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SumTools[Hypergeometric]

  

PolynomialNormalForm

  

construct the polynomial normal form of a rational function

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

PolynomialNormalForm(F, n)

Parameters

F

-

rational function of n

n

-

variable

Description

• 

Let F be a rational function of n over a field K of characteristic 0. The PolynomialNormalForm(F,n) command constructs the polynomial normal form for F.

• 

The output is a sequence of 4 elements z,a,b,c where z is an element of K, and a,b,c are monic polynomials over K such that: F=zaEcbc.  gcda,Ekb=1for allnonnegative integersk. gcda,c=1,gcdb,Ec=1.

  

Note: E is the automorphism of K(n) defined by {E(F(n)) = F(n+1)}.

Examples

withSumToolsHypergeometric:

F32nn+23n+23n+4n12n+9n+42

F3nn+23n+23n+42n12n+9n+42

(1)

z,a,b,cPolynomialNormalFormF,n

z,a,b,c274,n+2n+23n+43,n+92n+42,n1

(2)

Check the results.

Condition 1 is satisfied.

evalbF=normalzabsubsn=n+1,cc

true

(3)

Condition 2 is satisfied.

LREtoolsdispersionb,a,n

FAIL

(4)

Condition 3 is satisfied.

gcda,c,gcdb,subsn=n+1,c

1,1

(5)

References

  

Gosper, R.W., Jr. "Decision procedure for indefinite hypergeometric summation." Proc. Natl. Acad. Sci. USA. Vol. 75. (1977): 40-42.

  

Petkovsek, M. "Hypergeometric solutions of linear recurrences with polynomial coefficients." J. Symb. Comput. Vol. 14. (1992): 243-264.

See Also

evalb

LREtools[dispersion]

subs

SumTools[Hypergeometric]

SumTools[Hypergeometric][Gosper]

SumTools[Hypergeometric][RationalCanonicalForm]