Confluent Hypergeometric Differential Equation

$$xy''+(b-x)y'-ay = 0,$$ (1)

where $y'\equiv dy/dx$ and with boundary conditions

$${}_1F_1(a;b;0)= 1$$ (2)

$$\left[{{\partial\over\partial x} {}_1F_1(a;b;x)}\right]_{x=0} = {a\over b}.$$ (3)

The equation has a Regular Singular Point at 0 and an irregular singularity at $\infty$. The solutions are called Confluent Hypergeometric Function of the First or Second Kinds. Solutions of the first kind are denoted ${}_1F_1(a;b;x)$ or $M(a,b,x)$.

See also Hypergeometric Differential Equation, Whittaker Differential Equation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 504, 1972.

Arfken, G. ``Confluent Hypergeometric Functions.'' §13.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 753-758, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 551-555, 1953.