

to

_G(x)

with

Further _suppose that there are N

_convergents

_{(1 j N)}

to

_F(x)

with 0

_n(j)

k and

_deg

&#x3E;

_{deg fg.}

Now since

we see from criterion

(6)

that

all are

_convergents

to G with

these

_convergents

_monopolise

most of the available

_{degree. Hence,}

as the

remaining

convergents

are at worst linear we see that

The bound follows at once since

_trivially

N k and t M .

Notice that we made no

_prior

_assumption

here about the nature of

_F(x);

for

_example

we

might

consider

products

with

to ensure that the ’truncations’

are

_convergents

(needless

_{to say} we shall do no such

thing

here).

4. The Transduction Process Justified

We saw in Theorem 1 that

_examples

with the desired

_property

could be

obtained

_by

_multiplying

a

product

known to have bounded _gaps

_(for

exam-ple

the

_2-apart

_products)

_by

a rational function. In this section we show

how the matrix formulation alluded to in

_§2

can be used to

_generate

the

new

_quotients

from the old. We suppose that

so that in terms of the matrix

_equivalence

Hence we _may obtain the

b2’s

from the

Ci’8

by passing

the

multiplying

matrix

_through

the and L~’s term

_by

term. At each

_stage

this amounts to

_multiplying

each RC or L’ on the left

by

a matrix of determinant

f g

of

the form

where

_deg

C

_deg

A .

_By

_applying

Euclid’s

_algorithm

to the rows of the

resulting

matrix

_(that

is _{to say}

_{by writing}

where d = or d’ =

[0/6]

and _so

on)

_we

_output

_a successsion of

Rd~’s

and

Ld’s until

we are left once

_again

with a matrix

In the

_special

case that all the

ci’s

i &#x3E; 1 have

deg

ci &#x3E;

_{deg fg}

_(as

will be the case in all of our

examples)

the

procedure

can be described

fairly

concretely:

As

_{deg co}

_may well be zero the first transduction can be

_atypical.

De-noting

by

pi/qi

the

_convergents

to

_coflg

=

[do ?

... ,

dr]

where

co f

=

a( x )Pr

and g

=

a(z)qr

we see that

The

_general

transition

_through

an Lc is

particularly

simple:

, , ,,/

with

_{deg C’}

=

deg

(Bci

₊ C - A

deg A .

The

_general

_passage

_through

an Rc is a little more

complicated

and breaks down most

_naturally

into two _parts:

We first let

_pilqi

denote the

_convergents

to

_A/C

=

[eo ,

_{el ,}

... ,

et]

where A =

ti(x)pt

and C = and observe that

where u = +

_(-l)’Bpt-,

and v =

(-1)t-lBpt;

of _course there is the

convention that

/-. -B

/ B / -. B

so that if C = 0 we and do

_nothing.

Hence if we let denote the

_convergents

to

_u/v

=

[ho ,

... , with u =

A(x)p’ s

and v = we obtain

where

_deg

C’

_deg

A’ .

So as we _pass

through

successive R’s and L’s the R’s of the

output

are

_always

followed

_by

L’s and vice versa. Moreover due to the condition

deg

ci &#x3E;

deg f g

=

deg

AB we see that

(with

the

_{possible exception}

of the

very first

output

do )

all our

_outputs

are of

degree

_greater

then or

equal

to one. Hence in this case we are indeed

_obtaining

(term

by

term)

the

legitimate

expansion bo, bl,

... of

G(x) .

Since our

output

is never less then our

_input

we can

(by

feeding

the

_output

back in as

input)

use this

approach

on

F(x)

satisfying

a functional

_equation

of the form:

We shall

_begin

the next section with

_just

such an

example.

Of course the described _process still functions if we allow the

possibility

of

_{input quotients}

with

_deg

ei =

deg f g .

