Confluent Hypergeometric Function of the First Kind
The confluent hypergeometric function of the first kind 
is a degenerate form of the hypergeometric function 
which arises as a solution the confluent hypergeometric differential equation. It is also known as Kummer's function of the first kind. There are a number of other notations used for the function (Slater 1960, p. 2), including 
(Kummer 1836), 
(Airey and Webb 1918), 
(Humbert 1920), and 
(Magnus and Oberhettinger 1948). An alternate form of the solution to the confluent hypergeometric differential equation is known as the Whittaker function.
The confluent hypergeometric function of the first kind is implemented in the Wolfram Language as Hypergeometric1F1[a, b, z].
The confluent hypergeometric function has a hypergeometric series given by
 |
(1)
|
where 
and 
are Pochhammer symbols. If 
and 
are integers, 
, and either 
or 
, then the series yields a polynomial with a finite number of terms. If 
is an integer 
, then 
is undefined. The confluent hypergeometric function is given in terms of the Laguerre polynomial by
 |
(2)
|
(Arfken 1985, p. 755), and also has an integral representation
 |
(3)
|
(Abramowitz and Stegun 1972, p. 505).
Bessel functions, erf, the incomplete gamma function, Hermite polynomial, Laguerre polynomial, as well as other are all special cases of this function (Abramowitz and Stegun 1972, p. 509). Kummer showed that
 |
(4)
|
(Koepf 1998, p. 42).
Kummer's second formula gives
where 
, 
, 
, ....
REFERENCES:
Abad, J. and Sesma, J. "Computation of the Regular Confluent Hypergeometric Function." Mathematica J. 5, 74-76, 1995.
Abramowitz, M. and Stegun, I. A. (Eds.). "Confluent Hypergeometric Functions." Ch. 13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503-515, 1972.
Airey, J. R. "The Confluent Hypergeometric Function." Brit. Assoc. Rep. (Oxford), 276-294, 1926.
Airey, J. R. "The Confluent Hypergeometric Function." Brit. Assoc. Rep. (Leeds), 220-244, 1927.
Airey, J. R. and Webb, H. A. "The Practical Importance of the Confluent Hypergeometric Function." Philos. Mag. 36, 129-141, 1918.
Arfken, G. "Confluent Hypergeometric Functions." §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753-758, 1985.
Buchholz, H. The Confluent Hypergeometric Function with Special Emphasis on its Applications. New York: Springer-Verlag, 1969.
Humbert, P. "Sur les fonctions hypercylindriques." C. R. Acad. Sci. Paris 171, 490-492, 1920.
Iyanaga, S. and Kawada, Y. (Eds.). "Hypergeometric Function of Confluent Type." Appendix A, Table 19.I in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1469, 1980.
Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.
Kummer, E. E. "Über die hypergeometrische Reihe 
." J. reine angew. Math. 15, 39-83, 1836.
Magnus, W. and Oberhettinger, F. Formeln und Lehrsätze für die speziellen Funktionen der mathematischen Physik. Berlin, 1948.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 551-554 and 604-605, 1953.
Slater, L. J. Confluent Hypergeometric Functions. Cambridge, England: Cambridge University Press, 1960.
Spanier, J. and Oldham, K. B. "The Kummer Function 
." Ch. 47 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 459-469, 1987.
Tricomi, F. G. Fonctions hypergéométriques confluentes. Paris: Gauthier-Villars, 1960.
