An extension of Saalschütz’s summation theorem for the series \(_{r+3}F_{r+2}\). (English) Zbl 1286.33006
The authors offer an extension of Saalschütz’s summation theorem for the series \({_{r+3}F_{r+2}}\). The proof is based on a method of A. R. Miller and the third author [Rocky Mt. J. Math. 43, No. 1, 291–327 (2013; Zbl 1275.33009)] on a generalization of an Euler-type transformation. Various examples to the general theorem are pointed out.
Reviewer: József Sándor (Cluj-Napoca)
MSC:
| 33C05 | Classical hypergeometric functions, \({}_2F_1\) |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
Citations:
Zbl 1275.33009Digital Library of Mathematical Functions:
Pfaff–Saalschütz Balanced Sum ‣ §16.4(ii) Examples ‣ §16.4 Argument Unity ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-FunctionPfaff–Saalschütz Balanced Sum ‣ §16.4(ii) Examples ‣ §16.4 Argument Unity ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function
References:
| [1] | Kim YS, Int J Math Math Sci 309503 (2010) |
| [2] | Lavoie JL, Indian J Math 34 pp 23– (1992) |
| [3] | Lavoie JL, Math Comp 62 pp 267– (1994) |
| [4] | DOI: 10.1016/0377-0427(95)00279-0 · Zbl 0853.33005 · doi:10.1016/0377-0427(95)00279-0 |
| [5] | DOI: 10.1088/0305-4470/38/16/005 · Zbl 1068.33009 · doi:10.1088/0305-4470/38/16/005 |
| [6] | DOI: 10.1080/10652469.2010.549487 · Zbl 1241.33006 · doi:10.1080/10652469.2010.549487 |
| [7] | DOI: 10.4134/BKMS.2011.48.1.151 · Zbl 1214.33003 · doi:10.4134/BKMS.2011.48.1.151 |
| [8] | DOI: 10.1216/rmjm/1030539701 · Zbl 1046.33002 · doi:10.1216/rmjm/1030539701 |
| [9] | Slater LJ. Generalized hypergeometric functions. Cambridge: Cambridge University Press; 1966. |
| [10] | DOI: 10.1216/RMJ-2013-43-1-291 · Zbl 1275.33009 · doi:10.1216/RMJ-2013-43-1-291 |
| [11] | DOI: 10.1007/s00033-010-0085-0 · Zbl 1225.33005 · doi:10.1007/s00033-010-0085-0 |
| [12] | DOI: 10.1090/conm/471/09211 · doi:10.1090/conm/471/09211 |
| [13] | Rakha MA, Math Commun 14 pp 207– (2009) |
| [14] | Rakha MA, Internat J Math Modell Comput 1 (3) pp 171– (2011) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
