Conditional recurring sequences

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. IR. H. VAN BEKKUM, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, TE VERDEDIGEN OP WOENSDAG 9 JUNI I976 TE 16.00 UUR

DOOR

JOHANNES LEONARDUS SIMONS WISKUNDIG INGENIEUR GEBOREN TE • s-GRAVENHAGE.

the Andalus in the West cannot manage thirty-two chessmen in a space of two cubits by two"

LIST OF SYMBOLS

0 INTRODUCTION

1 POLYNOMIAL ARITHMETIC

2 CONDITIONAL RECURRING SB^UENCES WITH CONDITIONS ON THE ORDINAL NUMBER

2.1 Linear conditions 2.1.1 The two-way model 2.1.2 The multi-way model 2.1.3 Algebraic aspects 2.2 Non-linear conditions

3 CONDITIONAL RECURRING SEQUENCES WITH CONDITIONS ON THE ELEMENTS OF THE SEQUENCE

4 CONDITIONAL RECURRING SEQUENCES WITH MIXED CONDITIONS

5 SPECIAL SEQUENCES

5.1 Sequences generated by inhomogeneous difference equations

5.2 Non^linear sequences

5.3 Coupled recurring sequences

6 PERIODICITY PROPERTIES 6.1 General

6.2 Influence of the initial value set

7 REFERENCES

a . , b . , c . c o e f f i c i e n t s of polynomials J J J

a . . m a t r i x - e l e m e n t

N

characteristic sequence

A matrix

A ( Z ) , B(Z)... polynomials in the operator Z with coefficients in 2S

B„(x,Z)u linear combination of initial elements of the f^ ' ' n

sequence (u I generated by f(z)u =0.

c„, c inhomogeneous parts of difference equations C period of a sequence

C(n) class of polynomials f(z) d degree of a polynomial

K» part of equation in splitted form

f, g, ... polynomials in the operator Z with coefficients

f(z), g(z)

in Z. 2

f (z ) "even-powered" part of a polynomial

f«(Z ) part of polynomial in Z, consisting of all powers equal i modulo k. P. . determinant element k, i , m, n integer numbers L, L,, Lp moduli M, M^, M_... moduli (M) ideal P-, , Pp period of a sequence 2 3 abbreviation for p^(Z ) or P-|(Z ) etc. p(z), q ( z ) , r(z)... polynomials in Z

Q decision function

u , v

n n elements of a sequence ( s c a l a r ) / u i sequence

f u 1 v e c t o r - s e q u e n c e x ( z ) unknown polynomial i n l i n e a r e q u a t i o n

y a r b i t r a r y polynomial i n product of M polynomials Z b a c k - s h i f t o p e r a t o r Z / ( M ) r e s i d u s e t g e n e r a t e d by an i d e a l (M) 2 s e t of i n t e g e r s 2Zi[z] s e t of polynomials i n Z w i t h i n t e g e r c o e f f i c i e n t s •^ a r b i t r a r y element of TL 2 2 (p(Z ) polynomial i n Z p p r i m i t i v e r o o t of u n i t y •* c r o s s - m u l t i p l i c a t i o n

INTRODUCTION

The theory of linear recurring sequences has become classic: In the 13th century Fibonacci founded the theory by studying the

sequence u _ = 0 , u, = 1 , U , T = U + U , for n > 1. In later 0 ' 1 ' n+1 n n-1

years many papers have been published which reveal the theory in its general algebraic aspects as well as in its applications, such as the generation of pseudo-random sequences, [3] > [4] » [5] •

Sofar all these investigations deal with sequences, generated by one recurrence relation, an appropiate initial value set being given.

Here a new type of sequences will be introduced, which will be illustrated by a simple example. Consider a sequence which starts with u. = 0, u, = 1 and of which the elements are generated by alternate application of two different relations e.g. u ,^ = u + u

n+1 n n-1 if n is even and u ,, = 3u - u , if n is odd. This sequence is no

n+1 n n-1

longer generated by one relation of the above mentioned type, but requires, depending on the parity of the ordinal number, two relat-ions. Of course it is possible to write down one relation valid for all n:

( i ^ ^ " " " )(u ^ - u - u J + (i^^'')(u ^. - 3u + u J = 0 2 n+1 n n-1' ^ 2 n+1 n n-1 but this requires a relation with non-constant coefficient. The question remains whether or not there exists a linear recurrence

relation with constant coefficients valid for all n.

The new type of sequences will be called conditional recurring sequences and the main goal of this thesis is to study conditional recurring sequences with respect to the following questions:

(i) For what kind of conditions does there exist one linear re-curring relation with constant coefficients, valid for all n? (ii) What is the structure of such an "unconditional" recurrence

relation?

(iii) How does the unconditional recurrence relation depend on the genarating recurring relations?

The given example can be put in a more general algebraic framework. Consider in the integral domain Z a (principal) ideal ( M ) with the property that the residue set 2 ? / ( M ) is finite.

The classical theory of lineair recurring sequences deals with sequences |u } in Z , which are generated by one fixed

2)

recurrence relation f(z) u = 0 , which is often characterized ' n '

by the polynomial f(z) in 2Z[z] . Here Z [z] denotes the set of polynomials, with coefficients in Z , in the backshift-operator Z, defined by Z : u -*. u , for elements u of a sequence (u I in Z .

n n-1 n ^ n' The relation f(z) u = 0 yields all elements of the sequence in succession, an appropiate initial value set being given.

Here the classical theory is generalized in the following way: Instead of one polynomial f(z) in Z[z] a set S of M polynomials f^ ^ ( Z ) , f^-'"''(z), ... , f^^~-^\z) is introduced. Then a decision function Q: ( 2 , 2Z , ... , Z ) -*• Z / ( M ) is defined. This function is of polynomial type in n and u ... u , and decides

"^ ''^ n n-k

which of the polynomials f^ '^(z) ... f^^~"'"'^(z) of the set S generates the next element of the sequence:

i f Q ( n , u , u ^,... , u , ) = i , where i c Z / ( M ) then ^ ' n n-1' n-k

f^^^Z) u^^^ = 0.

The considered sequences will be called conditional recurring sequences because the generation of the elements is conditioned'by the function Q, i.e. by the ordinal number and some fixed number of elements of the sequence.

It will be proved that if Q(n, u , u ,,... , u , ) i s a poly-^ ' n' n-1' ' n-k

nomial in n and u , u ., ... u , with coefficients in Z , there n' n-1 n-k

exists, for conditional recurring sequences a polynomial P(z) with the property that F ( Z ) U = 0 for all n. This means that the

conditional recurring sequences can be considered to be classical. It will further be proved that the polynomial P(z) is a

poly-nomial in Z with J>1,

1 ) The reason for mentioning the finiteness of 2 / ( M ) will appear in the sequel.

2) Here and throughout the text the validity of such relations is restricted to n ^ d, where d is the degree of the polynomial

The relation between the elements of the set S and the poly-nomial F ( Z ) can algebraically be described in terms of an operation in Z [Z] . This operation will be the subject of an algebraic reflection.

Periodicity properties of conditional recurring sequences will be discussed in relation with the influence of the initial value

set and the value of J .

Application of these ideas to the earlier given example leads to f(z) = 1-Z-Z^, g(z) = 1-3Z+Z^, S = (f(z), g(z)} , M = 2,

Q(n) = n(mod M ) . Here Q is a function of n only. As a second (more general) example one can think of a sequence with f(z) = 1-Z-Z^, g(z) = 1-3Z+Z^, S = { f(z), g(z), g(z) ) , M = 3 and Q ( U ,n) = u - n(mod M ) . A S a result of the theory developed in the sequel there exists for both examples one fixed linear recurr-ence relation with constant coefficients. The reader is invited to verify that. Both examples will be treated in detail in the follow-ing sections.

POLYNOMIAL ARITHMETIC _ ,

In this section polynomials f(z) in Z[z] are considered, where 2Z and Z are defined according to the introduction. The main lemma's to be used later on will be given here. This list is far from

complete. More information can be found in [2] , [6]

Lemma 1: If a sequence |u } satisfies f(z)u = 0, it also satisfies g(z).f(z) u = 0, where u e Z and f(z) and g(z) are

elements of Z [Zj .

Proof : Obviously the polynomial operators form a linear space over Z , i.e. If q(z) u = 0 and r(z) u = 0, it follows that for a and b elements of Z , holds {a.q(z) + b.r(z)} u = 0. Further it follows from f(z) u = 0 that f(z) u , = 0 i.e.

n . n-k

Z f(z) u = 0. Now l e t g(z) = ?.c.Z then one has

success-^ ' n . 1 1

ively f(z) u = 0 ->Z^f(z) u = 0 - > c . Z ^ f ( z ) u = 0 for

_{u \ / ^ ^ ' n i n}

a l l c . - * ' ? c . Z ^ f ( z ) u = 0 -».g(z).f(z) u = 0.

Lemma 2 : If a sequence (u | satisfies f(Z ) u^^ t = 0 for k,2 fixed and all n, it also satisfies g(Z ).f(Z ) u, . = 0, where u is an element of Z and f(z) and g(z) are elements

of Z [Z] .

Proof : If f(Z^) \^^2 ^ ° ^'^-^ ^^^^^\n+i ^ ° "^^^^ *"°^ ^' ^ elements of Z , one has consequently

{a.f(Z^) + b.g(Z^) ) u^^_^ = 0. From f(Z^) u^^_^ = 0 it follows that f(Z^) Uj^^^^^^j = 0. Since ^(^^^^^2 =

^^\n+l °^^ ^^^ Z^.f(Z^) u ^ ^ ^ = 0. If g(z) = V ki

f* c.Z then one has, similar to lemma 1, successively

c ^ Z ^ f(Z^) u ^ + f for all c^ ->

Remark 1 :lemma 1 is a special case of lemma 2 with k = 1, 1 = 0 . Remark 2 :If a (conditional) recurrence sequence |u | satisfies

f(Z ) u, . = 0 for k and i fixed, it does in general not satisfy a relation of the kind g(z).f(z ) '\. + f- ^ where g(z) is an arbitrary polynomial in Z [zJ .

