scholarly journals Hypergeometric Series Acceleration Via the WZ method

10.37236/1318 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Tewodros Amdeberhan ◽  
Doron Zeilberger
Keyword(s):  
Zeta Function ◽  
Hypergeometric Series ◽  
Infinite Family ◽  
Series Acceleration ◽  
Wz Method

Based on the WZ method, some series acceleration formulas are given. These formulas allow us to write down an infinite family of parameterized identities from any given identity of WZ type. Further, this family, in the case of the Zeta function, gives rise to many accelerated expressions for $\zeta(3)$.

scholarly journals An Infinite Family of Graphs with the Same Ihara Zeta Function

10.37236/354 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christopher Storm
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Infinite Family ◽  
Ihara Zeta Function ◽  
One To One

In 2009, Cooper presented an infinite family of pairs of graphs which were conjectured to have the same Ihara zeta function. We give a proof of this result by using generating functions to establish a one-to-one correspondence between cycles of the same length without backtracking or tails in the graphs Cooper proposed. Our method is flexible enough that we are able to generalize Cooper's graphs, and we demonstrate additional families of pairs of graphs which share the same zeta function.

scholarly journals In Praise of an Elementary Identity of Euler

10.37236/2009 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Gaurav Bhatnagar
Keyword(s):  
Hypergeometric Series ◽  
Fibonacci Numbers ◽  
Basic Hypergeometric Series ◽  
Binomial Theorem ◽  
Pell Numbers ◽  
Generalized Hypergeometric Series ◽  
Pentagonal Number Theorem ◽  
Wz Method ◽  
Elementary Identity

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu–Vandermonde sum, the Pfaff–Saalschütz sum, and their $q$-analogues. We also give a proof of Jackson's $q$-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised $_8\phi_7$ sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for $q$-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.

scholarly journals Principal polarizations of abelian surfaces over finite fields

1980 ◽  
Vol 77 ◽  
pp. 1-4
Author(s):  
Stuart Turner
Keyword(s):  
Zeta Function ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Infinite Family ◽  
Abelian Surfaces ◽  
Jacobian Varieties

In § 1 of this note we construct abelian varieties of dimension two defined over Fpn, n odd, which admit infinitely many distinct principal polarizations. These polarizations determine an infinite family of geometrically non-isomorphic complete singular curves defined and irreducible over Fpn which have isomorphic Jacobian varieties. In § 2 we calculate the zeta function of these curves.

scholarly journals A Partial Theta Function Borwein Conjecture

2019 ◽  
Vol 23 (3-4) ◽  
pp. 561-572
Author(s):  
Gaurav Bhatnagar ◽  
Michael J. Schlosser
Keyword(s):  
Theta Function ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Infinite Family ◽  
Macdonald Polynomial ◽  
Partial Theta Function ◽  
Multiple Basic Hypergeometric Series

Abstract We present an infinite family of Borwein type $$+ - - $$+-- conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.

scholarly journals Generating Function Identities for $\zeta(2n+2), \zeta(2n+3)$ via the WZ Method

10.37236/759 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Kh. Hessami Pilehrood ◽  
T. Hessami Pilehrood
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Riemann Zeta Function ◽  
Generating Functions ◽  
Convergent Series ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Wz Method

Using WZ-pairs we present simpler proofs of Koecher, Leshchiner and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\{\zeta(2n+2)\}_{n\ge 0}$ and $\{\zeta(2n+3)\}_{n\ge 0}.$ By the same method, we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.

scholarly journals Infinite families of accelerated series for some classical constants by the Markov-WZ Method

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Mohamud Mohammed
Keyword(s):  
Hypergeometric Series ◽  
Series Representations ◽  
International Audience ◽  
Wz Method

International audience In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).

Certain Subclasses of Bi-Univalent Functions Involving Double Zeta Function

2015 ◽  
Vol 4 (4) ◽  
pp. 28-33
Author(s):  
Dr. T. Ram Reddy ◽  
◽  
R. Bharavi Sharma ◽  
K. Rajya Lakshmi ◽  
◽  
...  
Keyword(s):  
Zeta Function ◽  
Univalent Functions ◽  
Double Zeta
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