Some infinite sums identities (original) (raw)

On a New Class of Series Identities

Honam Mathematical Journal, 2015

The aim of this paper is to provide a new class of series identities in the form of four general results. The results are established with the help of generalizatons of the classical Kummer's summation theorem obtained earlier by Rakha and Rathie. Results obtained earlier by Srivastava, Bailey and Rathie et al. follow special cases of our main findings.

On some sum–to–product identities

Bulletin of the Australian Mathematical Society, 2001

We give new proofs of some sum-to-product identities due to Blecksmith, Brillhart and Gerst, as well as some other such identities found recently by us.

Some fascinating series and their sums

ArXiv, 2010

In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (this http URL) nor could we derive them using software for analytical computation like Maple. The identities are beautiful and involve finite series.

The evaluation of Tornheim double sums, Part 1

Journal of Number Theory, 2006

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function A k (q) := kζ ′ (1 − k, q).

A Family of Multiple Harmonic Sum and Multiple Zeta Star Value Identities

Mathematika, 2014

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehroodet al [Trans. Amer. Math. Soc.(to appear)]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.

The evaluation of Tornheim double sums. Part 2

The Ramanujan Journal, 2010

We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function A k (q) := k (1 − k, q).

Some basic bilateral sums and integrals

Pacific Journal of Mathematics - PAC J MATH, 1995

By splitting the real line into intervals of unit length a doubly infinite integral of the form ∞ −∞ F (q x ) dx, 0 < q < 1, can clearly be expressed as 1 0 ∞ n=−∞ F (q x+n ) dx, provided F satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his 1 ψ 1 sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised 2 ψ 2 sum as well as the very-well-poised 6 ψ 6 sum are also found in a straightforward manner. An extension to a very-well-poised and balanced 8 ψ 8 series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.