Hypersurface 2

8/10/2019 Hypersurface 2

1/11

DETERMINANTS- HYPERSURFACES

FOR

LUCAS SEQUENCES

OF

ORDER

r

AND

A

GENERALIZATION

A

F

HORADAM

University of New England, Armidale, Australia 2351

(Submitted December 1983)

1 .

INTRODUCTION

In Hoggatt and B ickn ell [1 ] , th e Fibon acci sequence {R

n

} of order v > 2)

was defined by

R

n+ r

= R

n+r-l

+R

n+r-2~*~

e

+

Rn

s

R

= 1

s

R

2

= 1, 1-1)

with

R-

(r

-2) =*- i-3 ) = = - ^ = /? = 0. 1.2)

Using themethod ofagenerating matrix for

{R

n

}

,they obtained thedeter-

minantal identity

R

n+ r- 1

"n+r-

2

R

n+l

R

n

R

n +r-2

R

n+ r-3 *

R

n

R

n- 1

R

n +

]_

n

" w

- r

+ 3

^w-r 2

Rn

R

n-

1

Rn- r +

R

n- r +

which is an extension of the Simson formula (identity)for thesimplest case

V

= 2

for Fibonacci numbers.

Carrying these numbers

R

n

through tocoordinate notation (writing

x-

=

R

n

*

x

2 ~Rn+i>

x

3

=

Rn +2

B

x

r ~Rn+r-.i)*

t n e

author [4]showed that (1.3)could

be interpreted as one ormore hypersurfac es inEuclidean spaceof r dimensions

(the number

of

hypersurface loci depending on

n).

Thecases

v = 2

9

3

9

4

were

delineated inalittle detail ([3 ], [4]).

Itis nowproposed to extend the resultsin[3] and [4] to thecaseofa

Lucas sequence {S

n

}or order r

s

i.e., toconstructadeterminant analogousto

(1.3)and tointerpret itgeometrically asalocus inr-space.

From experience

3

weshould expectthealgebraic aspects of {S

n

} toresemble

thoseof

{i?

n

}.

Nevert heles s, therearesufficient variations fromth eFibonacci

case tomake the algebraic maneuve rs, which constitute the main partofthis

article,

achallenging an dabsorbing exercise.

Becauseofcomplications associated with th efact that S

0

[tobedefinedin

(2.1)]isnonzero, whereas R

0

=0, themethod usedbyHoggatt andBicknell[1]

for

{R

n

}

is not

applied here

fo r

{S

n

},

However,

o ur

method

is

applicable

to

{i?

n

},asw eshall see, providedwe add to thedefinitionsin[1] theinjunction

R-(r-l) = 1.

Schematically, this paper consistsof two parts. Part

I

isorganized

to

secure results for theLucas sequence which correspond tothosefor theFibo-

nacci sequence. On thebasisofthis knowledge, inPart II webriefly gener-

alize the results fo rasequence which contains th eFibonacciandLucas(and

other) sequencesasspecial cases.

1986]

227

( - 1 )

(r-l)n+[(r-l)/2]

(1-3)


2/11

DETERMINANTAL HYPERSURFACES FOR LUCAS SEQUENCES OF ORDER r

PART I

2.

LUCAS SEQUENCE OF ORDER

r

Define {S

n

}, the Lucas seq uence of order r (^2) by

S

n

r ~ S

n

v

-i+ S

n

+

r

- 2+ * + S

n

, S

= 2, S

= 1,

with other initial conditions

\S_

l

= 5_

2

= =

&-(p-

2

)

=

^-(r-1)

_ 1

-

Simplest special cases of

{S

n

}

occur as follows:

J

n+ 2

^n+1

+

L

n>

L

o

- 2, L

x

- 1,

L_ -1;

Mn+3 =

M

n+ 2 + Mn+l + ^ n > ^ 0 =

2

'

M

l =

X

>

M

-1 = '

M

- 2 = " ^

2,^ = 1,

^ - 0,

N_

3

= -1.

The first few numbers of these sequences are:

K

L

i

L

2

L

3 h

L

s

L

e

L

7

L

s

L

s

L

io

|2 1 3 4 7 11 18 29 47 76 123. .

)M

Q

M

M

2

M

3

M

h

M

5

M

s

M

?

