Abstract
New q-series in the spirit of Jacobi have been found in a publication first published in 1884 written in Russian and translated into English in 1928. This work was found by chance and appears to be almost totally unknown. From these entirely new q-series, fresh lattice sums have been discovered and are presented here.
Highlights
A Short History of Lattice SumsThe Jacobian θ-functions ∞ θ2 = θ2 ( q ) = ∑ q(m−1/2)= 2q1/4 (1 + q2 + q6 . . . + qn(n+1) . . . , (1)= 1 + 2q + 2q4 + 2q9 + 2q16 . . . , (2) −∞ θ3 = θ3 ( q ) =
These sums have become known as lattice
Just as the q-series derived by Jacobi for products of θ-functions can be transformed into a lattice sum, so may the results of Nazimov
Summary
These sums have become known as lattice The Madelung constant is an example of a three-dimensional lattice sum. The representation of lattice sums by θ series was put on a more formal basis by use of the Mellin transform Ms defined by
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
