AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Special Values of the Hypergeometric Series
About this Title
Akihito Ebisu, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo, 060-0810, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 248, Number 1177
ISBNs: 978-1-4704-2533-3 (print); 978-1-4704-4056-5 (online)
DOI: https://doi.org/10.1090/memo/1177
Published electronically: March 15, 2017
Keywords: Hypergeometric series,
three term relation,
special value,
solving polynomial systems
MSC: Primary 33C05; Secondary 13P10, 13P15, 33C20, 33C65
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Derivation of special values
- 4. Tables of special values
- A. Some hypergeometric identities for generalized hypergeometric series and Appell-Lauricella hypergeometric series
- Acknowledgments
Abstract
In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series $F(a,b;c;x)$; we show that values of $F(a,b;c;x)$ at some points $x$ can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of $F(a,b;c;x)$ that can be obtained with this method. We find that this set includes almost all previously known values and many previously unknown values.- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
- George E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441–484. MR 352557, DOI 10.1137/1016081
- Moa Apagodu and Doron Zeilberger, Searching for strange hypergeometric identities by sheer brute force, Integers 8 (2008), A36, 6. MR 2438527
- W.N.Bailey, Generalized hypergeometric series, Cambridge Mathematical Tract No. 32, Cambridge University Press, (1935).
- Akihito Ebisu, Three term relations for the hypergeometric series, Funkcial. Ekvac. 55 (2012), no. 2, 255–283. MR 3012577, DOI 10.1619/fesi.55.255
- Akihito Ebisu, On a strange evaluation of the hypergeometric series by Gosper, Ramanujan J. 32 (2013), no. 1, 101–108. MR 3105988, DOI 10.1007/s11139-013-9483-1
- S.B.Ekhad, Forty “strange” computer-discovered[and computer-proved(of course)] hypergeometric series evaluations, The Personal Journal of Ekhad and Zeilberger, http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/strange.html, (2004).
- A.Erdélyi (editor), Higher transcendental functions Vol.1, McGraw-Hill, (1953).
- L.Euler, Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, typographeo petri galeatii, 1755: available at http://www.math.dartmouth.edu/~euler/docs/originals/E212sec2ch4.pdf.
- C.F.Gauss, Disquisitiones generales circa seriem infinitam, Comm. Soc. Reg Gött. II, Werke, 3, 123–162, (1812).
- Ira M. Gessel, Finding identities with the WZ method, J. Symbolic Comput. 20 (1995), no. 5-6, 537–566. Symbolic computation in combinatorics $\Delta _1$ (Ithaca, NY, 1993). MR 1395413, DOI 10.1006/jsco.1995.1064
- R.W.Gosper, A letter to D. Stanton, XEROX Palo Alto Research Center, 21 December 1977.
- Henry W. Gould, Combinatorial identities, Henry W. Gould, Morgantown, W.Va., 1972. A standardized set of tables listing 500 binomial coefficient summations. MR 0354401
- Édouard Goursat, Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3–142 (French). MR 1508709
- Ira Gessel and Dennis Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), no. 2, 295–308. MR 647127, DOI 10.1137/0513021
- Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig, 1991. A modern theory of special functions. MR 1118604
- Per W. Karlsson, On two hypergeometric summation formulas conjectured by Gosper, Simon Stevin 60 (1986), no. 4, 329–337. MR 885204
- Wolfram Koepf, Hypergeometric summation, 2nd ed., Universitext, Springer, London, 2014. An algorithmic approach to summation and special function identities. MR 3289086
- Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
- Nobuki Takayama, Gröbner basis for rings of differential operators and applications, Gröbner bases, Springer, Tokyo, 2013, pp. 279–344. MR 3204385, DOI 10.1007/978-4-431-54574-3_{6}
- Nobuki Takayama, Gröbner basis and the problem of contiguous relations, Japan J. Appl. Math. 6 (1989), no. 1, 147–160. MR 981518, DOI 10.1007/BF03167920
- Raimundas Vidūnas, Algebraic transformations of Gauss hypergeometric functions, Funkcial. Ekvac. 52 (2009), no. 2, 139–180. MR 2547100, DOI 10.1619/fesi.52.139