Lemma 3 : For every couple of relatively prime polynomials f(z) and g(z) there exist two polynomials F ( Z ) and G ( Z ) with degree F ( Z ) < degree g(z) and degree G ( Z ) < degree f(z) for which holds P(z).f(z) - G(z).g(z) = k ^ 0, where k e Z .

Proof : Application of Euclids' algorithm yields for the GCD of f(z) and g(z), which is a non-zero constant,

F'(z).f(z) + G'(z).g(z) = k. Let degree f(z) = n and degree g(z) = m. Suppose further that degree P'(z) = p 2t m.

Consequently degree G'(z) = q 2 n. Putting P'(z) =

A(z).g(z) + F(Z) and G'(z) = B(z).f(z) + G ( Z ) with degree P(z) < m and degree G ( Z ) < n, one obtains:

l) The notation f(Z^) is only to indicate that a polynomial in Z is involved.

P(z).f(z) + G(z).g(z) + (A(z)+B(z)).f(z).g(z) = k. The third term, being the only tenn with powers 2 n+m, must be equal zero.

Lemma 4 s If a sequence |u ] satisfies f(z) u = 0 for n > n„, it also satisfies the relation g(z) u = 0 , where g(x) = f(x)/h(x) and h(x) is the GCD of B (x,Z) u and f(x), and

^ "0 B„(x,z) u is defined by the formula B„(x,z) u =

f^ ' n^ "^ f^ ' -^ n_ f(x)-f(z) , . -^ ^ -, . n .

' 'y' ' u and IS considered as a polynomial m x.

Proof : A proof can be found in [3J and [7] • Remark : For the majority of initial value sets one has

h(x) = (B (x,z) u , f(x)) = 1 and consequently g(x) = f(x).

t n ^

Lemma 5 ! If k sequences |u | , | v j , ... , | w j satisfy k relat-ions of the form

^11^2^) \n+p^^l2(^^)^kn+q-' * *' + ^•

^21^2^) ^+p-'^22(2^)^kn+q'' ' " ^^

Ik^^"") -kn+r = ° ^k^^'^) -kn+r = °

^1^^^) \n+p^^2(^'')"kn+q^ ' " ^ ^^^""^ \ n + r then all sequences also satisfy the relation

= 0 det.A X = 0 n where A = ^11 kl ^Ik ^ /

Proof : According to lemma 2, each of the k relations remains k

valid after multiplication by a polynomial in Z . Since all elements of A are polynomials in Z , the subdeter-minants are such polynomials too. Multiplication of the i-th relation by the subdeterminant of a. . and addition over i proves this lemma for all j.

Lemma 6 : I f a sequence l u } s a t i s f i e s f ( z ) u = 0 f o r n <• n , and I n ' n ~ o h(z) u = 0 for n > n with h(z) = f(z).g(z) then it also satisfies f(z) u = 0, where u e Z and f(z), g(z) and h(z) are elements of .Z[z] . Proof T5 ( u\ h(z)-h(x)

V^'2) \ = ^ Z-x^

' \

f(z)g(z) - f(x).g(x) __ Z-x '^0 ^ f(z)g(z) - f(z)g(x) + f(z)g(x) - f(x).g(x) ^ Z-x

0

= B (x,Z).f(z) u + B (x,Z)g(x) u = B^(x,z)g(x) u^ .

So, for an arbitrary initial value set, the expression B, (x,Z)u , considered as a polynomial in x, and h(x)

n n^

possess the common factor g(x). Then lemma 4 shows that the sequence ju | satisfies a recurrence relation with

nJ / >, characteristic polynomial f(z) = ) ( .

CONDITIONAL RECURRING SEQUENCES WITH CONDITIONS ON THE ORDINAL NUMBER

In this section the restriction is made that the decision function Q is a function of the ordinal number only, and not of the elements of the sequence {u | .

2.1 Linear conditions

2.1.1 The two-way model

The general case in which, for the generation of the sequence |u I , two polynomials f(z) and g(z) are applied alternatively, is

given by the two relations

f(z)

_u, ^2n S (Z) u _2n+l = 0 = 0 (2.1)

Here one has S = { f(z), g(z) } , M = 2, Q(n) = n-1 (mod 2 ) .

For further investigations it appears useful to split f(z) and g(z) in "even and odd powered terms" as follows:

f(z) = fQ(z^) + Zf^(z^) g(z) = g^rz^) + zg^(z2) J

(2.2) 1)

where the four polynomials on the right hand side are polynomials

•J

in Z . Equation (2.l) can be rewritten in the form [fQ(z2) + Zf^(Z^)]u2^ = 0

[go(z') +Zg^(z2)]u2^^^ = 0

(2.3)

Using Up = ZUp , equation (2.3) can be further rewritten: fo(z2) u^^ + 'L\{J^) U^^^^ = 0 •

%(2') ^2n ^ ^0(2') 2n+l = ° -Application of lemma 5 yields

[fQ(z2).gQ(z2) - Z2f^(z2)g^(z2)] u^ = 0

(2.4)

(2.5)

Thus by construction, there exists for this case a polynomial F ( Z ) with P(z) u = 0 for all n. This polynomial F ( Z ) is a polynomial in

2 ^

Z . The relation between F ( Z ) , f(z) and g(z) will be written in the form

P(Z) = f(z) * g(z),

where F ( Z ) is of the mnemotechnical determinant form

(2.6) F(Z) = fo(z) zg^(z) zf^(z) . 0 ( 2 ) (2.7)

Equation (2.5) always gives a non-trivial polynomial F ( Z ) . For f(z) and g(z) one has, according to (2.l), f(0) = g(o) = 1. Combination 1) It is noted that the notation ff>(Z ) is for mnemotechnical

of this result with (2.7) yields F ( 0 ) = 1.1 - O.f^(o).g^(o) = 1, so F ( Z ) is not identical zero.

Consider as a first example a sequence |u | , generated by f(z) u = 0 if n 3 1 (mod 2) and by g(z) u = 0 if n = 0 (mod 2) with UQ = 0, u, = 1, f(z) = 1-Z-Z^ and g(z) = 1-3Z+Z^. Here one has

S = {f(z), g(z) } , M = 2, Q(n) = n-l(mod 2). After some computation one obtains the sequence

0, 1, 1, 2, 3, 7, 10, 23, 33 ... Equation (2.7) leads to P(Z) = 2 1 - Z - Z -3Z 1 + Z^

= i-3z2-z4,

so this sequence is also generated by the linear relation

u = 3u o "^ li /I for n 2 4« The existence of this relation establishes n n-2 n-4

the existence of an unconditional recurrence relation for which the characteristic polynomial has minimal degree. The question arises

2 whether this minimal degree polynomial is still a polynomial in Z . This will be answered by

Theorem 1 : Consider a sequence ju ] , characterized by S = {f(z), g(z) } , M = 2, Q(n) = n-l(mod 2 ) .

If (f(z), g(z)) = 1 then the minimal degree polynomial, for which holds p(z) u = 0 for all n, is a polynomial in

2 "

Proof : Let p(z) = P Q ( Z ^ ) + Zp (z^) and satisfy p(z) u = 0. Suppose that p(z) has minimal degree. Now one has

f(z) u^^ = 0 - _ PQ(Z2) f(z) u^^ = 0 (2.8)

p(z) u^ = 0 -*. p (Z) f(z) u^ = 0 -*•

p (Z) f(z) u^^ = 0 (2.9)

Subtraction of (2.8) from (2.9) yields

Zp^.(Z^) f(z) u^^ = 0 (2.10) Using ZUp = u _, (2.10) can be rewritten in the form

Similar to (2.8), one also has

p^(Z^) f(z) u^^ = 0 (2.12) So, for all n one has

p^(Z^) f(z) u^ = 0 (2.13) Similarly, one obtains

p^(Z^) g(z) u^ = 0 (2.14) Combination of (2.13) and (2.I4) yields

P^(Z^) . (f(z), g(z)) u^ = p^(z2) u^ = 0 (2.15) 2

Since p-, (Z ) has lower degree than p(z) =

p^(z ) + Zp (Z ), equation (2.I5) contradicts the minimal p

degree assumption of p(z), unless P-, (z ) = 0. Consequent-p

ly p(z) = p„ (Z ) which proves the theorem. For the case where (f(z), g(z)) yt 1 one can prove

Theorem 2: Consider a sequence [u } , characterized by

S = (f(z), g(z) } , M = 2, Q(n) = n-1(mod 2). There exists 2

a polynomial p(z), which is a polynomial in Z , with minimal degree and for which holds p(z).(f(z), g(z)) u =0 for all n.

Proof : Let (f(z), g(z)) = q(z). Then one has f(z) = f(z).q(z) and g(z) = g(z), q(z). Define a new sequence

|v I by V. = q(z) u. . For the sequence {v | theorem 1 holds, from which it follows that

,2N /„2,

P (Z ) v„ = p(Z^) . (f(z), g(z)) u^ = 0.

Consider as an example the sequence |u | generated by f(z) Up = 0

and g(z) U2^_^^ = 0 with f(z) = I-3Z + 2Z^ g(z) = 1 - 4Z + 3Z^.

If u^ = 1 and U-. = 1 one obtains the sequence 1, 1, 1, 1, 1, 1, 1 . . .

If u^ = 1 and u^ = 2 one obtains the sequence 1, 2, 4, 10, 22, 58, 130 . . .