M,

M

n

M,

9

i J

1 0

10 19 35 64 118 217 399

N

0

N

N

2

N

3

N

Li

N

5

N

Q

N

?

N.

t

1

9 10

12 22 43 83 160 308 594

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.3)'

(2.4)'

(2.5)'

The determinant of order r (which we may here call the Lucas determinant

of order r, corresponding to that in (1.3) for the Fibonacci sequence of order

r) is

Ay,

=

Sn + r -1

Sn + r - 2

S

n

+1

S

n

S

n

+ v - 2

Sn+ v -3

S

n

S

n

- 1

S

n

+ 1

S

n

S

n

- r +3

n

-

r

+ 2

S

n

S

n

~ l

Sn - r+ 2

Sn-r+ 1

(2.6)

Notice the cyclical nature of the elements in the columns of

A

r

.

Conse-

quently, there is symmetry about the leading diagonal of A

r

. Both of these

properties for {S

n

} are also features of the Fibonacci sequence {R

n

}

Special notation: We use the symbol

r^

to mean the operation of subtract-

ing from row i the sum of all the otherrows,in a determinant of arbitrary

order.

An operation such as r^ may be called a basic operation. Clearly,r^f

utilizes the defining recurrence (2.1) with (2.2).

228

[Aug.


3/11

DETERMINANTS HYPERSURFACESFORLUCAS SEQUENCESOFORDER r

Itis nownecessarytointroducetheconceptof abasic Lucas determinant.

3- BASSC LUCAS DETERMINANTS

Letusdefinethe basic Lucas determinant

of

order

r, 6

r9

as

(3.1)

All elements in agiven upward slanting lineare thesame,e.g., all the

elementsin thereverse (upward) diagonalare

S

0

(=2). Exceptfor theelement

(=-1) in thebottom right-hand corner,all the elements belowthereverse

diagonalarezero.

Observethecyclical natureofelementsin the

columns,

remembering initial

conditions(2.2)applyingtosymbols with negative suffixes.

Of course,(3.1) isonlythespecial caseof (2.6)when

n =

0.

Concerning basic Lucas determinants,we nowprovethefollowing theorem(a

determinantal recurrencerelation).

Theorem:

(-l)[W2]

2

r

+

(-i)*-i,s

2

(3.2)

Proof:

Expand6 in (3.1)alongthebottomrow toobtain

6

r

- (-l)

[r/2]

2

r

-l)

W21

f

(S

1

=

1,

S

0

= 2)

by

r'

=(-l)

[r/2]

2

r

- (-l)

r

2


4/11

DETERMINANTAL HYFERSURFACES FOR LUCAS SEQUENCES OF ORDER

r

Thus,we

[r =

2]

[r = 3]

have,for

6

2

=

s =

1

2

3

1

2

P

>2,

2

-1

1

2

0

= - 2

2

- 1 = -5

= -(2

3

- 3)

2

0

-1

= - 2

3

+ 6

2

= - 2

3

-

= -13 =

(3.3)

(3.4)

-(2* - 3)

[r

= 4] 6^ =

6 3 1 2

3 1 2 0

1 2 0 0

2 0 0 - 1

2*+ 2

d

+ 2

Z

+ 1

29 = 2

5

- 3

(3.5)

[p = 5]

[p = 6 ] 6

6

=

12 6

6 3

3 1

1 2

2 0

24 12

12 6

6 3

3 1

1 2

2 0

3

1

2

0

0

6

3

1

2

0

0

1

2

0

0

0

3

1

2

0

0

0

2

0

0

0

-1

1

2

0

0

0

0

=

2

0

0

0

0

-1

2

5

+ = 2

5

+ 2

4

+ 2

3

+ 2

2

+ 1

= 61 = 2

6

- 3

-2

b

- 6

5

_26 _ 25 - 2

1

* - 2

3

-125 = - (2

7

- 3)

(3.6)

(3.7)

and so on.

The emerging summation pattern by which the 8

r

may be evaluated is clearly

discernible. Notice that the term 2

1

(i.e.,

2) does not occur in any 6

P

summa-

tion.

However, before establishing the value of

S

Y9

we display the following

tabulated information, for all possible values of P:

[r/2]

(1.

