Three Adventures: Symbolically Discovered Identities for Zeta(4n+3) and like Matters
نویسنده
چکیده
I will describe three sets of results in which computer search and computer algebra played a large role in the discovery/and or proof of results each ultimately relating to hypergeometric functions. 1. Ap ery: A generating function for (4n+3). 2. Ramanujan: Integrals, means and transformations of the hypergeometric function 2F1(1=3;2=3;1; 1 x): 3. Euler: Multivalued {function values and a product of hypergeometric 2F1's. y In each case, my emphasis is on the role of disciplined experimentation and computation. 2
منابع مشابه
An exotic shuffle relation for multiple zeta values
In this short note we will provide a new proof of the following exotic shuffle relation of multiple zeta values: ζ({2}x{3, 1}) = ( 2n+m m ) π (2n+ 1) · (4n+ 2m+ 1)! . This was proved by Zagier when n = 0, by Broadhurst when m = 0, and by Borwein, Bradley, and Broadhurst when m = 1. In general this was proved by Bowman and Bradley. Our new idea is to use the method of Borwein et al. to reduce th...
متن کاملNew Identities Involving Sums of the Tails Related to Real Quadratic Fields
In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of Q( √ 2) and Q( √ 3). Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields a...
متن کاملEmpirically Determined Apéry-Like Formulae for ζ(4n+3)
Research supported by NSERC, the Natural Sciences and Engineering Research Council of Canada. Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for (4n+3) that generalizes Apéry’s series for (3), and appears to give the best possible series relations of this type, at least for n< 12. The formula reduces to a finit...
متن کاملOn Plouffe’s Ramanujan Identities
Recently, Simon Plouffe has discovered a number of identities for the Riemann zeta function at odd integer values. These identities are obtained numerically and are inspired by a prototypical series for Apéry’s constant given by Ramanujan: ζ (3) = 7π3 180 −2 ∞ ∑ n=1 1 n3 (e2πn−1) Such sums follow from a general relation given by Ramanujan, which is rediscovered and proved here using complex ana...
متن کاملExperimental Determination of Apéry-like Identities for ς(2n + 2)
We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others.
متن کامل