Some families of infinite series (original) (raw)
Communications of the Korean Mathematical Society, 2001
The main object of this paper is first to give two contiguous analogues of a well-known hypergeometric summation formula for 2 F 1 (1/2). We then apply each of these analogues with a view to evaluating the sums of several classes of series in terms of the Psi (or Digamma) and the Zeta functions. Relevant connections of the series identities presented here with those given elsewhere are also pointed out.
Some hypergeometric and other evaluations of ζ(2) and allied series
Applied Mathematics and Computation, 1999
Numerous interesting solutions of the problem of evaluating the Riemann f2 X I k1 k À2 , which was of vital importance to Euler and the Bernoulli brothers (James and John Bernoulli), have appeared in the mathematical literature ever since Euler ®rst solved this problem in 1736. The main object of this note is to present yet another evaluation of f2 and related sums by applying the theory of hypergeometric series. Some other approaches to this problem are also indicated.
Some series of the zeta and related functions
Analysis, 1998
We propose and develop yet another approach to the problem of summation of series involving the Riemann Zeta function s, the Hurwitz's generalized Zeta function s; a, the Polygamma function p z p = 0 ; 1 ; 2 ; , and the polylogarithmic function Li s z. The key ingredients in our approach include certain known integral representations for s and s; a. The method developed in this paper is illustrated by n umerous examples of closed-form evaluations of series of the aforementioned types; the method developed in Section 2, in particular, has been implemented in Mathematica Version 3.0. Many of the resulting summation formulas are believed to be new.
Summation of some infinite series by the methods of Hypergeometric functions and partial fractions
BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS
In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].
A certain family of series associated with the zeta and related functions
Hiroshima Mathematical Journal
The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690-1764) and Leonhard Euler (1707-1783). Many di¤erent techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Zeta function can be evaluated by starting with a single known identity for the generalized (or Hurwitz) Zeta function. Some special cases and their connections with already developed series involving the Zeta and related functions are also considered.
Certain families of series associated with the Hurwitz–Lerch Zeta function
Applied Mathematics and Computation, 2005
The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690-1764) and Leonhard Euler (1707-1783). Many different techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Hurwitz-Lerch Zeta function can be evaluated by starting with a single known identity for the Hurwitz-Lerch Zeta function. Some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the Hurwitz Zeta function.
On Some Sums of Digamma and Polygamma functions
Many formula involving sums of digamma and polygamma functions, few of which appear in standard reference works or the literature, but which commonly arise in applications, are collected and developed. Along the way, a new evaluation for some particular members of the family of hypergeometric functions - 4F3(1) - is presented, and a connection is made with Euler sums. All proofs are sketched.
Certain Classes of Infinite Series
Monatshefte f�r Mathematik, 1999
The authors evaluate some interesting families of in®nite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject.
Hypergeometric series associated with the Hurwitz-Lerch zeta function
The present work is sequel to the papers [3] and [4], and it aims at introducing and investigating a new generalized double zeta function involving the Riemann, Hurwitz, Hurwitz-Lerch and Barnes double zeta functions as particular cases. We study its properties, integral representations, differential relations, series expansion and discuss the link with known results.