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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

Sum1kbinomialk+n,k,k=1..

k=11kk+nk

(1)

convert&comma;hypergeomassuming0<n

1n

(2)

BesselJa&comma;z

BesselJa&comma;z

(3)

=convert&comma;hypergeom

BesselJa&comma;z=zahypergeom&comma;a+1&comma;z24Γa+12a

(4)

LegendrePa&comma;b&comma;z

LegendrePa&comma;b&comma;z

(5)

=convert&comma;hypergeom

LegendrePa&comma;b&comma;z=z+1b2hypergeoma&comma;a+1&comma;1b&comma;12z2z1b2Γ1b

(6)

KummerUa&comma;b&comma;z

KummerUa&comma;b&comma;z

(7)

=convert&comma;hypergeom

KummerUa&comma;b&comma;z=Γ1+bhypergeomab+1&comma;2b&comma;zz1+bΓa+Γ1bhypergeoma&comma;b&comma;zΓab+1

(8)

For negative a we have

=convert&comma;hypergeomassuminga::negint

KummerUa&comma;b&comma;z=pochhammerab+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)

=convert&comma;hypergeom&comma;include=all

sinz=zhypergeom&comma;32&comma;z24

(12)

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

SSumxnn&comma;n=1..

Sn=1xnn

(13)

convertS&comma;hypergeom

n=1xnn

(14)

convertS&comma;hypergeomassumingabsx<1

ln1x

(15)

_EnvFormaltrue

_EnvFormaltrue

(16)

convertS&comma;hypergeom

ln1x

(17)

See Also

convert

convert/MeijerG

convert/to_special_function

hypergeom

simplify/hypergeom