Compound Confluent hypergeometric function of two variables
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:
phi1(a, b, c, x, y)
a |
a > 0 |
b |
arbitrary |
c |
c > 0 |
x |
x > 0 |
y |
0 <= y < 1 |
phi_1(a,b,c,x,y) = Beta(a,b) 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.
Code for phi1 provided by Gordy.
Merlise Clyde (clyde@stat.duke.edu)
Gordy 1998
Other special functions:
hypergeometric1F1(),
hypergeometric2F1()
# 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) hypergeometric2F1(2, 1, 1.5, 1 / 100, log = FALSE) # phi1(a,0,c,x,y) is the same as 1F1(a,c,x) phi1(1, 0, 1.5, 3, 1 / 100) hypergeometric1F1(1, 1.5, 3, log = FALSE)
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