The Hankel Transform of the Sum of Consecutive Generalized ...

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12In our case it is enough to take d = 0 to get the final weightHencer −1 = −(L + 1),w(x) = ˘w(x)x .(r n = − ᾰ n +˘β )nr n−1Lemma 4.1. The parameters r n have the explicit form(n = 0, 1, . . .). (37)r n = − ψ n+1ψ n+2· Lψ n+2 + ξϕ n+2Lψ n+1 + ξϕ n+1(n ∈ N 0 ). (38)Proof. We will use the mathematical induction. For n = 0, we really get theexpected valuer 0 = − L2 + 2L + 2(L + 1)(L + 2) .Suppose that it is true for k = n. Now, by the properties for ϕ n and ψ n , we have˜α n+1 · r n + ˜β n+1 = − ψ n+1ψ n+3· Lψ n+3 + ξϕ n+3Lψ n+1 + ξϕ n+1.Dividing with r n , we conclude that the formula is valid for r n+1 .Example 4.3. For L = 4 we getwherefromα 0 = 24r −1 = −5, r 0 = − 1315 , r 1 = − 5152 , r 2 = − 356357 ,5 , β 0 = 5, α 1 = 32365 , β 1 = 10425 , α 2 = 1104221 , β 2 = 680169 ,just the same as in the Example 4.1.Proof of the main result. Let us start from Krattenthaler formulah 1 = a 0 , h n = a n 0β1 n−1 β2 n−2 · · · βn−2β 2 n−1 (n = 2, 3, . . .). (39)Here is a 0 = β 0 = L + 1. This formula can be also written in the formh 1 = a 0 , h n = β 0 β 1 β 2 · · · β n−2 β n−1 · h n−1 . (40)From the theory of orthogonal polynomials, it is known thatwherefromHere,‖Q n−1 ‖ 2 = β 0 β 1 β 2 · · · β n−2 β n−1 (n = 2, 3, . . .), (41)h 1 = a 0 , h n = ‖Q n−1 ‖ 2 · h n−1 (n = 2, 3, . . .). (42)‖Q n−1 ‖ 2 = β 0r n−2r −1n−2∏k=0˘β k = Ln−12·Lψ n + ξϕ nLψ n−1 + ξϕ n−1. (43)
We will apply the mathematical induction again. The formula for h n is true forn = 1. Suppose that it is valid for k = n − 1. Thenh n = Ln−12wherefrom follows that the final statement·Lψ n + ξϕ n· L(n−1)(n−2)/2· (LψLψ n−1 + ξϕ n−1 2 n n−1 + ξϕ n−1 ) ,ξ13is true.h n = Ln(n−1)/22 n+1 ξ· (Lψ n + ξϕ n ) (n ∈ N)References[1] P. Barry, On Ineger Sequences Based Constructions of Generalized Pascal Triangles, Preprint,2005.[2] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York,1978.[3] A. Cvetković, P. Rajković and M. Ivković, Catalan Numbers, the Hankel Transform andFibonacci Numbers, Journal of Integer Sequences, 5, May 2002, Article 02.1.3.[4] W. Gautschi, Orthogonal polynomials: applications and computations, in Acta Numerica,1996, Cambridge University Press, 1996, pp. 45–119.[5] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Clarendon Press -Oxford, 2003.[6] C. Krattenthaler, Advanced Determinant Calculus,at http://www.mat.univie.ac.at/People/kratt/artikel/detsurv.html.[7] P. Peart and W. J. Woan, Generating functions via Hankel and Stieltjes matrices,Journal of Integer Sequences, Article 00.2.1, Issue 2, Volume 3, 2000.[8] W. J. Woan, Hankel Matrices and Lattice Paths, Journal of Integer Sequences, Article 01.1.2, Volume 4, 2001.[9] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electronically athttp://www.research.att.com/∼njas/sequences/.(Mentions sequences A005807, A001906, A001519.)Predrag Rajković, Marko D. Petković,University of Niš, Serbia and Montenegroe-mail: pedja.rajk@gmail.com, dexterofnisgmail.comPaul BarrySchool of Science, Waterfall Institute of Technology, Irelande-mail: pbarry@wit.ie
Page 1 and 2: The Hankel Transform of the Sum ofC
Page 3 and 4: 3is given by−n)/2h n =L(n2 2 n+1
Page 5 and 6: 5Now,Also,∞∑T (2n, n; L) t n =n
Page 7 and 8: 7Example 3.2. From (??), we have:{
Page 9 and 10: 9Example 4.1. For L = 4 we getwhere
Page 11: 11Example 4.2. For L = 4 we getwher

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