genhypergeo: The generalized hypergeometric function
In hypergeo: The Gauss Hypergeometric Function

Description Usage Arguments Details Note Author(s) References See Also Examples

The generalized hypergeometric function, using either the series expansion or the continued fraction expansion.

genhypergeo(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE, series=TRUE)
genhypergeo_series(U, L, z, tol=0, maxiter=2000, check_mod=TRUE,
    polynomial=FALSE, debug=FALSE) 
genhypergeo_contfrac(U, L, z, tol = 0, maxiter = 2000)

`U,L`	Upper and lower arguments respectively (real or complex)
`z`	Primary complex argument (see notes)
`tol`	tolerance with default zero meaning to iterate until additional terms to not change the partial sum
`maxiter`	Maximum number of iterations to perform
`check_mod`	Boolean, with default `TRUE` meaning to check that the modulus of `z` is less than 1
`polynomial`	Boolean, with default `FALSE` meaning to evaluate the series until converged, or return a warning; and `TRUE` meaning to return the sum of `maxiter` terms, whether or not converged. This is useful when either `A` or `B` is a nonpositive integer in which case the hypergeometric function is a polynomial
`debug`	Boolean, with `TRUE` meaning to return debugging information and default `FALSE` meaning to return just the evaluate
`series`	In function `genhypergeo()`, Boolean argument with default `TRUE` meaning to return the result of `genhypergeo_series()` and `FALSE` the result of `genhypergeo_contfrac()`

Function genhypergeo() is a wrapper for functions genhypergeo_series() and genhypergeo_contfrac().

Function genhypergeo_series() is the workhorse for the whole package; every call to hypergeo() uses this function except for the (apparently rare—but see the examples section) cases where continued fractions are used.

The generalized hypergeometric function [here genhypergeo()] appears from time to time in the literature (eg Mathematica) as

[omitted; see PDF]

where U=(u_1,...,u_i) and L=(l_1,...,l_i) are the “upper” and “lower” vectors respectively. The radius of convergence of this formula is 1.

For the Confluent Hypergeometric function, use genhypergeo() with length-1 vectors for arguments U and V.

For the 0F1 function (ie no “upper” arguments), use genhypergeo(NULL,L,x).

See documentation for genhypergeo_contfrac() for details of the continued fraction representation.

The radius of convergence for the series is 1 but under some circumstances, analytic continuation defines a function over the whole complex plane (possibly cut along (1,inf)). Further work would be required to implement this.

Robin K. S. Hankin

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

hypergeo,genhypergeo_contfrac