It is well known that for the following identity holds $$ \lim_{b\to\infty}{}_2F_1(a,b;c;z/b)={}_1F_1(a,c;z) $$ where ${}_pF_q$ is an hypergeometric function. Is there a similar identity for $$ \lim_{b\to-\infty}{}_2F_1(a,b;c;z/b) $$ ?
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$$\phantom{}_2 F_1\left(a,b;c;\frac{z}{b}\right) = \sum_{n\geq 0}\frac{(a)_n}{n! (c)_n}\cdot\frac{(b)_n}{b^n} z^n$$ and for a fixed $n$, $\frac{(b)_n}{b^n}\to 1$ as $b\to +\infty$ due to Gautschi's inequality, so the first claim is just a consequence of the dominated convergence theorem. In the general case, $$ \frac{(b)_n}{b^n} = \frac{\Gamma(b+n)}{\Gamma(b)\,b^n}=\frac{b(b+1)\cdot\ldots\cdot(b+n-1)}{b^n}$$ still has limit $1$ as $b\to -\infty$, hence we may say that
$$ \lim_{b\to\pm\infty}\phantom{}_2 F_1\left(a,b;c;\frac{z}{b}\right)=\phantom{}_{1} F_1(a;c;z).$$
Jack D'Aurizio
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does something similar hold for ${}_3F_1(a,b,c;d;z/b)$? – PhoenixPerson May 31 '16 at 14:08
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@AnarchistBirdsWorshipFungus: by the same argument you get $\phantom{}_2 F_1(a,c;d;z)$ in that case. – Jack D'Aurizio May 31 '16 at 14:10
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what you say that $\lim_{b\to-\infty}{}3F_1(a,b,c;d;z/b)={}_2F_1(a,b,c;d;z/b)$ is wrong. I tried the numerical values in mathematica for say $a=1$, $c=1$, $d=2$ and $z=b/(b-1)$ where i take the value $b=-1000$. I obtain $4.10\neq{}6.91$. Actually, I suspect the real limit is $\lim{b\to-\infty}{}_3F_1(a,b,c;d;z/b)=\frac{1}{2}({}_2F_1(a,b,c;d;z/b)+\log2+\gamma_E)$ where $\gamma_E$ is the Euler Mascheroni constant. The numerics supports the claim, but I cannot prove it. – PhoenixPerson May 31 '16 at 15:40
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@AnarchistBirdsWorshipFungus: but you cannot fix the value of $z$ when taking the limit, don't you? – Jack D'Aurizio May 31 '16 at 15:41
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and what if do? do you know how to get the limit for my choice of $z$? – PhoenixPerson May 31 '16 at 15:42
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That you compute another thing, namely $$\lim_{b\to \pm\infty}\phantom{}_2 F_1(1,b;1;1).$$ – Jack D'Aurizio May 31 '16 at 15:44
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Do you know how to tackle $\lim_{b\to-\infty}{}_3F_1(b,1,1;2;1/(b-1)$? – PhoenixPerson May 31 '16 at 15:46
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@AnarchistBirdsWorshipFungus: that deserves a different question. – Jack D'Aurizio May 31 '16 at 16:02
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I just asked it in case you are interested – PhoenixPerson May 31 '16 at 16:14