-

1)+ p ^ i ]

p = hk

2k \

>

even

6k - 2)

p = 4& + 1

2fc)

>

even

6k)

v = kk + 2

2fc + 1)

>odd

6fe +1j

p = kk + 3

2fc + 1)

>odd

6k + 3)

(3.8)

From (3.8),we deduce

(_1)[W2]

=

(_l)

r

" l+t(r-l)/2]

(3.9)

Invoking this result and applying (3.2) repeatedly, we may now calculate

the value of $.

Theorem: 6

P

= (-l)

[p/2J

(2

P +

l

- 3).

(3.10)

230

[Aug.


5/11

DETERMINANTS. HYPERSURFACES FOR LUCAS SEQUENCES OF ORDER

r

Proof :

S

r

= ( - l ) [ " / 2 ]

2

r

+ ( - l ) " -

1

{ ( - l )

[ ( r

-

1 ) / 2

] 2

r

-

1

+

(-1)

1

2

su m min g t h e f i n i t e g e o m e t r i c p r o g r e s s i o n

= ( - 1 ) ^ / 2 ]

(2

r+ l

- 3 ) .

Checking back shows that the special cases of S

r

listed in (3.3)-(3.7) have

values in accord with

(3.10),

as expected.

4.

EVALUATION OF LUCAS DETERMINANTS

Next,

we show that [cf.(2.6),(3.1)]

A

P

= 6

P

.

To illustrate the ideas involved in the proof we shall give for this con-

nection between A

P

and 6

P

, suppose we take r - 5, n -3, i.e., r is odd, This

implies that we are dealing with the integer sequence

P_if -3 ^-2 ^-1 ^0 ^1

S

2 $3

^4 ^5 ^6

S

7

^8

0 0 0 1 12 24 46 91 179

(4.1)

Perform the basic operations r^, rJJ

r

l successively on the determinant A

5

when n =3 to derive:

91

46

24

12

6

46

24

12

6

3

24

12

6

3

1

12 6

6 3

3 1

1 2

2 0

1

3

1

2

12

6

1

2

0

6

3

2

0

0

3

1

0

0

0

1

2

0

0

-1

2

0

12

6

3

1

2

6

3

1

2

0

3

1

2

0

0

1

2

0

0

0

2

0

0

0

-1

[= 61,

see(3.6)]

(4.2)

i

In A

5

, the leading term 91 (= S

7

) is reduced to the leading term 12 (= S

h

)

in 6

5

by the 7- 4 = 3 (= n) basic operations specified. Because of the cycli-

cal nature of A

5

, these basic operations act to produce a determinant &

5

= A

5

whose rows are the permutation

[

2 6 3 1 2 "

3 1 2 12 6.

of the rows of 6

5

. Due to the fact that r is odd, this cyclic permutation is

even. Expressed otherwise, 6* is transformed to 8

5

by an even number of row

interchanges, so the sign associated with S

5

is +, i.e., 65 =

65.

Hence, A

5

= 6

5

.

1986]

231


6/11

DETERMINANTAL HYIPERSURFACES

FOR

LUCAS SEQUENCES

OF

ORDER

r

If

n

> 5, the cyclic process of basic operationsiscontinued untilthe

basic determinant 6

5

isreached. Obviously,theactual valueof nisirrele-

vant.

The reasoning inherent

in the

case

r= 5

applies equally well

to the

case

when

ris

arbitrarily

odd.

Consequently,

for rodd, A

P

= 6

r

always.

If

ris

even,

the

situation

is a

little more complicated.

For illustrative purposes,

let us

examine

the

case

r= 4.

Substituting

the numbers

in

(2.5)' into

(2.6)

when

n -3, 4, 5, 6 in

turn,

we

readily

cal-

culate thatA^ isreducedto thefour6*whose rowsarerespectively theper-

mutations

[6

3 1 2] [6 3 1 21 [6 3 1 21 [6 3 1 2l

|_3 1 2

6_T

|_6 3 1

2J

5

L

2

6 3 lj

[l

2 6 3J

of the rowsof S

h

(andthisistrue herefor n= 4& - 1, 4/c, 4& + 1, 4& + 2,

respectively

(fc = 1, 2, . . . .

