Compute the Confluent Hypergeometric function of two variables, also know as a Horn hypergeometric function or Humbert's hypergeometric used in Gordy (1998) with integral representation:
Details
phi_1(a,b,c,x,y) = [(Gamma(c)/Gamma(a) Gamma(a-c))] Int_0^1 t^(a-1) (1 - t)^(c-a-1) (1 - yt)^(-b) exp(x t) dt https://en.wikipedia.org/wiki/Humbert_series Note that Gordy's arguments for x and y are reversed in the reference above.
The original `phi1` function in `BAS` was based on `C` code provided by Gordy. This function returns NA's when x is greater than `log(.Machine$double.xmax)/2`. A more stable method for calculating the `phi1` function using R's `integrate` was suggested by Daniel Heemann and is now an option whenever $x$ is too large. For calculating Bayes factors that use the `phi1` function we recommend using the `log=TRUE` option to compute log Bayes factors.
See also
Other special functions:
hypergeometric1F1(),
hypergeometric2F1(),
trCCH()
Examples
# special cases
# phi1(a, b, c, x=0, y) is the same as 2F1(b, a; c, y)
phi1(1, 2, 1.5, 0, 1 / 100, log=FALSE)
#> [1] 1.013495
hypergeometric2F1(2, 1, 1.5, 1 / 100, log = FALSE)
#> [1] 1.013495
# phi1(a,0,c,x,y) is the same as 1F1(a,c,x)
phi1(1, 0, 1.5, 3, 1 / 100)
#> [1] 10.13001
hypergeometric1F1(1, 1.5, 3, log = FALSE)
#> [1] 10.13001
# use direct integration
phi1(1, 2, 1.5, 1000, 0, log=TRUE)
#> [1] 996.4253