My problem refers to the asymptotic formula for Gauss hypergeometric function $F(n, b; 2n; z)$, where $n$ is a fixed positive integer, $z$ is a fixed positive real number less than unity, and the large complex parameter is $b$, $|b|\gg1$. Specifically, I am interested in the case of fixed $Re(b)$ and $|Im(b)|\gg1$. This case is supposed to be covered by Eqs. 15.7.2, 3 in the Handbook of Mathematical Functions edited by M. Abramowitz and I. A. Stegun (available on the Web), same as the asymptotic formula (14) in A. Erdelyi et al., Higher Transcendental Functions, Vol. 1, Section 2.3.2. The asymptotic expression in question is the same as for a confluent hypergeometric function of large argument, $F(n; 2n; b z)$ (which for our choice of parameters can be expressed via a Bessel function), the leading terms of which in our case are$${(2n)!\over {2n!}}(b z)^{-n}[(-1)^n+e^{b z}]. $$ Indeed, the absolute value of the function $F(n, b; 2n; z)$ decreases with $|b|$ as ${(2n)!\over {2n!}}(b z)^{-n}$. But this average reduction is multiplied by a factor that oscillates with increased $|Im(b)|$ while $Re(b)$ is fixed. The above asymptotic formula predicts this factor to be $(-1)^n+e^{b z}$. By direct calculation I have found that both the frequency and the amplitude of the actual oscillations of the real and imaginary parts of ${2n!\over {(2n)!}}(b z)^{n} F(n, b; 2n; z)$ differ from those given by this expression. The difference is of order unity, not $O(1/|b z|)$, as Erdelyi’s derivation states. For example, taking $n=2$, $z=0.2$, and reducing the value of $1/|b z|$ to $10^{-3}$ one does not see improved agreement with the asymptotic formula. In this range the asymptotic formula for the confluent hypergeometric function works fine, no matter where on the complex plane its large argument $b z$ is.

Is there an accurate asymptotic formula for $F(n, b; 2n; z)$, one that captures the amplitude and frequency of its oscillations?


1 Answer 1


Using http://functions.wolfram.com/ and http://dlmf.nist.gov/15.12.ii and a standard asymptotic formula for ratios of gamma functions (see e.g. http://dlmf.nist.gov/5.11.E12), I found for large $|Im b|$ $$ _1F_2(n,b; 2 n;z) = i^n (z \ Im{(b)})^{-n}\frac{\Gamma(2 n)}{\Gamma(n) }\left(1+(-1)^n (1-z)^{n-b}\right)(1+O(b^{-1})). $$ which did a fairly good job in my numerical checks.


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