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Transactions of the American Mathematical Society

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On the summation formula of Voronoi


Author: C. Nasim
Journal: Trans. Amer. Math. Soc. 163 (1972), 35-45
MSC: Primary 10.43
DOI: https://doi.org/10.1090/S0002-9947-1972-0284410-9
MathSciNet review: 0284410
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Abstract: A formula involving sums of the form $ \Sigma d(n)f(n)$ and $ \Sigma d(n)g(n)$ is derived, where $ d(n)$ is the number of divisors of $ n$, and $ f(x),g(x)$ are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi's summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between $ f(x)$ and $ g(x)$ is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class $ {L^2}(0,\infty )$. We use $ {L^2}$-theory of Mellin and Fourier-Watson transformations.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0284410-9
Keywords: Arithmetic function, Voronoi's summation formula, $ {L^2}(0,\infty )$, convergence in mean, Parseval's theorem, Mellin transform, Fourier kernel
Article copyright: © Copyright 1972 American Mathematical Society