76,571 research outputs found
Generalized Twin Prime Formulas
Based on Golomb's arithmetic formulas, Dirichlet series for two classes of
twin primes are constructed and related to the roots of the Riemann zeta
function in the critical strip.Comment: 17 pages, no figure
Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula
Starting from the Rodrigues representation of polynomial solutions of the
general hypergeometric-type differential equation complementary polynomials are
constructed using a natural method. Among the key results is a generating
function in closed form leading to short and transparent derivations of
recursion relations and an addition theorem. The complementary polynomials
satisfy a hypergeometric-type differential equation themselves, have a
three-term recursion among others and obey Rodrigues formulas. Applications to
the classical polynomials are given.Comment: 13 pages, no figure
Regularities of Twin, Triplet and Multiplet Prime Numbers
Classifications of twin primes are established and then applied to triplets
that generalize to all higher multiplets. Mersenne and Fermat twins and
triplets are treated in this framework. Regular prime number multiplets are
related to quadratic and cubic prime number generating polynomials.Comment: 21 pages; p=21557 in x2+x+p deleted in (vi) of Theor. 4.1 on p.15; to
appear in Global J. Pure Applied Math. 8 (2012
Proton Spin Based On Chiral Dynamics
Chiral spin fraction models agree with the proton spin data only when the
chiral quark-Goldstone boson couplings are pure spinflip. For axial-vector
coupling from soft-pion physics this is true for massless quarks but not for
constituent quarks. Axial-vector quark-Goldstone boson couplings with {\bf
constituent} quarks are found to be inconsistent with the proton spin data.Comment: 8 pages, Latex, 1 table, no figure
Connections between Romanovski and other polynomials
A connection between Romanovski polynomials and those polynomials that solve
the one-dimensional Schr\"odinger equation with the trigonometric Rosen-Morse
and hyperbolic Scarf potential is established. The map is constructed by
reworking the Rodrigues formula in an elementary and natural way. The
generating function is summed in closed form from which recursion relations and
addition theorems follow. Relations to some classical polynomials are also
given.Comment: 17 pages, no figures, to appear in Central European J. Math. (2007
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