As these permutations

are

successively odd,

even,

odd,

even,it

follows that

S*

=

~&k>

5^,

-6^,

6

4

in

turn.

Thus,

A

4

=

S

h

,

depending

on n,

namely,

A

4

= 6

4

when

n

iseven, whileA^ =

-6

h

whenn is odd.

Armed with this knowledge,we can nowattack the general problem,i.e.,

when vand narearbitrary integers.

First,we

establish

the

following result:

Theorem:

A

r

= (-l)

m

6

P

,

where

m = n

r

(r - n

r

), n

f

= nmod r. (4.3)

Proof: In (2.6), the leading term S

n+r

_i in A

r

isdiminishedto theleading

term

S

r

in 6

P

by n+ r - 1 - (p - 1) - n

basic operations

r[, P

2

r

,

...,

r

which

produce

the

determinant 6%. Simultaneously,

5

P

_

drops

n

places

in the

first

columnof 6*.

To restorethecyclical orderin thefirst columnof 6* to thebasic cycli-

cal order

of the

first column

in 6

r

,

beginning with

S

r

_

l5

it is

necessary

to

effect

n(r - n)

interchanges

of

sign

to

account

for the v - n

terms below

and

including

S

r

- i,

and the

n

terms above

S

r

_

j.

When

n

> r, wereduce

n

mod

r.

Each interchange accounts

for a

change

of

sign

in the

value

of the

deter-

minant

.

When

r

is odd, theproduct

n(r

-

n)

is always even,nomatter whatthe

value

of nis.

But when

r is

even,

the

product

n(r - n) is odd if nis odd, and

even when

niseven. [Thatis,when

r

is agivenoddnumber thereisonlyonevalueof

A

r

, whereas for a given even v thereare twovaluesof h

v

dependingon the

value

of n.]

Thus,

A

r

=

(-l)

m

S

r9

where

m = n

r

(r - n

r

)

,

n

'E

n

mod

v.

Combining (3.10)and (4.3),wehavethefollowing theoremas animmediate

deduction.

Theorem:

A

r

= (-lf

+ [r/2]

(2

r+

x

-

3 ),where

m= n'(r - n

F

),n

T

E nmod r. (4.4)

For example,

when

r= 5, n =3: A

4

= (-l)

3x2+2

(2

6

- 3)

from

(4.4)

=

+61

232

[Aug.


7/11

DETERMINANTAL HYPERSURFACES

FOR

LUCAS SEQUENCES

OF

ORDER

r

in conformity with

(4.2) and

(3.6).

Ontheother hand,

when

v

= 5,

n

= 4: A

4

= (-l)

lx3+2

(2

5

- 3) from(4.4)

=-29

as

we

have seen

in the

discussion preceding (4.3).

Applying(4.4) to arandom choice

r

= 6,

n

= 8 (but not sorandom thatthe

computationsareunmanageable ),wediscoveronsubstitution that

A

6

= (-l)

2x4+3

(2

7

- 3) - -125,

asmay beverifiedbydirect calculation.

Our result

(4.4) for a

Lucas sequence

of

order

r

should

be

compared with

the Hoggatt-Bicknell result

(1.3) for the

corresponding Fibonacci case.

5- HYPERSURFACES

FOR THE

LUCAS SEQUENCES

Geometrical interpretationscan now begivento theidentity(4.4) and its

specializationsforsmall valuesof r. Thereaderisreferredto [3] and [4]

for details

of the

geometry relating

to

Simson-type identities

for

Fibonacci

sequences

of

order

r.

As this corresponding workforSimson-type identitiesforLucas sequences

of order

v

parallels

the

results

in [3] and[4],we

will content ourselves here

with

a

fairly brief statement

of the

main ideas.

Write

Xi = S

n9

x

2

= S

n

+i

9

.

..,

x

v

= S

n+r

~i.

Represent

a

point

in

r-dimen-

sional Euclidean space

by

Cartesian coordinates

(x

ls

x

29

.

..,

x

r

).

Then

9

we interpret (4.4) as theequationof alocusofpointsinp-space

whichhas maximum dimension r -1 in thecontaining space. Suchalocusis

called

a hyper surf ace.

Each

of the

loci given

by the

simple linear equations

x

x

-0, x

2

-0, ...,

x

r

=0 is a

"flat" (linear) space

of

dimension

r - 1, and is

called

a

(coordi-

nate)

hyperplane.