Both sequences are unconditional generated by resp. h(z) = 1-Z and h(z) = 1-Z-6Z^+6Z^ = (l-6z^).(l-Z) in agreement with theorem 2.

The multiway model

Consider a sequence ju | generated by M (M ^ 3) polynomials in the following way:

f(2) ^ = 0

^(2) ^ + 1 = 0

h(z) ^„+M-1 = 0

(2.16)

Here one has S = {f(z), g(z), ... , h(z)) , M > 3, Q(n) = n-1(mod M ) , In order to construct an unconditional polynomial, each polynomial

is splitted in "powers with exponents mod M" in a form similar to equation (2.2):

f(z) = f^(Z^) + Zf^(z™) + ... + Z»-^f^_^(Z^ . (2.17) Using this form, equation (2.l6) can be rewritten:

[f<,(z»).zf^(z«).... .z''-\.^(z»)]>^„

[.,(z»).zg,(z")...z>-Vi(^")]>^„

[Vz»)*zh^(z»)...z«-Vi(^'')]^„«,-i-o

Substitution of Z u^ = u^ for k = 1, 2, ... , M-1 yields the system. 1+1 = 0 = 0 ( 2 . 1 8 ) ' / M\ M / M\ M / Mv\

f^iz'') z'fM-i(z ) . . . z'f]_(z )>

g i ( z ^ go(z^)

.Vi(^') V2(^')

z\(z^

h , ( Z ^ . ' ^ n + 1 = 0 ( 2 . 1 9 ) ^ ^ + M - 1 ' By a p p l i c a t i o n of lenmia 5» one h a s f o r a l l n : F ( Z ) U = 0 , (2.20) M

g^(Z^ gQ(Z^) ... z\(z^)

V i ( ^ ' ) V2(^')

^o(^')

(2.21)

As before F will be rewi-itten in a mnemotechnical form. Non-diagonal elements of this determinant are described by

F. i»J M i •j+i P. i.J i-J i < j , i t j = 0 , 1, ... M-1 i > j, i,j = 0, 1, ... M-1

where E. denotes the i-th part in the splitted form in equation

(2.17) of the generating polynomial of u^ +v_i• Multiplication of the

k k k-th row by Z and division of the k-th column by Z does not change

the value of the determinant and transforms all non-diagonal elements;

^i,j - ^^^"~'4-,-^i i < j i'^ = °' 1' ••• ^-1

p . .

1 » J

z"~J E "

1-J i > j

i,j = 0, 1, ... M-1

Hence P(z) can be written in the form

f, (z^) z^-\.,(z^) ...zf^(z^ Z g, (/) Eo (Z^ ... Z%(Z^)

,M-1,

V i ( ^ ) ^"V.(^")

»o(^-)

(2.22)

The relation between P(z) and f(z), g(z) ... h(z) will be written in the symbolic form

F ( Z ) = f(z) * g(z) * ... * h ( z ) . (2.23) Remark: Again F ( Z ) i s a non-trivial polynomial in Z since F ( O ) =

Consider as an example a sequence ju \ generated by f(z) u, = 0 , g(z) u^^^^ = 0 and h(z) u^^^^ = °' ^^^^^ f(^) = 1-Z-Z^,

g(z) = 1-2Z+Z^-Z-^ and h(z) = 1-Z-Z^-Z^-Z'^. With u = 0 and Up = 1 one obtains the sequence

0, 1, 1, 1, 3, 4, 6, 14, 20, 32, 72 ...

the above theory shows that all elements satisfy p(z) u = 0 where

(z)

i s given by 1 - 2Z

- z ^

-Z2

l - z 3

-z-z4

-z

z2

1-Z^ = l-6z^ + 4Z (2.24)

which is in agreement with the found sequence.

As in the two-way model the question arises whether the poly-nomial p(z) is the minimal degree polypoly-nomial. In order to investigate the structure of the minimal degree polynomial some introductory definitions and lemmas will be given.

Definition: For a natural number n > 1 a class C(n) of polynomials f(z) in Z [Z] is defined. f(z) e C(n) if and only if the following two conditions are satisfied:

(i) f(z) does not contain a non-trivial factor in Z . (ii) There exist two rational integers i and j, with

0 < i < j < n, for which (f(p^Z), f(p''z)) =^ 1, where p is a primitive root of the equation p = 1 . Let f(z) satisfy condition (i) of the definition and let p be a primitive root of the equation p = 1 . For f(z) one can prove Lemma 7: If f(p^) = 0 with 0<k<n and k^|-, then f(z) e C(n). Proof : f(p^) = 0, thus f(z) = (Z-p^) . (Z-p"~^) . f(z).

Prom t h i s , one e a s i l y obtains

f(p^z)-=p^Z-l)(pVp"-^) . f (p^)

Lemma 8;

which shows that Z-l/(f(p 7,), f(p"~^Z)).

Since k ^ —, f(z) also satisfies condition (ii) and consequently f(z) e c(n).

If f(z) is a cyclotomic poljmomial, with respect to odd n, then f(z) e C(n).

Proof : Since f(z) is a cyclotomic polynomial f(z)/z -1.

Thus f(z) contains at least one factor Z-p , hence f(p ) = 0 and lemma 7 applies. Consequently f(z) e C(n).

Lemma 9' The class C(2) is empty.

Proof : Let f(z) e c(2). Hence f(z) satisfies condition (ii) of the definition. Since p = -1, one has (f(z), f(-z)) ?t 1. Let Z-a be such a common factor, then one has

. f(z) = (Z-a)(Z+a) f(z) = (Z^-a^).f(z).

Thus condition (i) cannot be satisfied by f(z), which proves this lemma.

With this definition of C(n) a theorem concerning the structure of the minimal polynomial in the multiway model can be proved. For reasons of simplicity the three-way model will be discussed in detail, whereas the cases n >2> will be left to the reader since they do not require new techniques.

Theorem 3: Consider a sequence |u \ generated by f(z) u, = 0,

S^^^ % + l = ° ^"'^ ^^^^ ''3n+2 = °* ^^ ^ ^ * ^ ^ ) ' S^^^' ^(^)^ = ^ then there exists a minimal degree polynomial p(z) for

which holds p(z) u = 0 and which satisfies one of the n

following properties

(i) p(z) = q(z^) . p(z) where B(z) e c(3) (ii) p(z) is a polynomial in Z .

Proof : Equation (2.23) proves the existence of a polynomial F(z ) with F(Z^) U = 0.

Let p(z) = q(Z-^) . p(z) be a divisor of P(z ) with minimal degree and for which also holds

Then one has:

p(z) u^ = 0 -^ p(z).f(z) u^ = 0 -•p(z)f(z) u^^ = 0 p(z)f(z) u^^_^ = 0 p(z)f(z) u^^_2 = 0

Splitting p(z) = P Q ( Z ^ ) + Z P (Z^)+Z Pp(Z^) this equation can be rewritten in the form:

^)f(z) P;^(Z^ Z^p-(Z^)f(z) p„(Z^)f(z) p,(z^)f(z) u (2.25) Po(z^)f(z) p^(Z^)f(z) P2(z3)f(z)\/u,„ \ ^1' \z^p^(Z^)f(z) Z^P2(Z^)f(z) pQ(Z^)f(z)/\u^^_2 3n 3n-l = 0 (2.26) = 0

Since f(z) u-, = 0 equation (2.26) can be simplified:

/ P3^(Z^)f(z) P2(Z^)f(z) W^3^_i^

Po(z^)f(z) p^(Z^)f(z)

z3p2(z^)f(z) pQ(Z^)f(z) /\U3^_2/

Elimination of u^ „, respectively u-, , yield

3n-2' 3n-l

^

[9/(Z^)-Po(Z^).P2(Z^)] q(Z^).f(z) u^^_^ = O"

[5i^(Z^)-Po(Z^) P2(Z^)] q(z3) f(z) U3^_2 = 0 J

where q(z^) = ( P Q ( Z ^ ) , p^(Z^), P2(z^)) and

p.(Z^)= q(Z^) . p.(Z^), i = 0, 1, 2.

By lemma 2, one derives from f(z) u^ = 0 the relation [ ; / ( Z ^ ) - ; Q ( Z ^ ) P 2 ( Z ^ ) ] . q(z2).f(z) U3^ = 0 .

Combination of (2.28) and (2.29) yields P^(Z^)f(z) u^ = 0 where P^(Z^) = [^^^(Z^) - PQ(Z^).P2(Z^) ] . q(Z^). (2.27) (2.28) (2.29) (2.30)

Prom equation (2.27) one can obtain in similar way

PQ(Z^)f(z) u^ = 0 (2.31)

¥^{z^)f{z)

u^ = 0 (2.32)

where PQ(Z^) = [

l^^{z'^)

-

z\{Z^Tv^{z'^) ]

. q(Z^)

V^{Z^)

= [ z V ( 2 ^ ) - ;o(Z^)p^(z^)] . q(Z^).

Combination of (2.30), (2.3l) and (2.32) yields

P(Z^) . f(z) u^ = 0 (2.33)

where p(z^) - ( P Q ( Z ^ ) , P^(Z^), P2(z^)).

