Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593678 | Journal of Number Theory | 2015 | 30 Pages |
Abstract
We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Res=1/2. The polynomials with zeros only on the critical line are identified in terms of certain F23(1) hypergeometric functions. These polynomials possess the functional equation pn(s)=(−1)⌊n/2⌋pn(1−s)pn(s)=(−1)⌊n/2⌋pn(1−s). Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Mark W. Coffey, Matthew C. Lettington,