Shannon, Anthony G.; Uysal, Mine; Özkan, Engin

In this paper, we introduce some feature of the Fermatian numbers. The finite sum formulas of these numbers is calculate. The exponential generating function of Fermatian numbers is found and some of its identities is calculated. Another number sequence is obtained from the partial row sums of these numbers and these numbers were examined. At the same time, another polynomial has been defined as a generalization of these numbers, depending on powers of z.

Shannon, Anthony G.; Akkus, Hakan; Aküzüm, Yesim; Deveci, Omur; Özkan, Engin

The purpose of this note is to develop a conjecture for a Fibonacci number generating function in terms of the elements of a second-order two parameter partial recurrence relation which arose in an operations research problem on Poisson distributed lead time in inventory control.

Akkus, Hakan; Deveci, Omur; Özkan, Engin; Shannon, Anthony G.

Recursive sequences with gaps have been studied previously. This paper considers some elementary properties of such sequences where the gaps have been created on a regular basis from sequence to sequence - 'discatenated' (systematic gaps) and 'lacunary' (general gaps). In particular, their generating functions are developed in order to open up their general terms and relations with other properties.

Özkan, Engin; Akkus, Hakan

In this study, we define the Bronze Leonardo, Bronze Leonardo-Lucas, and Modified Bronze Leonardo sequences, and some terms of these sequences are given. Then, we give special summation formulas, special generating functions, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, we find the special relations between the terms of the Bronze Leonardo, Bronze Leonardo-Lucas, and Modified Bronze Leonardo sequences. Moreover, we examine the relationships among the Bronze Fibonacci and Bronze Lucas sequences of these sequences. Finally, we associate these sequences with the matrices.

Kuloglu, Bahar; Ozkan, Engin; Shannon, Anthony G.

In this study, a binomial sum, unlike but analogous to the usual binomial sums, expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell-Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called 'incomplete' number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.

Shannon, Anthony G.; Özkan, Engin

This paper builds on Roettger and Williams' extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations with a variety of numerical illustrations which demonstrate that mathematical notation is a tool of thought.

Özkan, Engin; Uysal, Mine

In this work, we investigate the hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas octonions. We give Binet's Formula, Cassini's identity, Catalan's identity, d'Ocagne identity, generating functions of the hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas octonions. Also, we present many properties of these octonions.

Kuloglu, Bahar; Ozkan, Engin

We introduce new kinds of k-Pell and k-Pell-Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k, r)-Pell and (k, r)-Pell- Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k, r)-Pell and (k, r)-Pell-Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k, r)-Pell and (k, r)-Pell-Lucas sequences from the generating function, in connection to graphs.

Kuloglu, Bahar; Özkan, Engin; Shannon, Anthony G.

In this paper, we introduce the incomplete Vieta-Pell and Vieta-Pell-Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

Tastan, Merve; Özkan, Engin; Shannon, Anthony G.

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini's Identities for the families and their polynomials. Finally, we suggest avenues for further research.