Kummer's Formulas

Kummer's first formula is

$${}_2F_1({\textstyle{1\over 2}}+m-k,-n;2m+1;1) = {\Gamma(2m+1... ...}}+k+n)\over\Gamma(m+{\textstyle{1\over 2}}+k)\Gamma(2m+1+n)},$$ (1)

where ${}_2F_1(a,b;c;z)$ is the Hypergeometric Function with $m\not= -1/2$, $-1$, $-3/2$, ..., and $\Gamma(z)$ is the Gamma Function. The identity can be written in the more symmetrical form as

$${}_2F_1(a,b;c;-1)={\Gamma({\textstyle{1\over 2}}b+1)\Gamma(b-a+1)\over\Gamma(b+1)\Gamma({\textstyle{1\over 2}}b-a+1)},$$ (2)

where $a-b+c=1$ and $b$ is a positive integer. If $b$ is a negative integer, the identity takes the form

$${}_2F_1(a,b;c;-1)=2\cos({\textstyle{1\over 2}}\pi b){\Gamma(... ... b\vert)\Gamma(b-a+1)\over\Gamma({\textstyle{1\over 2}}b-a+1)}$$ (3)

(Petkovsek et al. 1996).

Kummer's second formula is

$${}_1F_1({\textstyle{1\over 2}}+m;2m+1;z) = M_{0,m}(z) = z^{m... ...^\infty {z^{2p}\over 2^{4p} p!(m+1)(m+2)\cdots (m+p)}}\right],$$ (4)

where ${}_1F_1(a;b;z)$ is the Confluent Hypergeometric Function and $m\not= -1/2$, $-1$, $-3/2$, ....

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 42-43 and 126, 1996.