Difference between revisions of "Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\lim_{c \rightarrow -m} \dfrac{1}{\Gamma(c)} {}_2F_1(a,b;c;z)= \dfrac{(a)_{m+1} (b)_{m+1}}{(m+1)!} z^{m+1} {}_2F_1 \lef...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Hypergeometric 2F1|next=2F1(1,1;2;z)=-log(1-z)/z}}: 15.1.2 | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Hypergeometric 2F1|next=2F1(1,1;2;z)=-log(1-z)/z}}: $15.1.2$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:15, 12 July 2016
Theorem
The following formula holds: $$\displaystyle\lim_{c \rightarrow -m} \dfrac{1}{\Gamma(c)} {}_2F_1(a,b;c;z)= \dfrac{(a)_{m+1} (b)_{m+1}}{(m+1)!} z^{m+1} {}_2F_1 \left( a+m+1, b+m+1; m+2; z \right),$$ where $\Gamma$ denotes the gamma function, $(a)_{m+1}$ denotes the Pochhammer symbol, and ${}_2F_1$ denotes the hypergeometric 2F1.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $15.1.2$
