Combinatorial Identities for Incomplete Tribonacci Polynomials

The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize the usual tribonacci polynomials T_n(x) and were introduced in [10], where several algebraic identities were shown. In this paper, we provide a combinatorial interpretation for T_n^{(s)}(x) in terms of weighted linear tilings involving three types of tiles. This allows one not only to supply combinatorial proofs of the identities for T_n^{(s)}(x) appearing in [10] but also to derive additional identities. In the final section, we provide a formula for the ordinary generating function of the sequence T_n^{(s)}(x) for a fixed s, which was requested in [10]. Our derivation is combinatorial in nature and makes use of an identity relating T_n^{(s)}(x) to T_n(x)

Incomplete Generalized (p; q; r)-Tribonacci Polynomials

In this paper, we consider an extension of the tribonacci polynomial, which we will refer to as the generalized (p; q; r)-tribonacci polynomial, denoted by Tn;m(x).We find an explicit formula for Tn;m(x)which we use to introduce the incomplete generalized (p; q; r)-tribonacci polynomials and derive several properties. An explicit formula for the generating function of the incomplete generalized polynomials is determined and a combinatorial interpretation is provided yielding further identities

On the self-convolution of generalized Fibonacci numbers

We focus on a family of equalities pioneered by Zhang and generalized by Zao and Wang and hence by Mansour which involves self convolution of generalized Fibonacci numbers. We show that all these formulas are nicely stated in only one equation involving a bivariate ordinary generating function and we give also a formula for the coefficients appearing in that context. As a consequence, we give the general forms for the equalities of Zhang, Zao-Wang and Mansour

Diophantine triples in linear recurrence sequences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialized generalizations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1602.0823