511 research outputs found
A short note on a extended finite secant series
In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions
On a class of distributions stable under random summation
We investigate a family of distributions having a property of
stability-under-addition, provided that the number of added-up random
variables in the random sum is also a random variable. We call the
corresponding property a \,-stability and investigate the situation with
the semigroup generated by the generating function of is commutative.
Using results from the theory of iterations of analytic functions, we show that
the characteristic function of such a -stable distribution can be
represented in terms of Chebyshev polynomials, and for the case of -normal
distribution, the resulting characteristic function corresponds to the
hyperbolic secant distribution. We discuss some specific properties of the
class and present particular examples.Comment: 12 pages, 1 figur
The Euler and Springer numbers as moment sequences
I study the sequences of Euler and Springer numbers from the point of view of
the classical moment problem.Comment: LaTeX2e, 30 pages. Version 2 contains some small clarifications
suggested by a referee. Version 3 contains new footnotes 9 and 10. To appear
in Expositiones Mathematica
Human and automated approaches for finite trigonometric sums
We show that identities involving trigonometric sums recently proved by
Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta
functions, were either already in the literature or can be proved easily by
adapting results that can be found in the literature. Also we prove two
conjectures given in that paper. After mentioning many other works dealing with
identities for various trigonometric sums, we end this paper by describing an
automated approach for proving such trigonometric identities
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