ARBITRARY ORDER RECURSIVE SEQUENCES AND ASSOCIATED CONTINUED FRACTIONS

The purpose of this paper is to demonstrate the existence of a one-to-one correspondence between a certain class of square matrices of arbitrary order and a related extension field. The elements of these matrices are eventually obtained from certain basic linear recursive sequences by generalizations of the ordinary continued fraction algorithm. This paper incorporates generalizations of Horadam’s work [1] to show the connections among arbitrary order linear sequences. This accounts for the on-conventional use of superscripts.


INTRODUCTION
The purpose of this paper is to demonstrate the existence of a one-to-one correspondence between a certain class of square matrices of arbitrary order and a related extension field. The elements of these matrices are eventually obtained from certain basic linear recursive sequences by generalizations of the ordinary continued fraction algorithm. This paper incorporates generalizations of Horadam's work [1] to show the connections among arbitrary order linear sequences. This accounts for the on-conventional use of superscripts.
Ward [2] established the existence of a one-to-one correspondence between a certain class of square matrices of order three (with elements from the field F of the auxiliary polynomial

DEFINITIONS
In the notation of this paper, Lucas studied the sequences      Horadam [3] developed properties of the more general sequence   ) 2 ( n w which satisfies the same recurrence relation (2.1) but has arbitrary initial conditions. Williams [4] extended Lucas' work both to arbitrary order and to the third order sequences, particularly   ) 3 ( ,n s u , s = 0,1,2, n>2, which satisfy the third order recurrence relation (2.3) in which the P3,i are given complex numbers, and with appropriate initial conditions.
In this paper we consider a generalization of Horadam's sequences to arbitrary order r, we define r basic sequences of order r,   the Kronecker delta.
The adjective "basic" is used by analogy with the corresponding third order sequence of Bell [5]. One of the basic sequences,  , Generally in the literature only one basic second order sequence is mentioned, namely the fundamental one, but Gootherts [8] has shown the need for two basic second order sequences as well as the primordial sequence. The fundamental nature of the sequence,  , which agrees with Equation (3.14) of Horadam [1]. This is mentioned in order to emphasize that the results in this paper are quite generalizable.
Modifying Philippou [10] we now define for notational convenience. This is similar to the A matrix of [11] and is a generalization of the Q (companion) matrix in the Fibonacci and Lucas number generators [12]. As in the second order (r = 2) case the following matrix properties are easily proved.

PRIMORDIAL AND BASIC SEQUENCES AND CONTINUED FRACTIONS
We can see in Table 1 some values for the first few terms of the second order (r = 2) primordial (s = 0) and basic (s = 1,2 ) sequences. This helps to visualize their roles in sequences of higher order than 2.
for m = 0,1,...,r-2, and j = 0,±1,±2,.... The proof follows symbolically from trivial but tedious matrix multiplication, but more importantly what follows from (4.5) is that if we extend it so that it is valid for m = r -1, then the (r-1)-dimensional Euclidean space of (r-1)-tuples of real numbers (r≥2) with k a non-negative rational integer. Following Bernstein [17] we define the transformation   (4.10) which in effect shows the connection between the arbitrary order sequences and the JPA which are themselves generalizations of the Euclidean algorithm; this means that in the way Bernstein has defined them they are also generalizations of the ordinary continued fraction algorithm. This can be seen for r = 2, when the elements of Table 1 are inserted in (4.8) [18].