I have obtained a specific function of x, involving the gaussian hypergeometric function 2F1. While evaluating the function for large values of x (x>10), I found a clear logarithmic trend, which I empirically fitted by linear regression with R2=1. The equations and the plot of the linear regression are shown in the attached file. Can anyone provide an analytic demonstration of such equality?
On the basis of the convention 2F1(a,b;c;z) ≡ F(a,b;c;z) (§15.1.1, p. 556, in [1]), your particular function is related to F(a,a;1+a;z), where a = 1+ 8/√π and z = -2x. You will have to consider the expression for F(a,a;c;z) in §15.3.13, p. 560, of Ref. [1]. You may also wish to consult Ref. [2].
[1] M Abramowitz, IA Stegun, Eds, Handbook of Mathematical Functions, 9th printing (Dover, New York, 1972).
[2] http://dlmf.nist.gov/15
http://dlmf.nist.gov/15.12
If we sample without replacement from a set of N items and there are K items that are different and we take a sample of size n without replacement, the probability of finding k (which is < K) different items is described by a hypergeometric distribution such as the 2F1 :)
Andrew Paul, this is not surprising, as almost all, if not all, polynomials can be identified as being the generating functions of various combinatorial problems.
Thanks Behnam Farid. I'm not a mathematician, but I did a lot of seed dispersal modelling. You know much more than me lol :)
You are welcome Andrew Paul, and thank you for your kind compliment. If interested, you might like to consult the following most excellent book:
J Riordan, Introduction to Combinatorial Analysis (Dover, New York, 2002).
The equation given has logarithmic growth if one considers the leading order asymptotics of the hypergeometric function F(a, a, c; z) as z goes to negative infinity. I have attached a file with some workings. I should state that I don't find the same coefficients of the logarithmic fit.
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