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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&comma;k<0.  Then if f1f2f3&equals;n, an integer, the product is multiplied by

binomialf1&comma;f2cf2&excl;f3&excl;f1&excl;

  

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

  

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

cf3&excl;f1&excl;f2&excl;binomialf2&comma;f1

• 

Transformation 2: given a product

f1&excl;if2&excl;j

  

where i&comma;j are integers and f1f2 is a rational constant r.

  

Case 1: 1<r.  Multiply by   f2&excl;binomialf1&comma;f2f1f2&excl;f1&excl;

  

Case 2: r<1.  Multiply by   f2&excl;f1&excl;binomialf2&comma;f1f2f1&excl;

Examples

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

a:=n&excl;k&excl;nk&excl;

(1)

converta&comma;binomial

binomialn&comma;k

(2)

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

a:=nn2k&plus;m&plus;2n2&plus;m&excl;k&excl;n2k&plus;m&plus;2&excl;

(3)

converta&comma;binomial

nbinomialn2&plus;m&comma;kn2k&plus;m&plus;1

(4)

am&excl;33m&excl;

a:=m&excl;33m&excl;

(5)

converta&comma;binomial

1binomial3m&comma;mbinomial2m&comma;m

(6)

a&Gamma;m&plus;32&pi;&Gamma;m

a:=&Gamma;m&plus;32&pi;&Gamma;m

(7)

converta&comma;binomial

mm&plus;1binomialm&plus;12&comma;12

(8)

See Also

convert/factorial

convert/GAMMA

 


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