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We derive, using matrices, new classes of linear recurrence identities linking the general term of a degree three linear recurrence sequence with those of an associated ‘cohort’ sequence that differs only in its initial values. The class description is seen to recover that previously established for the so called Horadam sequence and its cohort sequence set. [ABSTRACT FROM AUTHOR]

We state and prove a condition for an identity which involves the diagonal entries of any nth root of a general 2 × 2 matrix, and develop a method for obtaining all roots through a so called generator pair of parameters defined by its eigenvalues. [ABSTRACT FROM AUTHOR]

We state, and prove by a succinct matrix method deployed previously, a result for terms of a sequence generated by a fully general (linear) fourth order recurrence equation which--being characterised by four additional independent parameters--describes a new class of highly non-linear identities. [ABSTRACT FROM AUTHOR]

We derive, using matrices, new classes of linear recurrence identities linking the general term of the so called Horadam sequence with those of an associated 'cohort' sequence that differs only in its initial values. [ABSTRACT FROM AUTHOR]

We present--as an alternative to the standard technique deployed historically--a matrix based method that delivers both of the characteristic root dependent closed forms for the general term of the celebrated Horadam sequence. [ABSTRACT FROM AUTHOR]

Using an existing matrix method developed by the authors, a non-linear recurrence identity class is formulated for terms of a quasi Fibonacci sequence which--in addition to fundamental sequence variables--is characterised by four arbitrary parameters. The possibility for further systematic re-application of the technique is also discussed and illustrated. [ABSTRACT FROM AUTHOR]

We state and prove a non-linear recurrence identity for terms of the so called Ho- radam sequence, and then offer its generalisation which is available from the same methodology. We illustrate how the overarching idea may be used to sequentially produce extended versions, each in turn with an extra level of non-linearity and term index complexity. These identities can all be captured in matrix determinant form. [ABSTRACT FROM AUTHOR]

We state, and prove using matrices, two (related) linear recurrence identities for terms of the so called Horadam sequence; each result expresses the general term of the sequence as a linear combination of terms with particular initial values. First offered by A.F. Horadam himself in the 1960s, our approach to their formulation is quite different and, we believe, new. [ABSTRACT FROM AUTHOR]

We offer a formalised proof argument to establish the invariance, with respect to matrix power, of the n − 1 anti-diagonal ratios within a fixed n-dimensional tri-diagonal matrix. [ABSTRACT FROM AUTHOR]

We state and prove (first by construction, and then by induction) the invariance, with respect to matrix power, of the anti-diagonals ratio of a general 2 × 2 matrix, adding these proofs to others found previously. [ABSTRACT FROM AUTHOR]

We state and prove the invariance, with respect to matrix power, of both the diagonals and anti-diagonals ratio of a special case 2 × 2 matrix. The proof methodology is new, contrasting with those deployed previously in establishing anti-diagonals matrix invariants. [ABSTRACT FROM AUTHOR]

We state and prove an invariance property, with respect to matrix power, for those n - 1 immediate off-diagonal ratios of a tri-diagonal n-square matrix. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

We state and prove the invariance, with respect to matrix power, of the antidiagonals ratio of a general 2 x 2 matrix. The proof methodology--utilising basic properties of a so called (two vertex) quiver--is new, and adds to others found recently. [ABSTRACT FROM AUTHOR]

We introduce in this paper a straightforward but useful method for computing indefinite rational matrix products. The method is used to prove a certain identity involving definite sums and a definite integral. [ABSTRACT FROM AUTHOR]