Abstract
In this chapter we deal with hypergeometric identities.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A hypergeometric term is always the summand, not the sum!
- 2.
The right-hand sides \(a_{n}\) form \(m\)-fold hypergeometric terms. These are generalizations of hypergeometric terms satisfying a recurrence equation of the type \(a_{n+m}=R(n)\,a_{n}\) for some \(m\in {\mathbb {N}}\) with rational \(R(n)\).
- 3.
\({\mathbb {Q}}(x_{1},x_{2},\ldots ,x_{m})\) denotes the field of rational functions in the variables \(x_{1},x_{2},\ldots ,x_{m}\) over \({\mathbb {Q}}\).
- 4.
This fact is also expressed by the \(k!\)-term in the denominator of the right-hand sum (2.8).
- 5.
Note that simplify can easily handle the next question. However, simplify does not always simplify towards a rational function, even if the result is rational. Moreover, simplify is a combination of so many algorithms so that it is not even possible to describe its full mechanism. It is better to use simplification commands that have a clear description.
- 6.
The current updated version is hsum17.mpl. In future Maple sessions we will always assume that the hsum package is loaded by the read command.
- 7.
If the input terms have integer-linear arguments in \(k\), then the ratio \(a_{k+1}/a_{k}\) is clearly a rational function; if the input terms are rational-linear in \(k\), then this is not automatically the case, and the algorithm detects this.
- 8.
In such a case, \(\alpha _{k}\) and \(\beta _{k}\) are called similar hypergeometric terms, see p. 94.
- 9.
Rational factorization will be considered in more detail on p. 83.
- 10.
If parameters are involved, this might be undecidable, compare e.g. the Dixon case!
- 11.
The next two sums presented are not really “nice” and need an application of the Gamma duplication formula for conversion towards simple forms. Even Maple’s assume facility does not simplify appropriately.
- 12.
Note that basic hypergeometric functions and their properties were already considered in [Bailey35] where the definition of the \(q\)-hypergeometric function was given without this additional factor, though.
- 13.
This is a confluence process, again; hence the name of the function.
References
Andrews, G., Askey, R.A., Roy, R.: Special Functions: Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge-New York (1999)
Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Astérisque 61, 11–13 (1979)
Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935). Reprinted 1964 by Stechert-Hafner Service Agency, New York-London.
De Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)
Gasper, G., Rahman, M.: Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications) vol. 35, 2nd edn (2004). Cambridge University Press, London and New York (1990)
Gasper, G.: Elementary derivations of summation and transformation formulas for \(q\)-series. In: Ismail, M.E.H., et al. (eds.) Fields Institute Communications, vol. 14, pp. 55–70. American Mathematics Society, Providence (1997)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: a Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading, Massachussets (1994)
Rainville, E.D.: Special Functions. The MacMillan Co., New York (1960)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer-Verlag London
About this chapter
Cite this chapter
Koepf, W. (2014). Hypergeometric Identities. In: Hypergeometric Summation. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6464-7_2
Download citation
DOI: https://doi.org/10.1007/978-1-4471-6464-7_2
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-4471-6463-0
Online ISBN: 978-1-4471-6464-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)