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# Rummer’s function

• Wilhelm Magnus
• Fritz Oberhettinger
• Raj Pal Soni
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 52)

## Abstract

Rummer’s function1 1 F 1(a; c; z) is defined by the function, and all its analytic continuations, represented by the infinite series $$\sum\limits_{n = 0}^\infty {\frac{{{{(a)}_n}}}{{{{(c)}_n}}}\frac{{{z^n}}}{{n!}}}$$. That is,
$$_1{F_1}(a;c;z) = \sum\limits_{n = 0}^\infty {\text{ }} \frac{{{{(a)}_n}}}{{{{(c)}_n}}}\frac{{{{(z)}^n}}}{{(n)!}}, \ne 0, - 1, - 2 \ldots {\text{ }} = \frac{{\Gamma (c)}}{{\Gamma (a)}}\sum\limits_{n = 0}^\infty {\text{ }} \frac{{\Gamma (a + n)}}{{\Gamma (c + n)}}\frac{{{{(z)}^n}}}{{(n)!}}.$$

## Keywords

Asymptotic Expansion Hypergeometric Function Multiplication Theorem Independent Solution Infinite Series
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Literature

1. Buchholz, H.: Die konfluente hypergeometrische Funktion. Berlin/Göttingen/Heidelberg: Springer 1953.
2. Erdélyi, A.: Higher transcendental functions, Vol. 1. New York: McGraw-Hill Book Co. 1953.Google Scholar
3. Erdélyi, A., and C. A. Swanson: Asymptotic forms of Confluent hypergeometric functions, memior 25. Amer. Math. Soc., Providence, R. I., 1957.Google Scholar
4. Slater, L. J.: Confluent hypergeometric functions. Cambridge: Cambridge Univ. Press 1960.
5. Tricomi, F. G.: Funzioni ipergeometriche Confluenti. Rome: Edizioni Cremonese 1954.
6. Whittaker, E. T., and G. N. Watson: A course of modern analysis. Cambridge: Cambridge Univ. Press 1952.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1966

## Authors and Affiliations

• Wilhelm Magnus
• 1
• Fritz Oberhettinger
• 2
• Raj Pal Soni
• 3
1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA
2. 2.Oregon State UniversityUSA
3. 3.International Business Machines CorporationUSA