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convert/hypergeom

convert functions in an expression to hypergeometric form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

convert( expr, hypergeom, opt_1, opt_2, ... )

Parameters

expr

-

expression

opt_i

-

optional arguments, see convert/to_special_function

Description

• 

Converts the functions found in expr, including sum and Sum, to their hypergeometric forms when possible.

• 

For sums, no attempt is made to ensure that the resulting hypergeometric function is convergent or terminates.

• 

convert/hypergeom reacts to the setting of the environment variable _EnvFormal (for more information, see sum/details). When that variable is set to true, the conversion will be attempted regardless of whether the original sum may be divergent or not.

Examples

k=1∞1kbinomialk+n,k

k=1∞1kbinomialk+n,k

(1)

convert&comma;hypergeomassuming0<n

1n

(2)

BesselJa&comma;z

BesselJa&comma;z

(3)

&equals;convert&comma;hypergeom

BesselJa&comma;z&equals;zahypergeom&comma;1&plus;a&comma;14z2&Gamma;1&plus;a2a

(4)

LegendrePa&comma;b&comma;z

LegendrePa&comma;b&comma;z

(5)

&equals;convert&comma;hypergeom

LegendrePa&comma;b&comma;z&equals;z&plus;112bhypergeoma&comma;1&plus;a&comma;1b&comma;1212zz112b&Gamma;1b

(6)

KummerUa&comma;b&comma;z

KummerUa&comma;b&comma;z

(7)

&equals;convert&comma;hypergeom

KummerUa&comma;b&comma;z&equals;&Gamma;1&plus;bhypergeomab&plus;1&comma;2b&comma;zz1&plus;b&Gamma;a&plus;&Gamma;1bhypergeoma&comma;b&comma;z&Gamma;ab&plus;1

(8)

For negative a we have

&equals;convert&comma;hypergeomassuminga::negint

KummerUa&comma;b&comma;z&equals;pochhammerab&plus;1&comma;ahypergeoma&comma;b&comma;z

(9)

Elementary functions are not converted by default (see convert/to_special_function)

convertsinz&comma;hypergeom

sinz

(10)

To convert them use either of the optional arguments: include=elementary, or include=all

sinz

sinz

(11)

&equals;convert&comma;hypergeom&comma;include&equals;all

sinz&equals;zhypergeom&comma;32&comma;14z2

(12)

Parametric or divergents sums may not be converted by default. Set _EnvFormal to true, or use appropriate assumptions, to obtain the desired conversion:

Sn&equals;1&infin;xnn

S:=n&equals;1&infin;xnn

(13)

convertS&comma;hypergeom

n&equals;1&infin;xnn

(14)

convertS&comma;hypergeomassumingx<1

ln1x

(15)

_EnvFormaltrue

_EnvFormal:=true

(16)

convertS&comma;hypergeom

ln1x

(17)

See Also

convert

convert/MeijerG

convert/to_special_function

hypergeom

simplify/hypergeom

 


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