id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4854	Engalan, Genevieve B.; Latayada, Mary Joy	The Linear Algebra of (r, β)-Stirling Matrices	2023	17	.pdf	application/pdf	4625	284	90	βt+ r βt+ β + r βt+ 2β + r βt+ 3β + r (βt+ r)2 (βt+ β + r)2 (βt+ 2β + r)2 (βt+ 3β + r)2 (βt+ r)3 (βt+ β + r)3 (βt+ 2β + r)3 (βt+ 3β + r)3  = Vβ,r 4 (t). 2.4. − s]S(β,s)(4) =  1 0 0 0 r − s 1 0 0 (r − s)2 2(r − s) 1 0 (r − s)3 3(r − s)2 3(r − s) 1   1 0 0 0 s 1 0 0 s2 β + 2s 1 0 s3 β2 + 3βs+ 3s2 3β + 3s 1  =  1 0 0 0 r 1 0 0 −1 + 2r + (r − 1)2 b+ 2r 1 0 −2 + 3r + 3 (r − 1)2 + (r − 1)3 b2 + 3b+ 6r − 3 + 3b (r − 1) + 3 (r − 1)2 3b+ 3r 1  =  1 0 0 0 r 1 0 0 r2 β + 2r 1 0 r3 β2 + 3βr + 3r2 3β + 3r 1  = S(β,r)(4) Proposition 3. Let Pn[r] and S2(n)[β] be the n×n Pascal matrix and Stirling matrix of the second kind.	cache/ejpam-4854.pdf	txt/ejpam-4854.txt