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A general class of Voronoi's and Koshliakov–Ramanujan's summation formulas involving d k ( n ).
- Source :
-
Integral Transforms & Special Functions . Nov2011, Vol. 22 Issue 11, p801-821. 21p. - Publication Year :
- 2011
-
Abstract
- By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10652469
- Volume :
- 22
- Issue :
- 11
- Database :
- Academic Search Index
- Journal :
- Integral Transforms & Special Functions
- Publication Type :
- Academic Journal
- Accession number :
- 66401404
- Full Text :
- https://doi.org/10.1080/10652469.2010.540447