Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m

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.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $15.1.2$