Difference between revisions of "Hypergeometric 2F3"
From specialfunctionswiki
(Created page with "The hypergeometric series ${}_2F_3$ is defined by the series $${}_2F_3(a_1,a_2;b_1,b_2,b_3;z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k}{(b_1)_k(b_2)_k(b_3)_k}...") |
|||
| Line 5: | Line 5: | ||
=Properties= | =Properties= | ||
[[1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)]]<br /> | [[1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)]]<br /> | ||
| + | [[1F1(a;2a;z)1F1(b;2b;-z)=2F3(a/2+b/2,a/2+b/2+1/2;a+1/2,b+1/2,a+b;z^2/4)]]<br /> | ||
=References= | =References= | ||
{{:Hypergeometric functions footer}} | {{:Hypergeometric functions footer}} | ||
Latest revision as of 20:31, 17 June 2017
The hypergeometric series ${}_2F_3$ is defined by the series $${}_2F_3(a_1,a_2;b_1,b_2,b_3;z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k}{(b_1)_k(b_2)_k(b_3)_k} \dfrac{z^k}{k!},$$ where $(a_1)_k$ denotes the Pochhammer symbol.
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)