Hypersurfacesof thesimplest kind occurforsmall valuesof

r.

Inaccord

with ourtheory, there willbe onehyper surf ace when ris odd, and twowhen r

is even.

Examplesofhypersurfacesfor {S

n

} are:

r

= 2

(conic):

x\ + x

x

2

- x\ =

5(-l)

n

r= 3(cubicsurface):

xl

+

2x\

+

x\ +

2x\x

2

+ 2x

l

x\ - 2x

2

x

+

/y - rf _ /V

/y

_

O

fY rf* ry . ,_

In passing,

we

note that

(5.1)

expresses

the

well-known Simson-type iden-

tity

for

Lucas sequences

of

order

2,

namely,

^

+

A - i "

Ll

=5(-D"

+1

.

(5.1)'

Moreover,

the

matrix

^1

2

.2

-1.

whose determinant

62 is

associated with identity (5.1)

f

,has

several interesting

2

3

13.

( 5 . 1 )

( 5 . 2 )

1986]

233


8/11

DETERMINANTAL HYPERSURFACES FOR LUCAS SEQUENCES OF ORDER r

geometrical interpretations (relating to: angle-bisection, reflection, vector

mapping).

(See Hoggatt and Ruggles [2].)

Observe that if we replace

x

19

x

2

, x

3

by x,

y

9

2, respectively, in (5.1)

and (5.2),we obtain equations whoseforms,except for the numbers on the right-

hand sides, are identical to those given in [3] and [4]. However, this formal

structure camouflages the fact that the corresponding equations of the conies

and cubic

surfaces,

for Fibonacci and Lucas sequences, are satisfied by differ-

ent sets of numbers.

Carrying further our comparison with the results for Lucas and Fibonacci

sequences,

we obtain, in the case

r

= 4, a nasty equation (refer to [4]) for a

quartic hypersurface in four-dimensional Euclidean space. And so on for hyper-

surfaces in higher dimensions.

Sections of these loci by coordinate hyperplanes yield plane curves of

various orders (cubics, quartics, quintics, sextics, and, generally, curves of

order r). Refer here also to [4].

This completes our summarized outline of the geometrical consequences of

of the determinantal identity (4.4) for Lucas sequences paralleling those for

the Fibonacci sequences.

With the notions of Part I in mind, we are in a position to examine closely

a more general sequence of order rwhich has the Fibonacci and Lucas sequences

of order

r

as special cases.

PART II

Only a brief outline of the ensuing generalization, which parallels the

information in Part I, will be given.

6. A GENERALIZED SEQUENCE OF ORDER r

Let us now introduce the generalized sequence

{H

n

}

defined by the recur-

rence relation

tiyi+ p-1~"~ ^n+ p - 2 + t

n

, ZQ (2 , ri -^

(6.1)

with further initial conditions

-(r-2)

#_o

- l)

a.

(6.2)

Interested readers might wish to write out the first few terms of these

sequences for different values of r. For example, H takes on, in turn, the

values 5a+ 8b , 11a + 13b, 14a + 15b, 15a + 16b

9

and 16a + 16 bfor successive

values r =2, 3, 4, 5, and 6.

As in(2.6),the generalised determinant of order r is defined to be

n+ r -

1

n+r- 2

Hn+

1

2

?i+ v-

3

H

%n+

1

H

n

n - r +

3

n-

1

^n-

r+ 2

(6.3)

234

[Aug.


9/11

DETERMSNANTAL HYPERSURFACESFORLUCAS SEQUENCESOFORDER

r

Evaluation

ofD

r

is the

object

of

Part

II.

Define,as in

(3.1),

the basic generalised determinant of order r,d

r

, to

be obtained from (6.3) when

n

= 0.

That

is,

d

r

=

(D

r

)

n=0

.

