منابع مشابه
Weierstrass and Hadamard products
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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In this paper, assuming p/n → 0 as n → ∞, we will prove the weak and strong convergence to the semicircle law of the empirical spectral distribution of the Hadamard product of a normalized sample covariance matrix and a sparsing matrix, which is of the form Ap = 1 √ np (Xm,nX ∗ m,n − σnIm) ◦ Dm, where the matrices Xm,n and Dm are independent and the entries of Xm,n (m × n) are independent, the ...
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An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1992
ISSN: 0022-247X
DOI: 10.1016/0022-247x(92)90261-b