In this section, we will show that both hypotheses formulated at the end of Section 5 are false. The following proposition disproves Hypothesis 5.4 I.
6.1 Proposition. Each -recursive number sequence with is additionally
PROOF. Let be some -recursive sequence with .
To summarize, Proposition 6.1 tells us that each sequence of natural numbers that can be represented by a recursion formula from domain is additionally representable by recursion formulae from the domains and . This result for the task-type analyzed here corresponds to the result stated in Theorem 4.9. In spite of the strong Restriction (R) for permitted recursion formulae, for each natural number sequence there exist permissible recursion formulae of higher degrees whenever a permissible recursion formula of degree r=1 exists. Observe that this statement (as the proof to Proposition 6.1 reveals) holds for each arbitrary linear-recursive number sequence under (R) of at least four members. Thus, the uncovered non-uniqueness of solution does not depend on the length of the given sequence and thus cannot be prevented by prolonging the given sequence.
There also exist classes of number sequences of the form which are recursive of degree 2 but not of degree 1, and that can be described by recursion formulae of degree 3. This is demonstrated by the following examples.
6.2 Examples.
By the preceding Proposition 6.1 and the examples of 6.2 the Hypothesis 5.4 I. is unequivocally disproved. Admittedly, the formulae derived in the proof of 6.1 all lead to equal extensions of the given number sequence. That, of course, even this type of non-uniqueness may cause serious problems in various applications has been emphasized in connection with the uniqueness problem explicated in 3.3.
The cases we have been dealing with up until now might at least point to the correctness of Hypothesis 5.4 II. This hypothesis includes that, at least under the strong Restriction (R), alternative recursion formulae (should they exist for some sequence) all lead to identic extensions. Unfortunately, also this part of Hypothesis 5.4 proves to be false. The following proposition describes classes of number sequences that disprove this hypothesis.
6.3 Proposition. Let be a non-constant number sequence, and let , with for be the sequence of differences of each two subsequent members of . Then the following hold:
PROOF.
For n=6 from (6) we obtain the matrix
(a) For , we obtain from (7)
the matrix
where since the sequence is non-constant.
The corresponding LES in D has the
solutions
with arbitrary, and
For we obtain
hereby would be independent of if and only if
which is not the case since the sequence is
non-constant.
(b) For , the matrix (7) transforms to
where since the sequence is non-constant.
Thus, solutions of the corresponding LES in D are
formulae of type
with arbitrary, and
For we obtain
where is essentially dependent of because
since the sequence is non-constant.
(c) For , we obtain from (7)
the matrix
where since the sequence is non-constant.
Solutions of the corresponding LES in D are
formulae of type
with arbitrary, and
For we obtain
where is not independent of since
.
Remark. One easily shows that, in the case of our particular type of number sequence tasks, exactly the previously described classes of (non-constant) sequences have the demonstrated problematic property of multiple representability and extensionability.
6.4 Examples.
To summarize, Proposition 6.3 reveals that certain classes of number sequences can be characterized by infinitely many recursion formulae of the same type where each formula generates a different extension of the sequence. This result clearly disproves Hypothesis 5.4 II. Moreover, particularly the examples in 6.4 should illustrate that alternative solutions to a number sequence task need not differ considerably in their complexity, whereas they produce differing extensions of the given sequence. To make matters worse, observe that each sequence-type described in 6.4 is not representable by any recursion formula of some lower degree; thus, no formulae exist that can be deemed the ''simplest'' in a psychological sense.
After both hypotheses connected to the Restriction (R) have been proven false, the justified question remains:
Do 6-term sequences of natural numbers, that under the Restriction (R) can be represented uniquely through some recursion formula from D (and thus also have unique extensions), exist at all ?This question can be easily answered on the basis of the results from Section 4.
If the answer is yes: How can these sequences be characterized ?