I have to deal with an expression with some $_2F_1$ and take some limits for some values of the parameters. Let's call this parameter $m$. The issue is that I get a different result whether I take the limit of $m \to \infty$ or I set $m$ to some very large number. As an example
N[Hypergeometric2F1[1/(1 + m), 11/10 + 1/(1 + m), 2/(1 + m), 1/3] /. m -> 10^40, 50]
N[Limit[Hypergeometric2F1[1/(1 + m), 11/10 + 1/(1 + m), 2/(1 + m), 1/3], m -> Infinity], 50]
gives me two different results.
1.2810348079943079401346432576733365035908779571267
1.0000000000000000000000000000000000000000000000000
Mathematica warns me that it cannot decide whether some quantities are numerically zero or not. If instead I use the regularized 2F1, this problem is not there (at least not in this case)
N[Hypergeometric2F1Regularized[1/(1 + m), 11/10 + 1/(1 + m), 2/(1 + m), 1/3] /. m -> 10^40, 50]
N[Limit[Hypergeometric2F1Regularized[1/(1 + m), 11/10 + 1/(1 + m), 2/(1 + m), 1/3], m -> Infinity], 50]
give me
2.5620696159886158802692865153466730071817954806353*10^-40
0
I assume that this happens because in the $m \to \infty$ limit the third argument of the $_2F_1$ is zero, so the function is better behaved in the case where I regularize it. But is there a way to tell Mathematica to always use the hypergeometric regularized $_2F_1$? My expression has several $_2F_1$ and in general if I do something of the kind
Hypergeometric2F1[1/(1 + m), -(11/10) + 1/(1 + m), -(1/10), 2/3] /. Hypergeometric2F1[a_, b_, c_, z_] -> Hypergeometric2F1Regularized[a, b, c, z] Gamma[c]
it simplifies the regularized $_2F_1$ and the gamma function and gives me back the thing I started from
Hypergeometric2F1[1/(1 + m), -(11/10) + 1/(1 + m), -(1/10), 2/3]
Thanks