Equations, similar to equation (2.33) can be derived

for g(z) and h(z):

P(Z^) g(z) u^ = 0 (2.34)

P(Z^) h(z) u^ = 0. (2.35)

Since (f(z), g(z), h(z)) = 1, the equations (2.33), (2.34)

and (2.35) can be combined to

P(Z^) u^ = 0 . (2.36)

Since p(z) u = 0 and p(z) has minimal degree, one has

from equation (2.36)

P(Z)/P(Z^) (2.37)

and consequently

p(pz)/p(z^) (2.38)

p(p2z)/p(Z^). (2.39)

By assumption p(z) is not an elem,ent of C(3), which means

that

Combination of the equations (2.37), (2.38), (2.39) and (2.40) yields

q(Z^) . p(z) . p(pZ) . p(p^Z) / P(Z^) . (2.41) Let degree p(z) = m and degree q(z ) = 3i , then it follows

that

degree (q(Z^). p(z) . p(pZ) . p(p^Z)) = 3-^ + 3m degree (P(Z )) ^ 3^ + 2m, which leads to m = 0 and

consequently p(z) is a constant. Since p(z)/p(z)/p(z) one has p(z) = 1, and as a consequence

p(z) =. q(Z^). (2.42) Resuming,if p(z) is not an element of C(3), then p(z) = 1

and p(z) is a polynomial in Z , thus condition (ii) is satisfied. If p(z) is an element of C(3), then p(z) = q(z ) . p(z) and condition (i) is satisfied, which proves this theorem,.

Remark 1: As in the case M = 2 a more general result can be proved if (f(z), g(z), h(z))^ 1.

Remark 2: Theorem 3 can be generalized for the M-way model where M ^ 2. From this general theorem, which will not be proved here, theorem 1 follows as a special case, since according to lemma 9 "the class C(2) is empty and consequently condit-ion (ii) must be satisfied.

Consider as a first example a sequence ju | generated by n s 0 (mod 3)

n s 1 (mod 3) n = 2 (mod 3)

Here one has S = {f(z), g(z), h(z) } , M = 3, Q(n) = n-1 (mod 3 ) . Computation of f(z)* g(z)* h(z) according to equation (2.22) yields

n+1 ^n+1 % + l ^0 = = u n = u n = u n 0 , n - 1 ^ 2 % - i • ^ ^ - 2 u^ = l i f i f i f

^ + 1 ^ n + 1 % + l ^ 0 = = u n = - u n n-1 a

p(z) = 1 - 5Z + Z . So, all elements satisfy u = ^u -, - u /-.

^ ' ' "^ n -^ n-3 n-6

This is in agreement with the actual sequence 0, 1, 1, 1, 2, 4, 5, 9, 19, 24, 43, 91 . . .

Since p(z) is irreducibel in 2'[ZJ , condition (ii) of theorem 3 is satisfied.

Consider as a second example a sequence [u [ , generated by if n s 0 (mod 3)

u - u , if n = 1 (mod 3) n n-1 ^

if n = 2 (mod 3)

Here one has S = {f(z), g(z), h(z) } , M = 3, Q(n) = n-l(mod 3 ) . Computation of f(z) * g(z) * h(z) according to equation (2.22) yields p(z) = 1 - Z . So, all elements satisfy u = u ,. This is in

o i-\ y " n n - 3

agreement with the actual sequence:

a, a, tLdi,J a, a, *~^a, a, a, "-^la ... .

T 2

However, since p(z) = 1-Z = (l-z).(l+Z+Z ) it is possible here that p(z) is not the minimal degree polynomial, which generates this sequence. After straightforward verification one finds that all elements also satisfy u = - u ^ - u o > which means that

condit-^ n n-1 n-2

ion (i) of theorem 3 applies.

Remark 3: Since f(z) * g(z) * h(z) ^ f(z) * h(z) * g(z) this result does not necessarely imply that a similar degree reduct-ion will also take place if the roles of g(z) and h(z) are interchanged. For the given example it can be verified that f(z) * h(z) * g(z) = 1 + 2Z^ so degree reduction cannot exist then.

2.1.3 Algebraic aspects

In the two foregoing sections a polynomial F ( Z ) was introduced on the basis of an alternate use of polynomials f(z), g(z) ... h(z) for the generation of a sequence |u } . Algebraically spoken: If S = {f(z), g(z) ... h(z)} is a subset of Z [zJ , n-ary operations

(n = 2,3 ... ) are introduced, respectively denoted by f • g , f * g * h etc. With these operations together with the ordinary multiplication and addition the set S is a universal algebra. In this section some properties of the n-ary operations or

cross-multiplications will be derived. The binary cross-multiplication will be discussed first. Throughout the section A, a., b . etc. are

elements of Z . (i) f * g = g * f (ii) (Af) * g = f * (^g) = A . (f * g) (iii) f * ( g + h ) = f * g + f * h (iv) 1 * 1 = 1 1 * Z = 0 Z * 1 = 0 Z * Z = -Z^ (v) f * (Zg) = - (Zf) * g (2.43) (vi) f * (Z^g) = (Z^f) * g = Z^ . (f * g) (vii) f * (g * h) / (f * g) * h .

These p r o p e r t i e s may be v e r i f i e d s t r a i g h t f o r w a r d from t h e d e f i n i t i o n

f * g ± 0 Zgn Zf, Sr

= fo^o - z ^ 1 %

Thus the binary cross-multiplication is commutative (i),

non-associative (vii) and connected with the addition by the distributive law (iii). This distributive law represents both distributive laws, since the binary cross-multiplication is commutative. The basic information is contained in the properties (ii), (iii), (iv) and (vi). If f(z) = 2a.Z^ and g(z) = Zb.Z^ one has

J

l) Where there cannot be any ambiguity the notation f instead of f(z) is used to indicate the polynomial f(z).

f * g =

(^a Z^)

*

(?b Z^) I l J J = 2 a.Z^ * (Sb.zJ) (iii) = ? l ^ a . Z ^ *b.Z^} (iii) I J l J -^

=

? a .

{^b. .

(Z^

* Z^)} (ii)

1 1 J J

= Z ^a..b^.sign (j,i+l).Z^'^'^ (iv), (vi) (2.44)

where sign (k,i ) denotes the signum function, defined by Eign(k,i) = 1, if the numbers k (mod 2 ) , i(mod 2) form an even

permutation of 0,1

=-1, if they form an odd permutation = 0, else.

The right-hand side of equation (2.44) then reduces to a polynomial 2

m Z which is equal to f * g. Some further properties are: (viii) f * (gh) = (l * h).(f * g) - 4- • (Z * h).(f * Zg) Z (ix) (fg) * (hk) = (f * h).(g * k) + -^ (Zf * h).(Zg * k) Z (x) (fg) * (fh) = (f * f) . (g * h) (xi) (f *• g) . (f * h) = (f * f).(g * h) + -i- . (zf * g).(Zf * h) Z (xii) (f * g) * h - f * (g * h) = -g * (i- . (f * Zh)) .

Formula (xii) shows explicitly that the binary cross-multi-plication is non-associative. The right-hand side of (xii) is equal to zero if and only if

- g is an "even powered" polynomial. In this case f and h can be arbitrary polynomials.

- f(z) and h(z) satisfy fQ(Z^).h^(Z^) = f^(Z^) . \{Z^) for all Z. Now g can be an arbitrary polynomial.

Only i n t h e s e two s p e c i a l c a s e s t h e b i n a r y c r o s s - m u l t i p l i c a t i o n i s a s s o c i a t i v e . ( x i i i ) f * (Zg • h) + g * (Zh * f ) + h * (Zf * g) = 0 ( x i v ) 1 * f gt f f * 1 :,t f (xv) 1 * ( l * f ) = 1 * f.

Now some properties of the ternary cross-multiplication will be discussed. Again these properties can be verified by straight forward applicated of the definition

f * g * h = ^0 ^ S l . \

z\

^0 Z h ^

Z f J

Z^g2 ^0 (xvi) f i k - g * h = h * f * g = g * h * f

(xvii) ( Af) • g* h = f * (Ag) •h = f * g * (Ah) = A . ( f * g * h ) (xviii) f * g * ( h + k ) = f * g * h + f * g * k

(xvi) and (xviii) yield also the two other distributive law^ f * (g+h) * k = f * g * k + f * h * k ( f * g ) * h * k = f * h * k + g * h * k (xix) 1 * 1 * 1 * 1 * 1 * 1 * 1 *

z *

z *

z *

z 2 * 1 * 1 * 1 * z •

z *

z^*

z 2 *

z *

z *

z^*

Z 2 . 1

z

z'

z

z'

z

Z2

z

z'

z2 z2 = 1 = 0 = 0 = 0 = 0

= -z

= 0

= z^

= 0 = 0

. z «

(xx) Z f * g * h + f * Z g * h + f * g * Z h = 0.

The basic relations listed under (xix) are covered by

Z^ * Z^ * Z^ = sign (k,j + 1, i+2) . Z^"^^"^^ (2.45) where the signum function is defined by

sign (ig» i , ••• iji^ ^

+ 1 if the numbers i„ (mod n), i^ (mod n) ... i (mod n) form an even permutation of the numbers 0, 1 ... n. - 1 if they form an odd permutation

0 else.

This result is in accordance with ordinary determinant theory [iJ . Thus the ternary cross-multiplication is cyclic commutative (xvi) and connected with the addition by the distributive laws (xviii). The ternary cross-multiplication is non-associative.

From equation (2.2l) it can easily be proved that the M-ary M

cross-multiplication yields a .polynomial in Z . The basic properties of the M-ary cross-multiplication are:

- cyclic commutativeness - distributiveness

- a M-ary cross-product is invariant under the exchange of a M

monomial Z from one factor to another factor

"^0 "^1 Vi-l - the M-ary cross-product of monomials i s Z * Z ...Z =

,

IM

sign {i^_-^, ijy[_2 + 1 , . . « » I Q + M-1) . Z where M-1

-•-M " k=0 ^k •

Consider a set S of elements of Z [ z ] for which two binary operations are defined

(i) the binary cross-multiplication (ii) the addition.