In that case the

_output

is still sustainable but _may well now contain _rogue constant terms -

although

techniques

do exist for

_{individually removing}

such

_{illegalities relying}

on the observations that

Input

containing

terms ei with

deg ei

deg( f g)

is more

problematical;

not

_only

is the above _program no

_longer

_quite

suitable but there

_genuinely

seems no _way of

_preventing

the

possibility

of R ’s followed R’s or L’s

_by

L’s so that

(with

the

_danger

of wholescale

_{back-tracking through}

cancellation)

we no

longer

have _any

right

to assume that the

_output

we

produce

from a

finite number of transductions bears much relation to the final

_expansion.

Note that with the

_exception

of the two

_{possibly large}

_quotients

_{l BCA+C J}

and

_ho

= the

_{remaining quotients}

produced

in

_passing

through

an are

(by

considering

the

degree

of the denominators C and

_{v )}

of

limited

_degree;

a

_phenomenon

that we have

_already

encountered in the case described in Theorem 1. In fact since the sum of their

_degrees

is at most

_{2 deg f g}

we see that each L’i RCi+l

produces

at most 2

_{deg f g}

+ 2 of

output.

More careful

_analysis

of common factors could no doubt be used

to recover the bound of Theorem 1 in this _way. When we are interested in truncations we can

(by

considering

the

potentially

large degree

of the

subsequent

partial

quotient)

use these observations to locate their

_position

in the

_output;

if the truncation is

_{(as indicated)}

the

_(n

+

_l)-st

convergent

to

_F(z)

then when n is even the

_expansion

for the truncation

convergent

fp,,Igq,,

to

_G(x)

_{corresponds simply}

to the

_output

_generated

by passing

through

When n is odd we need in addition the eo, el, ... , et

produced

(when

C ~

0)

in

partially passing through

Of course the former situation is

preferable

and indeed when

F(x)

is our

familiar

_2-apart

_product

we can _arrange for the truncations to be

_exactly

the ’even

_{convergents’}

_(at

the _expense of a

tolerably illegal

and in some

ways

preferable expansion

as we shall see

below).

Although

the cases are somewhat limited in form we are faced with

potentially infinitely

many different

possibilities

for the transition matrix

We would

_hope

that the transduction _process

_produces,

if not a

Cc

B)’

finite set, then at least

_only

matrices of a similar

enough

shape

that we can avoid

_simply

_performing

transitions ad infinitum. In the next section we

see this

_technique

in action.

5. The Matrices in Action

As mentioned

_earlier,

it was shown in

[2]

that _any infinite

_product

of the

form

where A _

(Ao, Ail, ... )

satisfies

has truncations

_2-apart

with

_expansion

and

We

_begin

this section

_{by using}

the method of transduction described above

to

_give

an alternative

_proof

of this fact: From the functional

_equation

we see that

The trivial observation

_{bo(x; A’)}

= 1 entails that the initial transition

yields

Now since

_{Aj )}

_I

we see that

F(x; A’)

is a

_polynomial

in

XÀl

and so

possesses

partial

quotients

with

Àl

&#x3E;

_2Ao

for all i &#x3E; 1.

Hence from our

_previous

comments we are allowed to read off the

_{bi(x; A)}

directly

from our

_output

as it is

generated

term

_by

term. Thus we obtain

bo(x;

A)

=

1,

b1(x;

A)

= and _we

proceed

to the next transition armed

with the

_knowledge

that

_A’)

=

XÀl

and the

assumption

that

2AO

I

À1:

Hence

_{b3(x; A)}

and

_{b4(x; A)}

are of the stated form.

From this

_point

onwards the transition matrix moves

through

the R’s and L’s

_unchanged.

In

_particular

under the

_assumption

_(as

we have

_just

seen for N =

1)

that

1)

divides

b2N.+.2(x; A’)

we obtain at the

N-th

_subsequent

_stage

and

A

_simple

induction

_argument

leads at once to the relations

and the stated formulae for

_{b2N+l(x; A)}

and

_A).

