33 research outputs found
Families of weighted sum formulas for multiple zeta values
Euler's sum formula and its multi-variable and weighted generalizations form
a large class of the identities of multiple zeta values. In this paper we prove
a family of identities involving Bernoulli numbers and apply them to obtain
infinitely many weighted sum formulas for double zeta values and triple zeta
values where the weight coefficients are given by symmetric polynomials. We
give a general conjecture in arbitrary depth at the end of the paper.Comment: The conjecture at the end is reformulate
Note on Legendre numbers
The definition and basic properties of Legendre Numbers are reviewed here. We then develop some new properties and identities involving sums of Legendre Numbers, including clarification of a statement in the paper of Haggard [1]
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Special Functions Related to Dedekind Type DC-Sums and their Applications
In this paper we construct trigonometric functions of the sum T_{p}(h,k),
which is called Dedekind type DC-(Dahee and Changhee) sums. We establish
analytic properties of this sum. We find trigonometric representations of this
sum. We prove reciprocity theorem of this sums. Furthermore, we obtain
relations between the Clausen functions, Polylogarithm function, Hurwitz zeta
function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum
T_{p}(h,k). We also give some applications related to these sums and functions