With respect to operation (i) the elements of S form a multiplicative ccxmnutative groupoid; with respect to operation (ii) the elements of S form an additive commutative group. According to the definition

S [••',*] is a (non-associative) ring. This ring contains an ideal.

p

This is the ideal of all elements y(z) with y (Z ) 5 0. The proof is straight forward. It is noted that this ideal has the property that for all a and b elements of S, a * b is an element of the ideal. In this ring one can consider linear equations of the type

f * y = h. (2.46) For this equation the following properties are easily verified:

(i) If h is not an element of the mentioned ideal, the equation is false.

2 (ii) If y satisfies (2.46), then also does y' = y + cp(z ) . Zf

since f * Zf = 0. Hence the degree of a solution of (2.46)

P

can be reduced by choosing cp(Z ) appropiate.

(iii) If degree (y) < degree (f) then the solution of (2.46) can be found by comparison of coefficients of equal power on both sides of the equation.

This will be illustrated by the following example: f(z) = 1-Z-Z^, h(z) = 1-3Z^-Z'^

f(z) * x(z) = h(z).

Let x(z) = a + a Z + a Z^.

One has f(z) • x(z) = (a + apZ^)(l-Z^) + a-^Z^ = 1-3Z^-Z'^, and thus ,2 „4

+ (-a^ + a^ + a^) Z'' - a^Z^ = 1-3Z -Z^. Hence a„ = 1, a-, = "3, ap = 1, and consequently

P

x(z) = I-3Z+Z , in accordance with the example of section 2.1.1. Remark 1: The above derived properties indicate that if a sequence

ju } is generated by f(z) Up = 0 and g(z) Up , = 0, the polynomial with the maximxmi degree, say f(z) can be reduced modulo Zg(z) in order to achieve degree f(z) s degree g(z) (mod 2 ) .

Consider as an example a sequence |u | for which f(z) = 1-Z-Z and g(z) = 1-Z-Z , u„ = 0, u = 1 . The actual sequence is

0, 1, 1, 1, 2, 3, 5, 7, 12, 17, 29, ..

p A

This sequence is also generated by (l-2Z -Z ) u = 0 and the P n

elements u„ ,, are also generated by (l-2Z+Z ) u„ ., = 0, where

1-2Z+Z^ = g(z) - Z.f(z).

Remark 2: This last example shows that there can be defined

equivalent products. These follow from nul-products, for instance

f(z)*zf(z) = o**f(z) *g(z) = f(z) * (g(z)+cp(z^).Zf(z)). For the ternary cross-multiplication one has

f(z) • g(z) * Z.g(z) 3 0 and f(z)*g(z)* Z^f(z)s 0.

For the M-ary cross-multiplication it can be verified that f(z) * ... • h(z) * ... * z\(z) * ... *g(z) ^ 0 if and only if the factor Z n(z) stands k places further than the factor h(z). This can be verified by consideration of definition

(2.21), with respect to row dependancy. As a result of these nul-products, there are in general M-1 reduction possibilities for the degree of y(z) in the equation: f ( Z ) * . . . * g(z) • y(z) = h ( z ) , where a M-ary cross-multiplication is considered. Consider for instance the ternary cross-multiplication f(z) * g(z) * h(z) =

(l-Z-Z^) * (l-2Z+Z^-Z^) * (l-Z-Z^-Z^-Z^). After the success-ive replacements h(z) -*-h(z) - Zg(z), g(z) -•g(z)-Zf (z), h(z) -»'h(z) - Z.g(z), this product appears to be equivalent with the product (l-Z-Z^) * (l-3Z+2Z^) * (l-Z-2Z^). It can be verified that the elements of the sequence of example

(2.24) also satisfy (l-3Z+2Z^) u , = 0 and (1-Z-2Z2) U3^^2 =

°-2.2 Non-linear conditions

In section 2.1 all conditions on the sequence numbers are linear. Here it will be proved that the developed theory remains valid if the

condition is of non-linear polynomial type.

Consider as an introduction the following example: S = {f(z), g(z), h(z)} , M = 3,Q(n) = n^-l(mod 3 ) . u_ = 0, u, = 1, and

If n s O (mod 3) •* Q(n) = '2 (mod 3) — h(z) u^^, = O If n = 1 (mod 3) •*• ft(n) = O (mod 3) -*• f(z) u^^^ = O If n = 2 (mod 3) •*• Q(n) = O (mod 3) — f(z) u^^.^^ = O . Thus, an alternative description of this sequence is

S = [ h(z), f(z), f(z) ] , M = 3, Q(n) = n-1 (mod 3).

Consequently all elements of this sequence satisfy p(z) u = 0 with

p(z) = h(z) * f(z) * f(z) = 1-6Z-^ + Z .

The first elements of this sequence are

0, 1, 1, 2, 5» 7» 12, 29, 41» 70 . . .

Remark: In this example the polynomial g(z) is not important since it is never used for the generation of new elements. If the function cp is a polynomial in n of degree > 2 , such dummy functions will always occur.

Therefore sequences iu | generated, subjected to non-linear condit-ions will be described in the following way:

f(z) u = 0 if Q(n) s 0 (mod M ) g(z) u = 0 if Q(n) ^ 0 (mod M ) .

Let the equation Q(n) = 0 (mod M ) possess k roots, r., i = l(l)k. Then it follows that in general these sequences are equivalent with conditional recurring sequences with linear conditions, since an alternative description of such sequences is given by

f(z) u = 0 if n = r. (mod M ) i=l(l)k n 1

g(z) u = 0 if n 34r. (mod M ) .

Without giving a full proof, it can be stated that any sequence subjected to a non-linear condition modulo M is equivalent with a sequence subjected to an appropiate linear condition, and consequent-ly equivalent with a classical sequence.

3 CONDITIONAL RECURRING SEQUENCES WITH CONDITIONS OK THE ELEMENTS OF THE SEQUENCE

be considered, governed by decision functions which are functions of the elements of the sequence. Since the value of the decision function is considered modulo M the number of different initial value sets is limited. If k is the maximum degree of the poly-nomials of the set S then there are M^ different initial

value sets. This means that after at most M^ new elements of the sequence an initial value set must occur which has occurred before, and consequently the pattern of polynomials, which generates the elements, is periodic. So, although of much larger order, these sequences are equivalent with sequences with linear conditions on the ordinal number only.

If after p new elements an initial value set occurs for the second time and p <. M^ then there is more than one periodic pattern of polynomials. For their lengths one has

Up.

^

M^. (3.1)

As an example consider the sequence |u | defined by f(z) =

1-Z-Z^-2Z^, g(z) = 1-Z-Z^-Z^, Q(u^) = u^ (mod 3), S = {f(z),

g(z), g(z)| . Here k = 3 and consequently the maximum cycle length is 27. It appears that, dependent of the initial value set, this sequence is equivalent with four sequences:

S, : generated b y g * g * g * g * g * g * f * f * g * g * g * g * g * g * f * f S_ : generated by g * g * f * g * f

^ (3.2) S^ : generated b y g * g * f * g * f

S. : generated by f .

This can be checked in a straightforward way. In correspond-ence with (3.1) one has:

^^^^i = l 6 + 5 + 5 + l = 2 7 i M ^ = 2 7 .

Remark : In the last formula the equality sign of (3.I) holds.

This cannot be proved in general since u = a(u , .. u , ) ^ n ^ n-1 n-k does not necessarely imply that u , = b(u ... u , ^^) if

'' ^ "^ n-k n n-k+1 the elements are considered mod M. However since M = 3 in tjiis case each coefficient has an inverse which means that the relation u , = b(u ... u •,,-,) always exists. As a

n-k ^ n n-k+1' "^ consequence the equality sign must hold.

In the case where Q ( U ... U , ) is a non-linear expression in n n-k

the elements, the same reasoning applies as in the considered case where Q ( U ... U _, ) is a linear expression. Therefore this case will not be treated further.

CONDITIONAL RECURRING SEQUENCES WITH MIXED CONDITIONS

This section deals with sequences |u | governed by a general non-linear decision function of polynomial type Q ( U ,n) = i (mod M ) . Let the maximum degree polynomial of S have degree k, then a

combination (n (mod M ) , u (mod M ) , ... , u , ,., (mod M ) ) must re-' re-' n re-' re-' re-' n-k+1

occur after at most M steps, which means that these sequences are equivalent with classical sequences. For the different polynomial cycles one has, similar to (3.I)

Z P . ^ M^-'^

(4.1)

J- .

As an example consider a sequence |u | generated by f(z) u ,-,=0 if Q ( U ,n) = 0 (mod M ) and by g(z) u = 0 if Q (u ,n) ^ 0 (mod M ) ,

ll Q _ I I - L II

with f(z) = 1-Z-Z , g(z) = 1-Z-2Z and Q ( U ,n) = u -n (mod 3) u„ = 0 and u.^ = 2 . The elements of this sequence are (mod 3) 0, 2, 2, 1, 2, 1, 2, 1, 0, 2, 2, 0, 1, 1, 2, 0, 2, 2 ... . One can verify that u^-. = Un s u, f. (mod 3), u.. s u„=u,/- (mod 3 ) , Up = u „ = u, „ (mod 3). Consequently the combination

(n, u T, u „, u -,) has a cycle of I5 and as a result this sequence

' n-1' n - 2 ' n-3 Ü y

is unconditional generated by

g * f * g * g * g • g • f * g * g • g * g * g * f * f•f .

A second example is given in the introduction. Here one has S = {f(z), g(z), g(z) ) , M = 3, Q(u^,n) = u^-n (mod 3 ) ,

f(z) = 1-Z-Z2, g(z) = l-3Z+z2, U Q = 0, U , = 1. Elementary calculations yield that this sequence is unconditional generated by f * g * f * f * f * g * g * g * g * f * g * f * g * g * g * g * f * f * g * g * g * g * g * g . This cross-product of 24 factors is equal to I-65229446Z '^ + Z^ , i.e. this sequence is for n > 48 also generated by

u = 65229446 u ^. - u .0 . n n-24 n-48

SPECIAL SEQUENCES

In the sections 2 to 4 "the conditional recurring sequences fitted in the general theory and consequently the results of the theory apply. In this section conditional recurring sequences will be introduced which do not meet the requirements of the

introduct-ion. Consequently the questions of the introduction cannot be answered without restrictions. It will however be shown that some special sequences can be described within the requirem.ents of the introduction with the aid of an appropiate transformation. No

attempt has been made to investigate the limitations of the theory. The only goal is to demonstrate that the assumptions of the

introduction are sufficient and not necessary.

