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First-kind confluent hypergeometric function

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Calculate the first-kind confluent hypergeometric function (Kummer function) M(a,b,z).
\(\normalsize Confluent\ Hypergeometric\ function\ of\ the\ 1st\ kind\\ \large \hspace{30px} {}_1F_1(a;b;z)=M(a,b,z)\\\hspace{105px}=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)}\frac{z^2}{2!}+\cdots=\displaystyle \sum_{\small n=0}^{\small \infty}\frac{(a)_n}{(b)_n}\frac{z^n}{n!}\\ \)
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M(a,b,z) 
\(\normalsize Confluent\ hypergeometric\ differential\ equation\\ (1)\ zy''+(b-z)y'-ay=0\\ \hspace{25px} y=c_1M(a,b,z)+c_2U(a,b,z)\\ (2)\ M(a,b,z)={}_1F_1(a;b;z)\\ \hspace{85px}=1+{\large\frac{a}{b}}z+{\large\frac{a(a+1)}{b(b+1)}\frac{z^2}{2!}}+\cdots={\large \displaystyle \sum_{\small n=0}^{\small \infty}}{\large \frac{(a)_n}{(b)_n}\frac{z^n}{n!}}\\ \)

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