confluent hypergeometric function

A degenerate form of the hypergeometric function that arises as a solution to the ordinary differential equation:

   2
  d y           dy
x --- + (b - x) -- - ay = 0
    2           dy
  dx

The confluent hypergeometric function is given in terms of the generalized hypergeometric function as ₁F₁(a; b; x), usually denoted M(a; b; x). If a and b are integers, then the series yields a polynomial. If b is an integer ≤ 0, then, M(a; b; x) is undefined.

It is related to the regular Gaussian hypergeometric function in that M(a;c;x) = lim_b->∞2F₁(a, b; c; x).

Bessel functions, the error function, the incomplete Gamma functions, Hermite polynomials, and Laguerre polynomials among many other special functions arise as special cases of this function, because the differential equations they satisfy may be transformed into the confluent hypergeometric equation.

As Ernst Eduard Kummer was largely responsible for the systematic study of these functions, the confluent hypergeometric functions are also known as Kummer's functions.