Sequences generated by inhomogeneous difference equations

So far all considered sequences were generated by homogeneous difference equations. As a further generalization, sequences

generated by linear inhomogeneous difference equations, will be considered. The most simple case is given by

f(z) u = c;^ if n s 0 (mod 2) n I

g(z) u^ = Cg if ri = 1 (mod 2)

For sequences generated by (5.I) one can easily prove Theorem /[: Consider a sequence generated by (5.I).

For all n holds F ( Z ) U = 0 where F ( Z ) = n

(l-Z^) . (f(z) * g(z)).

Proof : It will be assumed that (f(z), g(z)) = 1 and that c^ : c^ :^ f(l):g(l).

One has for n = 0 (mod 2)

(5.1) f(z) u^_2 = c^-*Z^f(z) u^ = c^, • (l-Z^) . f(z) u = 0 . n f(z) u^ = c^ f(z) u^_2 = c^

Similarly one obtains for n s 1 (mod 2) (l-Z^) . g(z) u^ = 0.

Remark: Theorem 4 can be generalized in an obvious way for more general decision functions.

c,

In the case where — = )' i one can prove a stronger result. If the polynomials of equation (5.I) are splitted up in the usual way, one obtains

^0(^') -2n ' ^1^^') -2n-l = ^f 1

z\iZ^)

u^^ + gQ(z2) U2^_, = c

g

(5.2)

Elimination of Up _^, respectively Up gives (f(z)*g(z)) u^^ = c^.gQ - c ^ ,

• (f(z)*g(z)) U2^_^ = Cg.fQ - c^,g,

where gQ(z).c,^ = gQ(l).c^ = gg.c^. etc. If C ^ Q - c^f^ = c f„ - c^g., = c then obviously for all n holds

g 0 f^"! "^

(f(z) * g(z)) u^ = c.

The relation c .g„ - c .f = c f„ - c ^ is equivalent with C^ fQ(l) + l.f,(l) ^/,N

r = g,(i) + i . g , ( i ) = i t r T ' - - - ^ h - f = '' f ( i ) ' Cg= •^•g(i)'

Thus for special values of the inhomogeneous parts of the 2

generating difference equations a polynomial P(z ) exists with p

the property P(Z ) u = c.

Remark: Theorem 1 can be considered to cover a special case, namely the case A = 0. In a similar way generalizations of theorems 2 and 3 can be derived.

5.2 Non-linear sequences

The theory of conditional sequences is still valid for sequences of the kind

u u n+1 « n+1 a c e g u n u n u n u n + + + + b d f h if n = 0 (mod 2) 3f n 3 1 (mod 2) (5.3)

This is because the substitution u = — results in lineair re-''n

currence relations for x and y . One may write n n Xo J.T = a x_ + h y„ 2n+l 2n ''2n 2n+l 2n '2n ^ ^2n •" ^ y2n = e Xp^ . + f yp^_ 2n-l 2n-l S Xp^_T + h yp^_ 2n-l 2n-l (5.4)

Strictly spoken: eq (5.4) is a sufficient system for eq (5.3). After the substitution

eq (5.4) becomes X w = —n n n - aZ cZ - eZ gZ - bZ\

i - J

-^"^i

= 0

1 - hz/ i^2n (5.5)

In order to show that |w J is also generated by an unconditional recurrence relation the following theorem is proved:

Theorem ^: Consider a vector sequence 1 11 I , generated by A ( Z ) U = 0 if n = 0 (mod 2) and by B ( Z ) U = 0 if n = 1 (mod 2 ) .

Here A and B are m x m matrices, with elements which are polynomials in Z with integer coefficients, and u is a m X 1 vector. If the matrices A and B have no common

(left or right-divisor) then there exists a polynomial p(z) which has minimal degree and for which holds

p ( z ) . I . u = 0 for all n > n , where I is the identity matrix.

Proof: Let A ( Z ) = A Q ( Z ^ ) + Z . A (z^) B(Z) = B Q ( Z ^ ) + Z . B,(Z^) .

From A ( Z ) u_ = 0 and B ( Z ) U ^ ^, = 0 it follows that 2n 2n+l AQ Z \ B. O ^2n ^2n+l. = O . (5.6)

And thus for all elements one has A.

0

z\

B, B

0

. I.u = 0 or p(Z ) . I . u = 0 . (5.7)

The proof is similar to the proof of theorem 1. In the same p

way the proof for the non-existence of divisors of p(z ) can be given.

I t seems therefore n a t u r a l t o define for matrices the binary cross-product:

A(Z)

*

B(Z)

A Q ( Z 2 ) Z \ ( Z 2 )

B ^ ( Z 2 ) B Q ( Z 2 )

(5.8)

Application of this theorem to the example

^ / 0 \ /l ' - 1 " 1 ^ 1 1 -Z-2Z 0 -Z 1+Z u = 0 if n = 0 (moe» 2) —n 1 -Z 0

-z

1-Z-Z 2 ^ u = 0 if n = 1 (mod 2)

gives p

(z')

1 0

z

z

0 1-Z^ 0

z

-z

0 2 1-2Z 0

-z

-z

0 1 = 1-4Z^ + 3Z^ - 2Z

which is in agreement with the actual sequence

1 /,\ 0 J\ 1 J,\ 1

4

1 13 , > 3 16 \ / 42 5 /,' 11 ) , ( 53 16

Coupled recurring sequences

Consider two coupled recujrring sequences Iv ] and Iw ] satisfy-ing

f(z) v^^^, + g(z) w^^^^ = 0

h(z) Vg^ + i(z) w^^ = 0 .

(5.9)

Again the question of an unconditional generating function arises. According to lemma 3 there exist two constants M.^ and M defined by

M. ^ = F(z).f(z) - G(z).g(z) M^ = H(z).h(z) - l(z).i(z)

(5.10)

Now consider two (new) characteristic sequences ap , and ap , defined by ^2n+l = ^(^) ^2n+l ^ ^^^^ "2n+l 2n = I ( Z ) V2^ + H(Z) w^^ ( 5 . 1 1 ) One can v e r i f y t h a t

f(z)a = {f(z).P(z)-g(z).G(z)}

"2n+l 0 (mod M.^ ) h ( z ) a 2 ^ = { h ( z ) . H ( z ) - l ( z ) . i ( z ) } w^^ = 0 (mod M^)

Let M = (M.^ , Mp), then one has

(5.12)

h(z) a^^ H 0 (mod M ^ ) f(z) a^^^^sO (modM^)

(5.13)

Consider as an example the sequence \ s \ of numbers in the usual decimal number system (g = lOv + w ) generated by

g: -, = 7v + 3w if n = 0 (mod 2) ^ + 1 n n ^ ^ g , = v + 7 w if n = l (mod 2) ^n+1 n n ^ ^

(5.14)

Since g , = lOv ,+w ^ for \v \ and Iw [ two coupled recurring

^n+1 n+1 n+1 I n » I n ) ^ ^

sequences of the type ( 5'9 ) exist. (10-7Z)V2^^^+ (1-3Z)W2^^^ = 0 (10-Z)v2^ + (l-7Z)w2^ = 0

(5.15)

Here one has

(3).(10-7Z) - (7).(1-3Z) = 23 = M^ (7).(10-Z) - (1).(1-7Z) = 69 = M^ h(z) * f(z) = 100-7Z^

Consequently one has a „ = -2a (mod 23)

n+2 n ^ '

Prom this it follows that the sequence (a | has a period

C = 2 . 22 = 44. l"t can be verified that the sequence jg^ | also has that period.

Resuming it can be stated that for coupled recurring sequence the theory does not apply fully, but applies to a characteristic sequence from which information about the original sequence can be obtained.

Remark: Zappa, Van der Poel and others studied a sequence ju | generated by: u s 0 (mod 2) n ^ ' u = 1 (mod 2) (5.16) U -, = — u n+1 2 n

V i = 3% +

UQ = a 1 i f i f

This sequence seems a generalization of the sequences of section 5»2. The generalization is that the function Q depends on

the elements of the sequence u instead of the ordinal number. How-ever because of this, and because of the interaction of the factor •p and the modulus 2, the in section 5.2 given substitution fails to put this equation in the frame of theorem 5. -A-s a consequence the theory is not applicable. To authors' knowledge there are five different cycles, depending on the given initial value u. =a.

a = 1 -+.U, = UQ a = 0 -».u = UQ a = - 1 - ^ u ^ = UQ a = - 7 H-u^ = UQ a = -n-^^^Q= UQ F(Z) = F(Z) = F(Z) = F(Z) = P(Z) = l - Z ^ 1-Z

i-z2

i-z5

i-z^S

This sequence clearly demonstrates the limitations of the theory, since none of these unconditional recurrence relations can be predicted.

Another possibility seems to be to write this sequence in the form

2 V - , + W T = V i f w s O (mod 2)

n + 1 n+1 n n ^ ^ ^^^^^^ 2v

n+ T + w = 6 v + 3 w + l i f w = l (mod 2) 1 n+1 n n n ^ '

The equation (5.17) seems to be a generalization of equation (5.9). However, because the decision function depends on w , w cannot bè

' n' n

eliminated in the derivation of the characteristic sequence. Conse-quently, no information concerning the period can be obtained in this way.

