hypergeomPFQ: Hypergeometric function of a matrix argument

In HypergeoMat: Hypergeometric Function of a Matrix Argument

Hypergeometric function of a matrix argument

Description

Evaluates a truncated hypergeometric function of a matrix argument.

Usage

hypergeomPFQ(m, a, b, x, alpha = 2)

Arguments

m	truncation weight of the summation, a positive integer
a	the "upper" parameters, a numeric or complex vector, possibly empty (or NULL)
b	the "lower" parameters, a numeric or complex vector, possibly empty (or NULL)
x	either a real or complex square matrix, or a numeric or complex vector, the eigenvalues of the matrix
alpha	the alpha parameter, a positive number

Details

This is an implementation of Koev & Edelman's algorithm (see the reference). This algorithm is split into two parts: the case of a scalar matrix (multiple of an identity matrix) and the general case. The case of a scalar matrix is much faster (try e.g. x = c(1,1,1) vs x = c(1,1,0.999)).

Value

A real or a complex number.

Note

The hypergeometric function of a matrix argument is usually defined for a symmetric real matrix or a Hermitian complex matrix.

References

Plamen Koev and Alan Edelman. The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument. Mathematics of Computation, 75, 833-846, 2006.

Examples

# a scalar x example, the Gauss hypergeometric function
hypergeomPFQ(m = 10, a = c(1,2), b = c(3), x = 0.2)
gsl::hyperg_2F1(1, 2, 3, 0.2)
# 0F0 is the exponential of the trace
X <- toeplitz(c(3,2,1))/10
hypergeomPFQ(m = 10, a = NULL, b = NULL, x = X)
exp(sum(diag(X)))
# 1F0 is det(I-X)^(-a)
X <- toeplitz(c(3,2,1))/100
hypergeomPFQ(m = 10, a = 3, b = NULL, x = X)
det(diag(3)-X)^(-3)
# Herz's relation for 1F1
hypergeomPFQ(m = 10, a = 2, b = 3, x = X)
exp(sum(diag(X))) * hypergeomPFQ(m = 10, a = 3-2, b = 3, x = -X)
# Herz's relation for 2F1
hypergeomPFQ(10, a = c(1,2), b = 3, x = X)
det(diag(3)-X)^(-2) *
  hypergeomPFQ(10, a = c(3-1,2), b = 3, -X %*% solve(diag(3)-X))

HypergeoMat documentation built on Feb. 16, 2023, 7:40 p.m.