Difference between revisions of "Hypergeometric 1F1"
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(Created page with "The hypergeometric function ${}_1F_1$ (sometimes denoted by $M$, sometimes called the confluent hypergeometric function of the first kind) is defined by the series $${}_1F_1(a...") |
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=Properties= | =Properties= | ||
| + | [[1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)]]<br /> | ||
| + | [[1F1(a;2a;z)1F1(b;2b;-z)=2F3(a/2+b/2,a/2+b/2+1/2;a+1/2,b+1/2,a+b;z^2/4)]]<br /> | ||
=References= | =References= | ||
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| + | {{:Hypergeometric functions footer}} | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 20:31, 17 June 2017
The hypergeometric function ${}_1F_1$ (sometimes denoted by $M$, sometimes called the confluent hypergeometric function of the first kind) is defined by the series $${}_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)
1F1(a;2a;z)1F1(b;2b;-z)=2F3(a/2+b/2,a/2+b/2+1/2;a+1/2,b+1/2,a+b;z^2/4)
