Singularity analysis, Hadamard products, and tree recurrences
نویسندگان
چکیده
منابع مشابه
Singularity Analysis, Hadamard Products, and Tree Recurrences
We present a toolbox for extracting asymptotic information on the coefficients of combinatorial generating functions. This toolbox notably includes a treatment of the effect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequence, it becomes possible to unify the analysis of a number of divide-and-conquer algorithms...
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1. Weierstrass products 2. Poisson-Jensen formula 3. Hadamard products Apart from factorization of polynomials, after Euler's sin πz πz = ∞ n=1 1 − z 2 n 2 there is Euler's product for Γ(z), which he used as the definition of the Gamma function: ∞ 0 e −t t z dt t = Γ(z) = 1 z e γz ∞ n=1 1 + z n e −z/n where the Euler-Mascheroni constant γ is essentially defined by this relation. The integral (E...
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Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1 1− ax − x2 ∗ 1 1− bx − x2 = 1− x2 1− abx − (2 + a2 + b2)x2 − abx3 + x4 , where ∗ denotes the Hadamard product. In a similar way, by considering tilings of a 2× n rectangle with 1× 1 and 1× 2 bricks in the top row, and 1× 1 and 1× n bricks in the bottom row, we find...
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Article history: Received 20 May 2015 Available online 6 November 2015 Communicated by Reinhard Laubenbacher MSC: 14T05 14N05 68W30 14M99
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ژورنال
عنوان ژورنال: Journal of Computational and Applied Mathematics
سال: 2005
ISSN: 0377-0427
DOI: 10.1016/j.cam.2004.04.014