Hypergeometric 1F1

The hypergeometric function ${}_1F_1$ (sometimes denoted by $M$, sometimes called the confluent hypergeometric function of the first kind) is defined by the series $${}_1F_1(a;b;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a)_k z^k}{(b)_k k!},$$ where $(a)_k$ denotes the Pochhammer symbol and $k!$ denotes the factorial.

Properties

1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)

References

Hypergeometric functions

Hypergeometric ${}_pF_q$

Hypergeometric 0F0

Hypergeometric 1F0

Hypergeometric 0F1

Hypergeometric 2F1