Our

_previous

comments or

simple degree considerations

show us that the truncations are

always

2-apart.

From now on we will assume such an

_expansion

which

(in

the manner of

_[2])

it will _prove convenient to write in the

_{slightly illegal}

form

, - - I

-with

where

with

From the functional

_equation

we obtain the matrix

_{correspondence}

where the ci all are

polynomials

in and hence of

degree deg ci

&#x3E;

Ao

&#x3E;

2r . In

_particular

_(from

our earlier

analysis)

it is clear that

(once

we have

safely passed through

the initial we are allowed to read off the ai from the

output,

transition

_by

transition. Of course in

_general

we

cannot

_expect

our transition matrix to remain constant as occurred in the above

_example;

however we can

hope

that it

changes

in a

_fairly

_predictable

manner.

THEOREM 3. Let

with

Ai

_satisfying

_(12).

Then the truncations of G are

eventually

_4-apart

for n &#x3E; 1. The first 5

irregular partial quotients

are

_{given implicitly}

below.

Proof. We first note the

_general

matrix transition formulae

Passing

the

_multiplier

_through

the first three terms

_gives

us the first five

partial quotients

ao , ... , a4 :

Now

_applying

the above formulae with

_{A = 4 ,}

D

_{= 4 ,}

_{Bo - - 2}

we see

B that passage

through

the n-th block LC2n+l RC2n+2 takes the form

with

_Bn

=

Bn-1 -

tC2nH( -1) = -

Hence the

_alleged

partial

quotients.

Moreover

_by

_previous

comments or

_{by simple degree}

considerations we see that the truncations coincide

exactly

with the finite continued fraction

_expansions

_produced

_by

these transitions and hence are all

_(bar

the

_first)

_four-apart.

THEOREM 4. Let

with

_Ai

_satisfying

_(12).

Then

(i)

if 4

_I

_Àl

then the truncations are

eventually

_2-apart

(initial

_gap

5)

with

_{partial quotients}

(ii)

then the truncations are

eventually

_6-apart

(initial

_gap

5~

with

_partial

_quotients

for n &#x3E; 1 where

The first 8 or 13

partial

quotients

can be discovered below. -, Proof. In both cases the initial transitions

_produce

the first 7

_quotients

after which the transition matrix remains

_unchanged

with

for n &#x3E; 1. The stated

_quotients

are then immediate. When

_{4 t Ai}

we

require

two further

irregular

transitions

before the process settles down with

for n &#x3E;

_2,

where

The

_{given partial}

_quotients

then follow

_{fairly easily.}

with

_Ai

_satisfying

_(12).

(i)

If

_{(3, Aj )}

= 1 for all i then the truncations of G all are

_5-apart

with

partial quotients

for n &#x3E; 1. The first seven

_{atypical partial}

_quotients

can be found below.

(ii)

If 3

_I

_AN

for some N &#x3E; 1

_(where

we take N to be

_minimal)

then the truncations of

_G(x)

are

eventually

(after

the N’th

truncation)

_{2 -apart}

with

_partial

_quotients

for n &#x3E; 1. The first

_(5N

-~-

₂₎

quotients

are as in case

(i) ,

and a5N+2 can be found below.

After the initial transitions

we have the

_following

identities: If n is odd

and

and

So each time we _pass

_through

a block we _output 5

_partial

quotients

and if n is even

(respectively

_odd)

either an A

(respectively C)

if

_{An *}

4

_{(mod 6)}

or a B

(respectively

D )

if 2

_{(mod 6).}

The stated

quotients

can be read off and then

_degree

considerations confirm that the

Now if 3

₁

_AN

for some N &#x3E; 1

(assumed minimal)

then the _process remains the same until we hit L C2N +1 . In

particular

the _{ao, a1, ... ,} a5N+1 are

generated

as above.

