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convert/binomial

Convert to Binomial Form

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

convert(e, binomial)

Parameters

e

-

expression

Description

• 

This option to convert converts the GAMMA function and factorials in an expression e to binomial coefficients. The code performs the following two transformations on products of factorials and the GAMMA function.

• 

Transformation 1: given a product

f1!if2!jf3!k

  

where i,j,k are integers. Note, the code handles the case where f1, f2, and f3 are GAMMA functions, and also the special case π=Γ12.

  

Case 1: 0<i and j,k<0.  Then if f1f2f3=n, an integer, the product is multiplied by

f1f2cf2!f3!f1!

  

where c is a correction factor depending on n and f3.

  

Similarly, CASE 2: where i,0<j, k<0.  This is the case where the binomial appears in the denominator.  Then if f3f1f2=n, an integer, the product is multiplied by

cf3!f1!f2!f2f1

• 

Transformation 2: given a product

f1&excl;if2&excl;j

  

where i,j are integers and f1f2 is a rational constant r.

  

Case 1: 1<r.  Multiply by   f2!f1f2f1f2!f1!

  

Case 2: r<1.  Multiply by   f2!f1!f2f1f2f1!

Examples

an&excl;k&excl;nk&excl;

an!k!nk!

(1)

converta&comma;binomial

nk

(2)

ann2&plus;mk&plus;2n2&plus;m&excl;k&excl;n2&plus;mk&plus;2&excl;

ann2k+m+2n2+m!k!n2k+m+2!

(3)

converta&comma;binomial

nn2+mkn2k+m+1

(4)

am&excl;33m&excl;

am!33m!

(5)

converta&comma;binomial

13mm2mm

(6)

aGAMMAm&plus;32PiGAMMAm

aΓm+32πΓm

(7)

converta&comma;binomial

mm+1m+1212

(8)

See Also

convert/factorial

convert/GAMMA