On Higher Dimensional Fibonacci Numbers, Chebyshev Polynomials and Sequences of Vector Convergents

We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integ

Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions

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.

Mellin transforms with only critical zeros: Legendre functions

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 zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)=(-1)^{lfloor n/2 rfloor} p_n(1-s)$. 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.

A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the zeta constants

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for zeta(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.