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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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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    I. W. Busbridge, A theory of general transforms for functions of the class ${L^p}(0,\infty ),(1 < p \leqq 2)$, Quart. J. Math. Oxford Ser. (2) 9 (1938), 148-160. A. L. Dixon and W. L. Ferrar, On the summation formulas of Voronoĭ and Poisson, Quart. J. Math. Oxford Ser. (2) 8 (1937), 66-74. A. P. Guinand, Summation formulae and self-reciprocal functions. I, Quart. J. Math. Oxford Ser. (2) 9 (1938), 53-67.
  • A. P. Guinand, General transformations and the Parseval theorem, Quart. J. Math. Oxford Ser. 12 (1941), 51–56. MR 4330, DOI https://doi.org/10.1093/qmath/os-12.1.51
  • N. S. Koshliakov, On Voronoĭi’s sum-formula, Messenger Math. (2) 58 (1928), 30-32.
  • John Boris Miller, A symmetrical convergence theory for general transforms, Proc. London Math. Soc. (3) 8 (1958), 224–241. MR 96948, DOI https://doi.org/10.1112/plms/s3-8.2.224
  • J. B. Miller, A symmetical convergence theory for general transforms, Proc. London Math. Soc. (3) 9 (1959), 451–464. MR 109995, DOI https://doi.org/10.1112/plms/s3-9.3.451
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
  • ---, Introduction to the theory of Fourier integrals, 2nd ed., Clarendon Press, Oxford, 1948. G. Voronoĭ, Sur une fonction transcendante et ses applications à la sommation de quelques séries, Ann. Sci. Ecole Norm. Sup. (3) 21 (1904). G. N. Watson, General transforms, Proc. London Math. Soc. (2) 35 (1933), 156-199.
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
  • J. R. Wilton, Voronoĭ’s summation formula, Quart. J. Math. Oxford Ser. (2) 3 (1932), 26-32.

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Keywords: Arithmetic function, Voronoi’s summation formula, <!– MATH ${L^2}(0,\infty )$ –> <IMG WIDTH="83" HEIGHT="43" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L^2}(0,\infty )$">, convergence in mean, Parseval’s theorem, Mellin transform, Fourier kernel
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