Voronoi-Nasim summation formulas and index transforms

• Type: Article in International Scientific Journal
• Date: 2012

Authors

Semyon Yakubovich (Author) FCUP

Journal

Title: Integral Transforms and Special Functions

Vol. 23

Pages: 369-388

ISSN: 1065-2469

Publisher: Taylor & Francis

Indexing

ISI Web of Knowledge - 2 Citations

Scopus - 1 Citation

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-002-GHW

DOI: 10.1080/10652469.2011.595092

Abstract (EN): Using L-2-theory of the Mellin and Fourier-Watson transformations, we relax Nasim's conditions to prove the summation formula of Voronoi. It involves sums of the form Sigma d(n) f(n), where d(n) is the number of divisors of n. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x --> infinity of the sum D(x) = Sigma(n <= x) d(n). In particular, we generalize Koshliakov's formula, which contains the modified Bessel function of zero-index f (x)= K-0(2 pi zx), (z is a parameter) on the modified Bessel function of an arbitrary complex index. Finally, we apply index transforms of the Kontorovich-Lebedev type to obtain a new class of summation formulas involving Dirichlet's function d(n).

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 20