12,958 research outputs found
Rational Convolution Roots of Isobaric Polynomials
In this paper, we exhibit two matrix representations of the rational roots of
generalized Fibonacci polynomials (GFPs) under convolution product, in terms of
determinants and permanents, respectively. The underlying root formulas for
GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier
paper by MacHenry and Tudose, make use of two types of operators. These
operators are derived from the generating functions for Stirling numbers of the
first kind and second kind. Hence we call them Stirling operators. To construct
matrix representations of the roots of GFPs, we use the Stirling operators of
the first kind. We give explicit examples to show how the Stirling operators of
the second kind appear in the low degree cases for the WIP-roots. As a
consequence of the matrix construction, we have matrix representations of
multiplicative arithmetic functions under the Dirichlet product into its
divisible closure.Comment: 13 page
A note on some constants related to the zeta-function and their relationship with the Gregory coefficients
In this paper new series for the first and second Stieltjes constants (also
known as generalized Euler's constant), as well as for some closely related
constants are obtained. These series contain rational terms only and involve
the so-called Gregory coefficients, which are also known as (reciprocal)
logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of
the second kind. In addition, two interesting series with rational terms are
given for Euler's constant and the constant ln(2*pi), and yet another
generalization of Euler's constant is proposed and various formulas for the
calculation of these constants are obtained. Finally, in the paper, we mention
that almost all the constants considered in this work admit simple
representations via the Ramanujan summation
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