Generalized hypergeometric function

Builds on Pochhammer symbol Power series Radius of convergence

Definition

Let $p, q \ge 0$ be integers, let $a_1, \ldots, a_p \in \C$ , and let $b_1, \ldots, b_q \in \C \setminus \set{0, -1, -2, \ldots}$ . Let $(x)_n$ be the Pochhammer symbol. The generalized hypergeometric series is the power series ${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z) = \sum_{n=0}^{\infty} \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} \, \frac{z^n}{n!}.$ The generalized hypergeometric function is its sum where it converges, together with its analytic continuation.

Remarks

Assume that no $a_i$ is a non-positive integer, so that the series does not terminate. By the ratio test, its radius of convergence is:

infinite when $p \le q$ , so the function is entire;
equal to $1$ when $p = q + 1$ , and then the function continues analytically to $\C \setminus [1, \infty)$ ;
equal to $0$ when $p > q + 1$ .

Examples: ${}_0F_0(;;z) = e^z$ and ${}_1F_0(a;;z) = (1 - z)^{-a}$ for $\abs{z} < 1$ . Also $J_\nu(x) = \frac{(x/2)^\nu}{\Gamma(\nu + 1)} \, {}_0F_1(;\nu + 1; -x^2/4)$ for the Bessel function of the first kind.

Non-example: ${}_2F_0(1, 1;; z) = \sum_n n! \, z^n$ is a formal series with radius of convergence $0$ ; it is only an asymptotic series and defines no function near $0$ .

A lower parameter equal to an upper one cancels: ${}_{p+1}F_{q+1}(a, c; b, c; z) = {}_pF_q(a; b; z)$ , with $a$ and $b$ standing for the lists of parameters.

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Pochhammer symbol concept
Power series concept
Radius of convergence concept
Generalized hypergeometric function definition

Tag	`HP9Z9HJ2SS`
URI	`m2://HP9Z9HJ2SS`
Kind	definition
Writing	W0 stub
MSC	33C20
Defines	Generalized hypergeometric function
Uses	Pochhammer symbol, Power series, Radius of convergence

Generalized hypergeometric function

Definition

Remarks

Reading path

Further reading

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