Determinantal forms and recursive relations of the Delannoy two-functional sequence

Abstract

In the paper, the authors establish closed forms for the Delannoy two-functional sequence and its difference in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence and its difference, and deduce closed forms in terms of the Hessenberg determinants and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers.

New families of Horadam numbers associated with finite operators and their applications

“…In this section, we present a closed formula for the Horadam finite operator numbers , in terms of a tridiagonal determinant and, as special cases of this newly‐established closed formula for the Horadam finite operator numbers , obtain closed formula for the k ‐Fibonacci finite operator numbers , the Fibonacci finite operator numbers , the Pell finite operator numbers , the k ‐Lucas finite operator numbers , the Lucas finite operator numbers , the Pell‐Lucas finite operator numbers , the Jacobsthal finite operator numbers , the Jacobsthal‐Lucas finite operator numbers , respectively. In the existing literature on the subject, one can find a fairly large number of papers relating the tridiagonal and Hessenberg determinants with special numbers and polynomials to algebra and combinatorial number theory, (see, for example, previous studies 20–29 ).…”

Three closed forms for convolved Fibonacci numbers

“…Remark 4. In [15,34,35], the authors have discussed the Cauchy product of central Delannoy numbers and other properties of the Delannoy numbers. Remark 5.…”

“…Remark 7. This paper is a companion of the electronic preprint [7] whose methods have been applied in [21,35,44,45] and closely related references therein. Remark 8.…”

Several recurrence relations and identities on generalized derangement numbers

“…Email addresses: mcihatdagli@akdeniz.edu.tr (Muhammet Cihat Da §l), qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com (Feng Qi) See the monograph [3] and the papers [18,20,21,23,24,25,26]. The generalized derangement numbers d n,r are introduced by Munarini [15] as…”