hypergeom - Maple Programming Help

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hypergeom

generalized hypergeometric function

 Calling Sequence hypergeom([n1, n2, ... ], [d1, d2, ... ], z) Hypergeom([n1, n2, ... ], [d1, d2, ... ], z)

Parameters

 [n1, n2, ...] - list of upper parameters (may be empty) [d1, d2, ...] - list of lower parameters (may be empty) z - expression

Description

 • Let $n=[\mathrm{n1},\mathrm{n2},...]$, $p=\mathrm{nops}\left(n\right)$, $d=[\mathrm{d1},\mathrm{d2},...]$ and $q=\mathrm{nops}\left(d\right)$. The hypergeom(n, d, z) calling sequence is the generalized hypergeometric function $F\left(n,d,z\right)$. This function is frequently denoted by $\mathrm{pFq}\left(n,d,z\right)$.
 • Formally, $F\left(n,d,z\right)$ is defined by the series

$\sum _{k=0}^{\mathrm{\infty }}\frac{{z}^{k}\prod _{i=1}^{p}\mathrm{pochhammer}\left({n}_{i},k\right)}{k!\prod _{j=1}^{q}\mathrm{pochhammer}\left({d}_{j},k\right)}$

 For the definition of the $\mathrm{pochhammer}$ symbol, see pochhammer.
 • If some ${n}_{i}$ is a non-positive integer, the series is finite (that is, $F\left(n,d,z\right)$ is a polynomial in $z$).
 If some ${d}_{j}$ is a non-positive integer, the function is undefined for all non-zero $z$, unless there is also a negative upper parameter of smaller absolute value, in which case the previous rule applies.
 • For the remainder of this description, assume no ${n}_{i}$ or ${d}_{j}$ is a non-positive integer.
 When $p\le q$, this series converges for all complex $z$, and hence defines $F\left(n,d,z\right)$ everywhere.
 When $p=q+1$, the series converges for $|z|<1$. $F\left(n,d,z\right)$ is then defined for $|z|>=1$ by analytic continuation.  The point $z=1$ is a branch point, and the interval (1,infinity) is the branch cut.
 When $q+1 the series diverges for all $z\ne 0$.  In this case, the series is interpreted as the asymptotic expansion of $F\left(n,d,z\right)$ around $z=0$.  The positive real axis is the branch cut.
 • Hypergeom is the unevaluated form of hypergeom (that is, it returns unevaluated because it is the inert form of this function). Use value to evaluate a call to Hypergeom, or evalf to compute a floating-point approximate value.  See also simplify and convert/StandardFunctions.

Examples

 > $\mathrm{hypergeom}\left(\left[\right],\left[\right],z\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{}\right]{,}\left[{}\right]{,}{z}\right)$ (1)
 > $\mathrm{hypergeom}\left(\left[\right],\left[\right],\mathrm{π}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{}\right]{,}\left[{}\right]{,}{\mathrm{π}}\right)$ (2)
 > $\mathrm{hypergeom}\left(\left[a\right],\left[\right],z\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{a}\right]{,}\left[{}\right]{,}{z}\right)$ (3)
 > $\mathrm{hypergeom}\left(\left[1,2\right],\left[2,3\right],z\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{1}\right]{,}\left[{3}\right]{,}{z}\right)$ (4)

To compute floating point values, use evalf or include a floating point number in the function call.

 > $\mathrm{evalf}\left(\mathrm{hypergeom}\left(\left[\right],\left[\right],\mathrm{π}\right)\right)=\mathrm{evalf}\left({ⅇ}^{\mathrm{π}}\right)$
 ${23.14069264}{=}{23.14069264}$ (5)
 > $\mathrm{hypergeom}\left(\left[1,1\right],\left[\right],1.\right)$
 ${0.6971748832}{-}{1.155727350}{}{I}$ (6)

The simplify function is used to simplify expressions which contain hypergeometric functions.

 > $\mathrm{simplify}\left(z\mathrm{hypergeom}\left(\left[\right],\left[\frac{3}{2}\right],-\frac{{z}^{2}}{4}\right),\mathrm{hypergeom}\right)$
 ${\mathrm{sin}}{}\left({z}\right)$ (7)
 > $\mathrm{simplify}\left(\mathrm{hypergeom}\left(\left[a\right],\left[\right],z\right),\mathrm{hypergeom}\right)$
 ${\left({1}{-}{z}\right)}^{{-}{a}}$ (8)

The inert form of Hypergeom can be evaluated by the function value.

 > $\mathrm{Hypergeom}\left(\left[1,2\right],\left[2,3\right],z\right)$
 ${\mathrm{Hypergeom}}{}\left(\left[{1}{,}{2}\right]{,}\left[{2}{,}{3}\right]{,}{z}\right)$ (9)
 > $\mathrm{value}\left(\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{1}\right]{,}\left[{3}\right]{,}{z}\right)$ (10)