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

Abstract

<p>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.</p>

Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences

“…Using the formula (2.2) together with (4.1), one can simply, elementarily, easily, standardly, and immediately obtain determinantal expressions in terms of some Hessenberg determinants and can simply, elementarily, easily, standardly, and immediately derive recursive relations of coefficients in power series expansions of functions in the form of a ratio of two infinitely differentiable functions. We believe that this approach should be useful and applicable in analytic combinatorics, analytic number theory, the theory of matrices, the theory of special functions, and other mathematical branches, as done in [8,18,20,22,23,24,25,26,28,29] and closely related references therein.…”

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.…”

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

“…The determinantal form (5.1) was established in [26,Theorem 1.1]. See also related texts in the papers [19]. 4.…”

“…In combinatorial number theory, it is significant to express concrete sequences or arrays of integer numbers or polynomials in terms of tridiagonal determinants or the Hessenberg determinants. In this respect, the Bernoulli numbers and polynomials [2,6,20,27,36,40,62], the Euler numbers and polynomials [29,32,64], (central) Delannoy numbers and polynomials [10,19,25,26,31,47,48,49], the Horadam polynomials [41], (generalized) Fibonacci numbers and polynomials [13,14,15,21,25,33,46,50], the Lucas polynomials [41,46], and the like, have been represented via tridiagonal determinants or the Hessenberg determinants, and consequently many remarkable relations have been obtained. For more information in this area and direction, please refer to [12,22,23,30,35,42,44,45,51,56,57] and closely related references therein.…”