Authors: | Stević, Stevo | Title: | On an extension of a recurrent relation from combinatorics | Journal: | Electronic Journal of Qualitative Theory of Differential Equations | Volume: | 2017 | First page: | 1 | Last page: | 13 | Issue Date: | 1-Jan-2017 | Rank: | M21 | ISSN: | 1417-3875 | DOI: | 10.14232/ejqtde.2017.1.84 | Abstract: | The following recurrent relation/partial difference equation wn,k = awn−1,k−1 + bwn−1,k, where k,n,a,b ∈ ℕ, appears in a problem in combinatorics. Here we show that an extension of the recurrent relation is solvable on, the, so called, combinatorial domain C={(n,k)∈ ℕ20:0≤k≤n}∖{(0,0)}, when its coefficients and the boundary values wj,0, wj,j, j ∈ ℕ, are complex numbers, by presenting a representation of the general solution to the recurrent relation on the domain in terms of the boundary values. As a special case we obtain a solution to the problem in combinatorics in an elegant way. From the general solution along with an application of the linear first-order difference equation is also obtained the solution of the recurrent relation in the case wj,j=c ∈ C, j ∈ ℕ. |
Keywords: | Boundary-value problem | Combinatorial domain | Equation solvable in closed form | Method of half-lines | Partial difference equation | Publisher: | Bolyai Institute, University of Szeged |
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