Writing

we see that

after which the transition matrix remains

_unchanged

with

for n &#x3E; N . The stated

_partial

_quotients

are then evident and

degree

considerations or otherwise show that the truncations are indeed

_2-apart

as claimed.

THEOREM 6. Let

with

_Ai

_satisfying

_(12).

The the truncations of G are

eventually

_4-apart

The first nine

_{partial quotients}

_{ao, ... , a8} can be found

Proof. If

_Ao

is odd and 6 !

_I

A,

then we have the

following

transition relations:

Here we have to endure some rather

_painful

initial transitions:

if

Ao T

1

_{(mod 6)}

and

if

Ao m

5

_{(mod 6),}

_where!3o

_{= - 3 (~1 ~~0~ .}

Hence after

_{passing through}

the k -th block we

_output

4

partial quotients

and a transition matrix of the form

where

_{f3k == f3k-l -}

_{3 czk+s(-1) _}

_{-Àk+1/2k3Ào.}

This makes the theorem evident.

Notice that we have covered here all cases of

(11)

with r = 1 or 2 and

a number of cases when r = 3. The

remaining

_cases with _r = 3 become

rather too

_unwieldy

to handle

_by

_locating

the

_quotients

in this way

(when

2 r

Ao

and

_{3 ~}

not

_only

do we have to force

_through

_{multiplication}

by

the two

_polynomials

_(x

+

₁₎

and

_{(x2 - x}

+

₁₎

but,

as we have seen

before,

the

_multiple

factor

_(x-~-1)2

in

_{(1-~x-3)(1-f-x-~°)}

_spawns

_unwieldy

constants).

These

_missing

cases will be

_given

in

_§6

and are found

_by

considering

convergents,

as

opposed

to

_evaluating

_{partial quotients.}

6. The

_Convergents

Considered

Given the

_convergents

and

_pn+h/qn+h

and the intermediate

_partial

quotients

an+1, ... , an+h it is

fairly

easy to retrieve the

missing

convergents

Pn+h-1 /qn+h-1

in the

_following

manner:

We define

and observe that

_{s(r, m)}

can be

_{straightforwardly}

written in terms of _aT+2,

... ,

by

means of the

easily proved

iteration relation

where,

as is well

known,

It is not hard to then show

_(by

induction on

r )

that the intermediate

_{pn+r ’s}

and

_qn+r’s,

1 r t~ - 1 are

_{given by;}

If the reader should

_object

that

_knowing

_{pn /qn}

and

_Pn+h/qn+h

can

_only

determine _{pn, qn} and _{pn+h, qn+ h up to} a constant we observe that to obtain we need

only

determine the ratio of the

_leading

coefficients of

and _Pn and this is

_simply

the

_product

of the

_leading

coeflicients of

In fact not

_only

must the

_missing

_convergents

_Pn+r/qn+r

be of a

special

form but

_{conversely given polynomials}

A and B with

_deg

AB

_deg

_S(n,

_h)

such that

.&#x26;...

satisfies

then

_p/q

is an intermediate

_convergent

for some 0 i h This fact is an _{easy consequence} of criterion

(6)

since

where

_clearly

Hence in order to find the

_(n

+

_i)-th

missing

convergent

it is necessary and sufficient to look for

_{polynomials A, B}

_(sharing

no

superfluous

common

factors)

with

and

giving

with

The limited form of the intermediate

_convergents

allowed

_by

₍₁₄₎

enables

TH.EOREM 7. Let

with the

Ai

E I~

_satisfying

Then the truncation

_convergents

are

(i)

2-apart

if and

_only

if

with intermediate

_convergent

(ii)

3-apart

only

in the

_special

case

where X =

We

_immediately

obtain the

_following:

COROLLARY 1. The infinite

_product

has truncations which are

eventually

_2-apart

(that

is,

from the truncation

_onwards)

if and

_only

if

There are no infinite

products

of the form

(15)

whose truncations

(except

possibly

the

_first)

are ever

three-apart.

