Mellin transforms with only critical zeros: Legendre functions

Mellin transforms of Whittaker functions

In this note we show that for an arbitrary semisimple Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We determine the possible poles.

Fourier Transforms Having Only Real Zeros

Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distribution functions F such that the Fourier transform H(z) = R∞ −∞G(it)e izt dF (t) has only real zeros.

Mellin transforms of p-adic Whittaker functions

Mellin Transforms: Harmonic Sums

My assignment is going to introduce the Mellin transform and its application on harmonic sums [1]. Hjalmar Mellin(1854-1933, [2] for a summary of his works) gave his name to the Mellin transform, a close relative of the integral transforms of Laplace and Fourier. Mellin transform is useful to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics, discrete prob...

Inverse Transforms of Products of Legendre Transforms

is obtained by applying successive integration by parts to the integral represented by the left-hand member of this equation so that the new integrand is the product of F by [(1 —x2)P„' ]', and by replacing this second factor by — n(n + l)Pn according to Legendre's differential equation [l]. Thus certain boundary value problems in ordinary and partial differential equations that involve the dif...