According to [Erdely A,1953; Higher Transcendental Functions, Vol I, Ch. VI.] the confluent hypergeometric equation
\frac{d^2}{d x^2} y + \left(c - x \right) \frac{d}{d x} y - a y = 0
has got eight solutions, which are the followings:
y_1=M[a,c,x]
y_2=x^{1-c}M[a-c+1,2-c,x]...