Proof. We _suppose

_{that p}

and _p are constants such that

so that

Now if h = 2 we see from

(14)

that the middle

_convergents

are

_necessarily

given by

For the first of these to be a

polynomial

we

require

that

jib,,

=

p;

sub-stituting

this in the second

_gives

the stated criterion.

_Conversely

if such a condition is satisfied then

by

appealing

to our

_previous

comments or to

criterion

₍₆₎

we see that the

_given

fraction is an intermediate

_convergent

and moreover

_by

_degree

considerations that it is the

_only

one.

Now if h = 3 the

equations

(13)

and

(14)

produce

From the

_equations

for _pm+2 we see that

for some

polynomials

A(x)

and

B(~).

However since

we see that

_{A, B}

are both in fact non-zero constants. Hence

_substituting

these

_expressions

for _am+2 and back into

₍₁₆₎

we obtain an

expres-sion of the form

In view of the

_rapid

increase of the

_Ài

&#x3E; 2

~2.- o ~~ ,

_comparison

of the number of terms on the

right

shows that such a relation is

only possible

if n = 1.

Finally

substituting

for

am+3 in q?.",.+1

gives

from which one

easily

deduces that

ao

I

Ai .

An obvious

change

of variable reduces this to the case

( 1

+

_{X-I )(1}

+

cX -~‘)

which as we have seen is

2-apart

and when

_{c ~}

_{(- 1)~}

is

_easily

checked to be

_3-apart

with

_expansion:

where b =

(-I) A c.

Our

_examples

have featured constant _gaps between the

_truncations;

lest the reader be misled into

_believing

that this is

_generally

so we

_give

the

THEOREM 8. Let

with the

_Ai

_satisfying

₍₁₂₎

and

Then the truncation

_convergents

are

_5-apart

when m - 3

(mod

4),

_6-apart

when rra - 2

(mod 4),

_{7 -apart}

when m - 0

(mod 4)

and

8-apart

when m - 1

_{(mod 4)}

with intermediate

convergents:

with

_u(x)

=

1 - x -~- x2 - x3 ,

and 1 - x .

Proof. The above

_expressions

_{may appear} formidable but the _process for

convergent

pN+k/qN+k .

Hence to find the

_convergents

between and

_PN+k/qN+k

it is

_enough

to find

_polynomials

_{Ai, Bi}

with

_{deg AjBj}

3

such that the stated denominators are

_polynomials.

Thus

starting

with

A1

= 1 we find successive with

deg Aj

= 4 -

deg Bj -1

until we

have filled in the _gap

_(that

_is,

_{E(4 -}

_degAjBj)

=

4).

Similarly

for the

Of course we have made

_things

here

simple by

choosing

the

Ai

such that for a

root

of we have

Note that it is not hard to see that a

judicious replacement

of a

subsequence

of the

Ai

by Am m 6 ~ 2"~+2

(mod

10)

can even

inject

an _appearance of utter randomness in these different

_spacings.

As

_promised,

we conclude with the

_remaining

r = 3 cases:

THEOREM 9. Let

with

_Ai

_satisfying

_(12).

Then the truncation

_convergents

are

where - and k are the constants

Proof. As

_before,

the

_proof

amounts to

_tediously

_finding

_Ai,

_BZ

such that the denominators are

polynomial

and so that the various

degrees

add _up.

REFERENCES

[1]

J.-P. Allouche, M. Mendès France and A. J. van der Poorten, An _{infinite product}

[2]

M. Mendès France and A. J. van der _Poorten, Some _explicit continued _fraction expansions, Mathematika, 38 _(1991), 1-9.

Christopher G. Pinner Alfred J. van der Poorten

Natarajan Saradha

Centre for Number Theory Research

Macquarie University

NSW 2109 Australia

pinner~mpce.mq.edu.au alftmpce.mq.edu.au saradha~mpce.mq.edu.au