Telescoping, rational-valued series, and zeta functions

Authors:
J. Marshall Ash and Stefan Catoiu

Journal:
Trans. Amer. Math. Soc. **357** (2005), 3339-3358

MSC (2000):
Primary 11J72, 11M41, 11A25, 40A25

Published electronically:
March 10, 2005

MathSciNet review:
2135751

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give an effective procedure for determining whether or not a series telescopes when is a rational function with complex coefficients. We give new examples of series , where is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set is linearly independent, where is the Riemann zeta function.

Some series of the form , where is a quotient of symmetric polynomials, are shown to be telescoping, as is . Quantum versions of these examples are also given.

**[A]**Roger Apéry,*Interpolation de fractions continues et irrationalité de certaines constantes*, Mathematics, CTHS: Bull. Sec. Sci., III, Bib. Nat., Paris, 1981, pp. 37–53 (French). MR**638730****[EMOT]**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****[GR]**George Gasper and Mizan Rahman,*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153****[HW]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR**568909****[KC]**Victor Kac and Pokman Cheung,*Quantum calculus*, Universitext, Springer-Verlag, New York, 2002. MR**1865777****[PBM]**A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,*Integrals and series. Vol. 1*, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR**874986**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11J72,
11M41,
11A25,
40A25

Retrieve articles in all journals with MSC (2000): 11J72, 11M41, 11A25, 40A25

Additional Information

**J. Marshall Ash**

Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Email:
mash@math.depaul.edu

**Stefan Catoiu**

Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Email:
scatoiu@math.depaul.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03699-8

Keywords:
Generalized zeta function,
generalized Euler phi function,
linear independence over the rationals

Received by editor(s):
August 12, 2003

Received by editor(s) in revised form:
February 21, 2004

Published electronically:
March 10, 2005

Additional Notes:
The first author’s research was partially supported by NSF grant DMS 9707011 and a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University.

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.