pFqhypergeometriset

The generalized hypergeometric function, commonly denoted pFq, is a broad family of special functions that extends many classical hypergeometric functions. It is defined by a power series with rising factorial coefficients:

pFq(a1, ..., ap; b1, ..., bq; z) = sum_{n=0}^∞ [(a1)_n ... (ap)_n] / [(b1)_n ... (bq)_n] · z^n / n!

Here (a)_n is the Pochhammer symbol (rising factorial), with (a)_0 = 1. The parameters a1, ..., ap are the upper (numerator) parameters and b1, ..., bq are the lower (denominator) parameters. If any upper parameter is a nonpositive integer, the series terminates and pFq becomes a polynomial in z.

Convergence and analytic properties: The radius of convergence depends on p and q. If p ≤ q, the series converges for all finite z and the function is entire. If p = q + 1, the radius of convergence is 1, so the series converges for |z| < 1 (with analytic continuation possible beyond this disk under suitable conditions). If p > q + 1, the series diverges for all z ≠ 0, though the function can often be defined by continuation or via integral representations.

Differential equation and representations: pFq satisfies a linear differential equation of order max(p, q + 1) with polynomial coefficients, generalizing the Gauss hypergeometric equation (p = 2, q = 1). In addition to the defining series, pFq admits Euler-type integral representations for suitable parameter ranges and Mellin-Barnes integral representations that enable analytic continuation and asymptotic analysis.

Special cases and applications: Many classical functions arise as special cases, such as 2F1 (Gauss), 1F1 (Kummer), and 0F1 (related to Bessel functions). The pFq family appears in solutions of differential equations, in mathematical physics, statistics, and combinatorics, providing compact expressions for sums, integrals, and generating functions.