Hypergeometric Functions

fhyper(a, b, c, x) Returns the value of the Gauss hypergeometric function, ₂F₁(a, b; c; x). Calculated by series expansion.

mhyper(a, b, x) Returns the value of the confluent hypergeometric function, ₁F₁(a; b; x) or M(a, b, x). Calculated by series expansion.

Arguments:

• a, b, and c are real, dimensionless scalars. If a and b are nonzero, c must also be nonzero.
• x is a real, dimensionless scalar. For fhyper, −1 < x < 1.

Notes:

• The hypergeometric functions are solutions to differential equations:

Gauss:

Confluent:

• Many functions are special cases of the hypergeometric functions. For example:

ln(1 + x) = x · fhyper(1, 1, 2, –x)	exp(x) = mhyper(1,1,x)
asin(x) = x · fhyper(.5, .5, 1.5, x2)	exp(x) · sinh(x) =  x · mhyper(1,2,2 · x)

as well as some of the more complicated functions, like Legendre polynomials.

QuickSheet