6 PERIODICITY PROPERTIES

6.1 General

Consider a sequence |u ^ in Z. A positive integer G is called a period of the sequence {ii } if an ordinal niunber n exists for which holds

u p = u if n i n^ (6.1) n+ C n C ^ ^

Sequences {u | in S generated by f(z)u =0, satisfy |u ] <A if and only if all roots of f(z)=0 are in absolute value ^1. In this thesis almost all polynomials do not satisfy this property, since f(z) is such that f(o)=l, which means that the product of all roots equals 1. In general at least one of the roots is less than 1 and consequenly these sequences are unbounded.

However, if the sequence iu \ is considered modulo an integer L, the sequence is bounded by definition. Then periodicity properties can be investigated.

Since the elements of conditional recurring sequences are in general considered with respect to a modulus M in relation to the decision function Q, at first the case where Q is applied to n only, will be considered.

For these sequences the results of section 2 are fully applicable, and consequently the periodicity properties of the sequence |u | for which holds

f(Z^)u = 0 (mod L) (6.2) n

are investigated. The degree of f(z ) is minimal. By the substitution u, =v equation (6.2) results in

f(z)v H 0 (mod L) (6.3)

Periodicity properties of the sequence [v | are discussed in detail in [2]. Here only the main result is given by

Theorem 6: Consider' a sequence ju | , satisfying f(z)v = 0 (mod L) d

f(z) = Y, 9--Z^ and (a L) = 1. For the period d

with

i=0

C(f(z),L) of this sequence one can prove:

In fact this is not quite true since if all roots lie on the unit circle the sequence is not necessary unbounded, (see the example at the end of section 2.1.2). In general however a polynomial f(z) with f(o)=l will possess roots inside the unit circle (see all the other examples).

Proof

(i) if L = L-j^L^ with (L^,L ) = 1 then C(f(z),L) = {c(f(z),L^), C(f(z),L2)| (ii) if L is a prime then

C(f(z),L^'^"^) = A.C(f(z),L^) with A=l or A=L (iii) if L is a prime and f(z)=g(Z).h(z) with

(g(z),h(z)) = 1 (mod L) then

C(f(z),L) = {c(g(z),L), C(h(z),L)}

(iv) If L is a prime an f(z) is irreducible (mod L) then C(f(z),L)iL'^-l and

C(f^"^"^(z),L) = (i.C(f^(z),L) with n=l or ]i=L.

see [2].

If the sequence |v | of equation (6.3) has period C than the sequence Iu | of equation (6.2) has period k.G.

The applicability of this theorem will be illustrated by the following

Example : Consider a sequence ju [ generated by u =u _-|+u _ if n = 1 (mod 3) and by u =u -, + 2u ^ if n # 1 (mod 3). What is the

n n-1 n-2 ^ '

period of his sequence if the elements are reduced mod 23? Prom theorem 3 it follows that this sequence is unconditional generated by 1 f* g•g = -Z2 _z -Z 1 -2Z ,2 -2Z -Z 1 - 6Z^ - 4Z^

In view of theorem 6, it remains to determine the constant C satisfying

z'" = 1 modd(l-6Z-4Z^,23) ^'

The notation f(x)=g(x) modd (h(x) ,M) means that a polynomial has to be considered modulo a given polynomial and as far as the coefficients concern, modulo a number M.

Since 1-6Z^-4Z^ = ( l + 3 Z ) ( l - 9 Z ) (mod 23) i t follows from theorem 7 t h a t C = LCM (C-,,C ) where C^ and C^ s a t i s f y ( - 3 ) ^ ^ H 1(23)

(9) ^ H 1(23)

Elementary calculations show that C-,=22 and since Cp/22 one has C=22, from which it follows that the period of the original sequence is equal to 3.22=66.

The elements of the sequence are 0 , 0, 0, 14, 9, 2 1 , 19, 1 10 13 10 8 22 1 , 1, 10, 13,

15,

3, 20, 16, 3, 20, 16, 12, 11, 18, 18. 4, 17, 19, 4, 17, 12, 11. 10, 11, 12, 5, 16, 7, 1, 18, 5, 4, 13, 10. 5, 4, 5, 18, 19, 22, 1. 18, 14,

9,

21, 16. 5, 18, 19, 11, 12,

7, 5,

The whole theory of determination of the period of conditional recurring sequences breaks down if the decision function Q depends on the elements of the sequence {u |. This is due to the interaction between the two moduli M and L. M is primarily related to the decision function, but since this function depends on the elements of ju ] , which are consider-ed modulo L the results of sections 3 and 4 are no longer appli-cable if ( L,M) ^ M. In fact one has to be more precise in defining the roles of the two moduli. Four cases will be distinguished if Q depends on u as well as on n; which includes the case where Q is a function of u only.

n '^ (i) Q(n , u ) = i (mod M )

(ii) Q(n , u^(mod L ) ) = i (mod M ) I ^g^^^ (iii) Q(n(mod L ) , u ) = i (mod M )

(iv) Q(n(mod L ) , U (mod L ) ) H i (mod M )

These four cases lead in general to different sequences, which is illustrated by the following example: Consider

a sequence \u \ satisfying u T=U - U ^ if I n ) "^ "= n+1 n n-1

Q ( U ,n) = U -n = 0 (mod 3) and by u ^ = u +u n if

^ n ' n ^ ^ n+1 n n-1

Q ( U ,n) = u -n jé 0 (mod 3) , with u„=0 and u^ = l. Here u and n will also be reduced mod 2, according to the four given possibilities. Elementary calculations yield that the sequence is unconditional generated by

( i ) f * g * f * f * g * g * g * g * g * g * g * g * g * - f * g * f * f * f* g * g * g 21 42 This c r o s s product e q u a l s 1+70Z +Z . This c r o s s - p r o d u c t e q u a l s 1-Z . A ft This c r o s s - p r o d u c t e q u a l s 1+Z +Z . ( i i ) f * f * g This cri ( i i i ) f * f * g * g This cross-( i v ) f * f * f * g * g * g 6 1? This c r o s s - p r o d u c t e q u a l s I+4Z -Z I t i s n o t e d t h a t t h e i n i t i a l v a l u e s e t p l a y s an important r o l e i n t h e d e t e r m i n a t i o n of t h e c r o s s - p r o d u c t .

This i s c l e a r from t h e c o n s i d e r a t i o n of case ( i i i ) where i f u ^ O and u-, = 2 one o b t a i n s t h e c r o s s - p r o d u c t

12 24 f * f * g * g * f * g * g * g * g * g * g * g = 1 + 24Z - Z ^ . 6.2 Influence of the initial value set

With the restriction that the decision function Q depends only on the ordinal number n, section 2 proves that there exist an un-conditional generating polynomial for which theorem 6 determines the period of the sequence.

In [3] it is shown that the period of theorem 6 is not always the minimal period. However, there a slightly ambiguous notation is used. Therefore a full proof is given here

Lemma 10: Consider a sequence ju | satisfying f(z)u = 0 (mod M ) . k

Let f(z) = "Y a.Z^ and (a. ,M) = 1. The following two i=0 ^

(i ) u ^„ = u (mod M ) for all n. (6.5) n+C n

(ii) B^(x,z)u ^^ = B^(x,Z)u (mod M ) for all n. {è.6)

f n+C f n k i-1 _. Proof: B^(x,Z)a = .•^, a. 4 ^ x^ ^ u

f^ ' ' n 1=1 1 j=0 n-j

From this it follows that

- B„(x,Z)u is a linear combination of the elements f n u , u , . . . u , , ^ n n-1 n-k+1 k-1 - the coefficient of x is a, u k n

Let C be a period of the sequence u . This is

equivalent with (i). Since B (x,Z)u is a linear combinat-ion of the elements u , u ^ . . . it follows that (ii)

n n-1

holds. Thus (i) implies (ii). - • Now suppose (ii) holds. Thus

B„(x,Z)u ^„ = B^(x,Z)u (mod M ) for all n. f n+C f n

Consider a fixed value for n. Then comparison of the k—1

coefficients of x in both parts leads to a, u ,„ = a, u (mod M )

k n+C - k n ^ '

and since (a, ,M) = 1-•• u ,^ = u (mod M ) . Application of _{k' ' n + C n /} i-j:-this result for all values of n yields

u ^_ = u (mod M ) . n+C n

Thus (ii) implies (i). Hence both statements are equivalent.

Now suppose that C is a period of the sequence ju I , satisfy-ing f(z)u = 0 (mod M ) ; then one has

xB^(x,Z)u^ = (x-Z+z) B^(x,Z)u^ = (f(x)-f(z) + ZB^(x,Z))u^ . (6.7) Thus ,

xB^(x,Z)u^= B^(x,Z)u^_^ (modd f(x),M) (6.8) and consequently

One has

B-(x,Z)u H B^(x,Z)u „ (mod M ) ( 6 . 1 0 )

Combination of (6.9) and (6.IO) yields

x'^B (x,Z)u^ = B (x,Z)u^ (modd f(x),M) (6.II) Now if ( B (x,z)u , f(x)) 5 1 (mod P) for all P/M then and only then

the relation (6.Il) is equivalent with

x^ = 1 (modd f(x),M) (6.12)

However i f the GCD > 1 then a polynomial h(x) e x i s t s with

h ( x ) . ( B ^ ( x , z ) n ^ , f ( x ) ) = f(x) (mod P)

for which hold x H 1 (modd h(x),M), M.P = M. Since h(x) is of lower degree than f(x) and M < M the period which follows from theorem 6 might be smaller for a special initial value set. For an example see [^3] . This lemma 10 covers the classical situation for which the initial value set can reduce the period.

