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

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.


Introduction
A tridiagonal determinant is a determinant whose nonzero elements locate only on the diagonal and slots horizontally or vertically adjacent the diagonal. Technically speaking, a determinant H = |h ij | n×n is called a tridiagonal determinant if h ij = 0 for all pairs (i, j) such that |i − j| > 1. For more information, please refer to the paper [11]. A determinant H = |h ij | n×n is called a lower (or an upper, respectively) Hessenberg determinant if h ij = 0 for all pairs (i, j) such that i + 1 < j (or j + 1 < i, respectively). For more information, please refer to the paper [18].
In combinatorial number theory, the Delannoy number D(m, n) for n, m ≥ 0 can be regarded as the number of lattice paths from (0, 0) to (m, n) in which only east (1, 0), north (0, 1), and northeast (1,1). The Delannoy numbers D(m, n) can be defined by means of the recurrence relation D(m, n) = D(m − 1, n) + D(m − 1, n − 1) + D(m, n − 1) with initial values D(0, n) = D(m, 0) = D(0, 0) = 1. The historic significance of these numbers D(m, n) was explained in the paper [1]. The Delannoy numbers D(m, n) can be computed by explicit formulas D(m, n) = n k=0 n k m k 2 k and D(m, n) = n =0 n m + n − n and can be generated by For more information on the Delannoy numbers D(m, n), please refer to [1,19,59,63] where are known as the rising and falling factorials respectively. The ideas, significance, and reasonability of these generalizations D a,b (k) and D a,b;λ (z) come from the papers [25,34,37,38,39,43,52,53,54,55] and closely related references therein. Inspired by the identity n k=0 n k n k x + k n in [61]. Various arithmetic properties and congruence relations for the Delannoy one-functional sequence D(x; n) have been studied in [9,16,17,59,60,63].
In [58], the Delannoy one-functional sequence D(x; n) was further generalized as D(x, r; n) = n k=0 x + r + k k was derived, and a plenty of identities for the Delannoy two-functional sequence D(x, r; n) were acquired. In [7], the Delannoy two-functional sequence D(x, r; n) was generalized again to the Delannoy twofunctional polynomials D(x, r; n; y) = n k=0 x + r + k k x − r n − k y k and, among other things, the generating function and recurrence formula for the Delannoy two-functional polynomials D(x, r; n; y) were derived. It is noted that D(x, 0; n) = D(x; n) and D(x, r; n; 1) = D(x, r; n).
In this paper, we will present closed forms [3] for the Delannoy two-functional sequence D(x, r; n) and its difference D(x, r; n) − D(x, r; n − 1) in terms of the Hessenberg determinants, derive recursive relations for the Delannoy two-functional sequence D(x, r; n) and its difference D(x, r; n) − D(x, r; n − 1), and deduce closed forms in terms of the Hessenberg determinants and recursive relations for the Delannoy one-functional sequence D(x; n), the Delannoy numbers D(m, n), and central Delannoy numbers D(k).

Determinantal forms of the Delannoy two-functional sequence and its difference
In this section, we will present closed forms for the Delannoy two-functional sequence D(x, r; n) and for its difference D(x, r; n) − D(x, r; n − 1) in terms of the Hessenberg determinants.
By the same arguments as in the derivation of the determinantal form (2.1), we can obtain the determinantal forms (2.2) immediately. The proof of Theorem 2.1 is complete.

Determinantal forms and recursive relations for the Delannoy one-functional sequence, the Delannoy numbers, and central Delannoy numbers
In this section, with the help of Theorems 2.1 and 3.1, we deduce closed forms in terms of the Hessenberg determinants and recursive relations for the Delannoy one-functional sequence D(x; n), the Delannoy numbers D(m, n), and central Delannoy numbers D(k).
Since D(x, 0; n) = D(x; n), when taking r = 0 in Theorems 2.1 and 3.1, we derive the following conclusions.
Theorem 4.1. The Delannoy one-functional sequence D(x; n) for n ≥ 0 can be determinantally expressed by where The difference D(x; n) − D(x; n − 1) for n ≥ 1 can be determinantally expressed by When taking x = m in Theorems 4.3 and 4.4, we derive the following conclusions. When taking m = n = k in Theorem 4.3, we derive the following conclusions.
Theorem 4.5. Central Delannoy numbers D(k) for k ≥ 0 can be determinantally expressed by