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hypergeom

generalized hypergeometric function

 

Calling Sequence

Parameters

Description

Examples

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=[n1,n2,...], p=nopsn, d=[d1,d2,...] and q=nopsd. The hypergeom(n, d, z) calling sequence is the generalized hypergeometric function Fn,d,z. This function is frequently denoted by pFqn,d,z.

• 

Formally, Fn,d,z is defined by the series

k=0zki=1ppochhammerni,kk!j=1qpochhammerdj,k

  

For the definition of the pochhammer symbol, see pochhammer.

• 

If some ni is a non-positive integer, the series is finite (that is, Fn,d,z is a polynomial in z).

  

If some dj 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 ni or dj is a non-positive integer.

  

When pq, this series converges for all complex z, and hence defines Fn,d,z everywhere.

  

When p&equals;q&plus;1, the series converges for &verbar;z&verbar;<1. Fn&comma;d&comma;z is then defined for &verbar;z&verbar;>=1 by analytic continuation.  The point z&equals;1 is a branch point, and the interval (1,infinity) is the branch cut.

  

When q&plus;1<p the series diverges for all z0.  In this case, the series is interpreted as the asymptotic expansion of Fn&comma;d&comma;z around z&equals;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

hypergeom&comma;&comma;z

hypergeom&comma;&comma;z

(1)

hypergeom&comma;&comma;&pi;

hypergeom&comma;&comma;&pi;

(2)

hypergeoma&comma;&comma;z

hypergeoma&comma;&comma;z

(3)

hypergeom1&comma;2&comma;2&comma;3&comma;z

hypergeom1&comma;3&comma;z

(4)

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

evalfhypergeom&comma;&comma;&pi;&equals;evalf&ExponentialE;&pi;

23.14069264&equals;23.14069264

(5)

hypergeom1&comma;1&comma;&comma;1.

0.69717488321.155727350I

(6)

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

simplifyzhypergeom&comma;32&comma;z24&comma;hypergeom

sinz

(7)

simplifyhypergeoma&comma;&comma;z&comma;hypergeom

1za

(8)

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

Hypergeom1&comma;2&comma;2&comma;3&comma;z

Hypergeom1&comma;2&comma;2&comma;3&comma;z

(9)

value

hypergeom1&comma;3&comma;z

(10)

See Also

convert

convert/elementary

convert/StandardFunctions

evalf

initialfunctions

pochhammer

simplify/hypergeom

value

 


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