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We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;beta)=(-1)^{lfloor n/2 rfloor} p_n(1-s;beta)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory.
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 Re $s=1/2$. The polynomials with
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of
The theory of zeta functions of fractal strings has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 20
In this work we verify the sufficiency of a Jensens necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast dec
A simple expression for the zeros of Weierstrass function is given which follows from a formula for relativistic orbits.