(6.4)

Then,the

following simplest basic generalized determinants

may be

readily

calculated

by

expanding along

the

bottom

row:

[ p = 2 ] d

2

= -a

2

+ (Z>- d)b ( 6 . 5 )

[r

= 3 ] d

3

= -a

3

- b - a)d

2

= - a

3

+ b - a)a

2

- b - a)

2

b 6 .6)

[r

= 4] d

h

= a

k

+ (b - a )< i

3

= a

4

- (Z? - a ) a 3 + (2? - a )

2

2

- (Z> - a)

3

Z? (6 .7)

[r = 5] d

5

= a

5

- (2? - a ) ^

=

a

5

- (2> - a ) a

4

+ (Z> -

a )

2

Z )

3

- (b - a )

3

a

2

+ (2> - a ) ^ (6 .8 )

[ r = 6] cZ

6

= - a

6

+ (Z> - a ) J

5

= - a

6

+ (2? - a ) a

5

- (b - a )

2

a^ + (Z? - a )

3

a

3

- Z? -

a)

h

a

2

+ b - a)

5

b 6 .9)

and so on .

A developing pattern

is

clearly discernible.

Calculationof

d

r

followsthemethod employedin(3.2).

Although only

an

outline

of the

theory

in

Part

II is

being offered,

it is

generally desirable for clarity

of

exposition

to

exhibit

the

main points

of the

calculation

of d

r

in a

little detail, even

at the

risk

of

some possibly super-

fluous documentation.

Theorem:

d

r

= (-1)

[r/2]

{a

r+ l

+

(-l)

r

~

*(a

+

b)(b

-a)

r

}/b

(6.10)

Proof:

Expand

d

r

in

(6.3)

and

(6.4) along

the

last

row to

obtain

d

r

=

( ~ l )

[ p / 2

^

-

b -

a)(-lf

l

d

r

_

l

. . . . . . . . i)

=

(-l)[W2]{

a

r

(h

_ a)a*'

1

} - (b- a)

2

d

r

_

2

by

(i),

(3.9)

= ( - l )

[ r / 2 3

{ a * - (Z> - a ) ^

1

+ (6 -

a)

2

a

r

'

2

- (b -

a)

3

a

r

~

3

+ . . .

+ ( l )

p

-

2

( Z ? - a )

p

"

2

a

2

+

{-l)

r

-

l

{b

- a r - i f c

+

[(~l)

r

~H b

- a )

p

"

x

a -

( - l ^ - ^ Z ?

-

a)

r

-

l

a}}

=

( - l )

[ p / 2 ]

{ a

p

( l - (b - a)a~

1

+ (Z> - a )

2

a "

2

- (Z> - a )

3

a "

3

+

+

(-l)

r

(b

-

ay~

2

a-^-2)+

( - i ) ^ - i ( f c - a )

p

~

1

a -

( p

-

1 )

)

+ ( - l ) ^ - i ( Z ? -

a)

r

}

= ( - 1 ) ^ / 2 ]

L r [ l - ( -( & -

a)a'

l

)

r

l

+

( . i j r - i ^

_

a r

l

i

1 - (-(Z? - a)a~

1

) )

- ( - l )

f p / 2 J

{ a

p + 1

+

(-l)

r

-Ha

+ 6)(fc - a )

p

} /Z> .

R e p e a t e d

use of ( i ) has

be e n ma de

in the proof .

A l s o ,

t h e

s u m m a ti o n f o r -

m u l a for a f i n i t e g e o m e t r i c p r o g r e s s i o n h as b e e n i n v o k e d .

1986]

35


10/11

DETERMINANTAL HYPERSURFACES FOR LUCAS SEQUENCESOFORDER r

Applying nextthearguments usedin theevaluationof theLucas determinant

or order

r

9

A

P

, we easily have

Theorem:

D

r

= (-l)

m

d

r

=

(~l)

rn+ [r/2]

{a

r + x

+

{-l)

T

'

l

{a

+ b) (b - a)

r

}/b,

(6.11)

, (

m

=

n

r

(r - n

r

)

where< , _ ,

\n ' = n

mod r.

Proof: As for (4.3).

Fo rtheLucas sequenceoforderp ,

{S

n

}

,

a= 2, b - 1 (so a + b = 3, b - a=

-1).

Itiseasytoverify that,inthis c ase,

d

r

-

S

P

,

D

r

= A

P

.

[Cf. (3.10),

(4.4).]

Coming

now to the

case

of the

Fibonacci sequence

of

order

r>, {R

n

}, we

have

0, b =1 (so a + b= 1, b

1).

