In mathematics, the confluent hypergeometric function is formed from hypergeometric series. It occurs in two forms, as Kummer's function (for Ernst Kummer) and as Whittaker's function (for E. T. Whittaker). Note also that there is a different Kummer's function bearing the same name but unrelated to this. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Edmund Taylor Whittaker (24 October 1873 - 24 March 1956) was an English mathematician, who contributed widely to applied mathematics, mathematical physics and the theory of special functions. ... In mathematics, there are several functions known as Kummers function. ...

Kummer's equation is

It has two independent solutions M(a,b,z) and U(a,b,z).

Kummer' function is given by

where (a)_n = a(a + 1)(a + 2)...(a + n − 1) is the rising factorial. In mathematics, the Pochhammer symbol is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial The empty product (x)0 is defined to be 1 in both cases. ...

and

Note that where the latter is a hypergeometric series. In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

The term confluent refers to the singular points of the differential equation, on the Riemann sphere. Where the usual hypergeometric equation has three singular points (in general position), confluence implies cases of degeneration by singularities being brought together by a limiting process. In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 See chapter 13 (http://www.math.sfu.ca/~cbm/aands/page_504.htm)