For conditional recurring sequences, reduced to unconditional recurring sequences the generating polynomial P(z) has no arbitrary initial value set u„ ... u,_, to start with. This initial value set follows partly from application of the original polynomials of the set S of the introduction»

Consider a sequence ju | generated by f(z)up = 0 and by g(z) g(z)u = 0, u_, u^,..,u,_^ given. Now the question arises whether or not there exists a special initial value set, which leads to a polynomial h(z) with h(z)u = 0 and degree (h(z)) < degree (p(z)), where p(z) = f(z)«g(z),

The classical theory requires for the existence of such a poly-ncanial a common factor of B (x,Z)u^ and p(x). The only difference

P ^

here is the restriction that this a relation with k instead of 2k degrees of freedom. The elements u„,,,u, _^ are free, but the elements u^..,Up,_-. are detennined by f(E) and g(z),

Consider as an example the sequence (u I , generated by

f(z)u2^ = 0 and by g(z)u2^^^ = 0, u^ = a, u^ = b and f(z) = 1-Z-Z^ and g(z) = 1-2Z^,

One has according to theorem 1 p(z) = 1-3Z^ + 2z4

and the elements of the sequence are ^ a, b, a+b, 2b, a+3'b, 41> . • • •

T p

Hence B (x,Z)u = -3xu -3u +2x-^u +2x u +2xu,+2u . If a=b then

B (l,Z)u =p(l)=p(-l)=B (-l,Z)u,=0 and consequently the sequence is also generated by

h(z) .l::d^=i-2z^

1-Z

The elements of the sequence are now

Rem.ark: This last example indicates that the factors 1-Z and l+Z vanish both as a result of one relation that exist between u^ and u, . This can be proved in general, since the class C

(2) is empty, which means that if 1-Z vanishes, because of a special initial value set, one has for all n p(z)u = 0,

" 2 ^ where p(z) is not a polynomial in Z . Then theorem 1 shows

that there must be another polynomial h(z) with smaller degree for which holds h(Z)u = 0.

n REFERENCES

1 Birkhoff, G. and Maclane, S. 2 Duparc, H.J.A.

A survey of modern algebra. New York, MacMillan, 1958.

Periodicity properties of recurring sequences I and II. Proceedings of the "Koninklijke Nederlandse Akademie van Wetenschappen".

A 57, III, pp. 331342, IV pp. 473

[3]

[4] Knuth, D.E.

[5] Laksov, D.

[6] Milne-Thompson, L.M.

[7] Willett, M.

Handboek der Wiskunde II en III (in dutch)

chapter on discrete mathematics.

The art of computer programming, vol. 2 Reading, Addison-Wesley, I969.

Linear recurring sequences over finite fields.

Math.Scand. vol. I6, pp. 18I - I96, 1965.

The calculus of finite differences. London, MacMillan, I95I.

The minimum polynomial for a given solution of a linear recursion. Duke Mathematical Journal 39 (l972) pp. 101 - 104.

SAMENVATTING

De theorie der lineaire recurrente rijen is klassiek geworden, Fibonacci, die in de 13e eeuw de naar hem genoemde rij, u„ = O, U T = 1 , U , - , = U + U T voor n ^ 1, bestudeerde wordt als de

grond-1 n+grond-1 n n-grond-1

legger van deze theorie beschouwd. In latere jaren zijn er vele ar-tikelen verschenen zowel ten aanzien van de algebraïsche eigenschap-pen als ten aanzien van de toepassingen. Bekende toepassingen zijn het genereren van een rij toevalsgetallen en het vercijferen van berichten,

Alle tot nog toe gepubliceerde artikelen hebben gemeen dat zij uitgaan van een rij getallen, welke voortgebracht wordt door één vaste relatie, nadat uiteraard een voldoend aantal beginvoor-waarden van zo'n rij gegeven is.

In deze dissertatie wordt een nieuw type rijen geïntroduceerd. Waarom van een nieuw type sprake is moge blijken uit het volgende (eenvoudige) voorbeeld. Beschouw een rij getallen voortgebracht door afwisselend twee verschillende relaties toe te passen, en wel u ,, = u + u T indien het rangnummer n even is en u ,, = 3u -u

n+1 n n-1 n+1 n n-1 indien dit getal n oneven is. Vanzelf is ook nog het duo u„ en u^ gegeven. Deze rij wordt dus niet voortgebracht door één vaste re-latie maar vraagt afwisselend één van de twee gegeven rere-laties. Sommigen zullen opmerken dat het altijd mogelijk is om één voort^ brengende relatie op te schrijven, waaraan alle elementen voldoen, bijvoorbeeld

1 - (-1 )"'•'• 1 - (-1)"

[%+l - % - %-lJ

•

*

• [ 2 ^jLVl - 3^n ^ ^n-lj = °

J ) f u , , - u - u , u , T - 3 u + u T I = 0 . Inderdaad is dit

I n+1 n n-lj I n+1 n n-1J

juist, maar zulke relaties behoren niet tot de klasse van de lineaire relaties met constante coëfficiënten. Niet triviaal is de vraag of er binnen deze klasse een relatie bestaat waaraan alle elementen van de in het voorbeeld genoemde rij voldoen,

Deze aldus gedefinieerde rijen zullen worden genoemd

"Conditionele recurrente rijen" en in deze dissertatie worden ten aanzien van deze rijen de volgende vragen gesteld en (niet altijd) afdoend beantwoord.

(i ) Welke condities laten het bestaan toe van één lineaire re-latie met constante coëfficiënten, waaraan alle elementen voldoen?

(ii ) Wat kan gezegd worden over de structuur van deze relatie, indien zij bestaat?

(iii) Wat is het verband tussen deze relatie en de relaties waar-. door de rij gedefinieerd is?

(iv ) Wat is de rol van de beginvoorwaarden in dit geheel? In deze dissertatie wordt aangetoond dat voor een ruime klasse van condities er inderdaad één vaste lineaire relatie met constante coëfficiënten bestaat, waaraan alle elementen voldoen. Als rode draad die door het antwoord op vraag (ii) loopt, kan ge-noemd worden dat deze relatie in het algemeen een relatie is tussen elementen die niet eikaars onmiddellijke opvolger zijn. De derde vraag wordt in deze dissertatie zowel constructief beantwoord als beschouwend; paragraaf 2.1.3 behandelt het verband tussen de ties die de rij definiëren en de (eventueel) bestaande vaste rela-tie in algebraïsche zin. Ten aanzien van vraag (iv) moet eerlijk-heidshalve gesteld worden dat een volledige theorie ontbreekt,

BIOGRAPÏÏY

J,L, SIMONS:

B o m in The Hague at february 7, 1946,

In 1964 he entered the Delft Institute of Technology and graduated in mathematics in 1971»

Since 1971 he is with National Aerospace Laboratory ( N L R ) , department of mathematics.

bij het proefschrift

C O N D I T I O N A L R E C U R R I N G

S E Q U E N C E S

van

J.L. SIMONS

9junil976

2. Bij conditionele recurrente rijen kunnen speciale beginvoor-waarden leiden tot graadverlaging van het karakteristieke poly-noom, zodanig dat het niet meer de in theorema 1 of 3 geponeer-de strcutuur bezit. In zulke gevallen kan dit polynoom op grond van genoemde theorema's verder in graad verlaagd worden.

(Vergelijk de laatste opmerking van dit proefschrift.) 3. Beschouw de inhomogene lineaire differentie vergelijking

f(z)u =r met Ir I^A. Indien alle wortels van f(z)=0 een

abso-^ abso-^ n n I n\ ^ '

lute waarde groter dan één bezitten, is de rij ju l begrensd door u < „^ V '^1^ \ 1 waarin m de graad van f(z) is.

4. Toevalsgetallen worden in de computer veelal gesimuleerd met behulp van zogenaamde "pseudo random generators". Voordat be-sloten wordt een bepaald type toe te passen, dient er grondig theoretisch inzicht in de wiskundige structuur van zo'n gene-rator te bestaan. In het bijzonder moet resonantie tussen deze structuur en die van het toepassingsgebied worden vermeden. 5. Kennis van de theorie der conditionele recurrente processen is

nuttig bij het beoordelen van sommige kryptografische technie-ken. (Zie H.A. van Ingen Schenau, L.J.M. Joosten, J.L. Simons: Image Data Security in the concept of the Agricultural Real Time Imaging Satellite System ( A R T I S S ) , NLR TR 76OIO C.) Het berekenen van de Besselfunktie J (z) (gehele orde, complex argument) geschiedt voor kleine orden zeer efficiënt met be-hulp van Poisson's sommatie formule. (Zie J.L, Simons: On the calculation of Bessel functions of the first kind, of integer order, for complex arguments, NLR MP 72018 U.)

Een positionele priemtafel volgens G.J. Hameetman eist minder papier dan een conventionele priemtafel. (Zie Mathematics of Computations, vol. 28, nr. 127.)

de roostersamensteller, doch deze belasting kan met behulp van heuristische programmatuur tot een overkomelijk bezwaar worden teruggebracht.

9. Introductie van een discreet tijdsbegrip in de logica elimi-neert een aantal paradoxen.

10. Om het kappen van oudere gezonde bomen tegen te gaan, dient ^r[ n </('

er een wet te komen die de (rechts) persoon die kapt verplicht ^^2. (••^^ >''?<•"

tot het planten van een equivalent boomquantum. Twee boom- ^ quanta zijn equivalent als hun leeftijdssommen gelijk zijn.

11. De dood dient uit de taboesfeer te worden gehaald.(Zie prof.dr. J.H.v.d. Berg, Het menselijk lichaam I en II.) 12. Het is merkwaardig dat er sportvissers bestaan die ethische

bezwaren koesteren tegen de jacht als recreatiesport.

13- Bewindslieden beantwoorden premature vragen van journalisten te weinig met de wedervraag: "Schaakt U?".