Itisimportant tonote th at,for ourtheoryto beusedfor

{R

n

}

9

thetermsof

{R

n

}

with negative suffixes haveto be extendedby one term in thedefinition

(1.1), (1.2)

given

by

Hoggatt

and

Bicknell [1 ], namely,

R

-(r-

1)~

l

'

(6.12)

Augmenting

{R

n

} by

this single element enables

us to

construct

basic Fibo-

nacci determinants

oforderr, V

P

, for

{R

n

}

derived from (1.3)analogouslyto

thosefor

{S

n

}

from (2.6). Computation yields

i . v,

i.

v

L

i . v ,

i . v

c

To give some appreciation of the appearance of the V

r

, choose r = 6, so

V

6

=

8 4 2 1 1 0

4 2 1 1 0 0

2 1 1 0 0 0

1 1 0 0 0 0

1 0 0 0 0 0

0 0 0 0 0 1

(= 1)

(6.13)

which is rather simpler than the corresponding form (3.7) for 6

6

.

Putting

a

= 0,

b

= 1 in (6.10),we have, with the aid of (3.9)

d r

=

(

_l)[W2]

+ 2-l

=

(

_

1 )

[(r-l)/2]

=

V p

.

Now it may be shown that

m + [(r -

l)/2],thepowerof -1 in (6.11),and

(p- l)n +

[(r -

l)/2], the corrected powerof -1 in theHoggatt-Bicknell[1]

evaluationin

(1.3),

areboth evenorbothodd. Thatis

(_]_)"*+[r/2]

=

(_ )(*-l)w+

[(r-D/2].,

(6.14)

236

[Aug.


11/11

DETERMINANTAL HYPERSURFACES FOR LUCAS SEQUENCES OF ORDER

r

Thus, a = 0, b = 1 in (6.11) with (3.9) give

D

=

(_i)W+[(r-l)/2]

=

(_

1

)(i"-

D n+ [(p-

l)/2]

= v

where V

r

is the symbol to represent the Hoggatt-Bicknell determinant (1.3).

Suppose we check for

{R

n

}

when

r

= 6,

n -

3, i.e., we are dealing with the

sequence

(6.15)

?-5 *-* *-3 *-2 *-l ^0 *1

R

2

R

3

^ ^5 ^6

R

7

R

B

(l 0 0 0 0 0 1 1 2 4 8 16 32 63 ...

Then

(n =

3

:

D

6

= (-l)

3 x 3 + 2

= -1 by (6.11), (3.9)

=(-1)5x3+2

=

_

x b y ( l 3 )

= -V

6

= -1 on direct calculation.

Geometrical considerations similar to those in [4] and in Part I of this

article are now applicable to the general case of {Hn} when aand bare unspe-

cified, and also to the multifarious special cases of {H

n

} occurring when aand

b are given particular values.

But we do not proceed ad infinitum* ad nauseam by discussing other classes

of sequences. Unsatiated readers, if such there be , may indulge to surfeit in

such an algebraic geometry orgy.

One further generalization might be contemplated if, in (6.1),we were to

associate with each H

n+r

-j (j = 1, 2,. . . , r) a nonzero, nonunity factor p..

However, the algebra involved makes this a daunting prospect.

REFERENCES*

1. V. E. Hoggatt, Jr., & Marjorie Bicknell. "Generalized Fibonacci Polyno-

mials." The Fibonacci Quarterly 11, no. 5 (1973):457-465.

2. V. E. Hoggatt, Jr., & I. D. Ruggles. "A Primer for the Fibonacci Numbers

Par t V." The Fibonacci Quarterly 1, no. 4 (1963):65-71.

3. A. F. Horadam. "Geometry of a Generalized Simson's Formula." The Fibonacci

Quarterly 20

, no. 2 (1982):164-168.

4. A. F. Horadam. "Hypersurfaces Associated with Simson Formula Analogues."

The Fibonacci Quarterly 24, no. 3 (1986):221-226.

5. T. G. Room. The Geometry of Determinan tal Loci. Cambridge: Cambridge

Uni-

versity Press, 1938.

"To the list of references, we take the liberty of adding the monumental

text [5] by Room, though its subject is determinantal loci in projective, not

Euclidean, spaces.

OfOf

1986]

237

Documents

Hypersurface 2