The linear algebra of ther-Whitney matrices

A new approach to the r-Whitney numbers by using combinatorial differential calculus

“…Then we obtain the following corollary. From the generating function (25) we obtain that the r-Whitney numbers of the first kind are given by the exponential Riordan array…”

“…Note that if (m, r) = (1, 0) we obtain the Stirling numbers of the second kind, if (m, r) = (1, r) we have the r-Stirling (or noncentral Stirling) numbers [4], and if (m, r) = (m, 1) we have the Whitney numbers [1,2]. Many properties of the r-Whitney numbers and their connections to elementary symmetric functions, matrix theory, special polynomials, combinatorial identities and generalizations can be found in [9,10,12,18,20,21,24,25,26,28,29,37].…”

Some Determinants Involving Incomplete Fubini Numbers

“…In [15], Li obtained three new Fibonacci-Hessenberg matrices and studied its relations with Pell and Perrin sequence. Xu and Zhou [26] studied a determinantal representation for a generalization of the Stirling numbers, called r-Whitney numbers, see also [19]. More examples can be found in [23,24,27].…”

A q-analogue of α־-Whitney numbers

“…. , r) the w q,m,ᾱ (n) is reduced to the r-Whitney matrix of the first kind [12]. Mansour et al [9] derived a closed formula for all sequences satisfying a certain recurrence relation as follows:…”

“…. , r) the L m,ᾱ (n) is reduced to the r-Whitney-Lah matrix[12].Theorem 11. Theᾱ-Whitney-Lah numbers L m,ᾱ (n, k) have the explicit formulaL m,ᾱ (n, k) = j + jm + α i + im) k i=0,i =j (α j + jm − α i − im),(5.6) and the recurrence relationL m,ᾱ (n, k) = n j=k L m,ᾱ (j − 1, k − 1) n−1 i=j (α i + im + α k + km).…”