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SumTools[DefiniteSum]

  

pFqToStandardFunctions

  

compute closed forms of definite sums using hypergeometric functions

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

pFqToStandardFunctions(f, k=m..n)

Parameters

f

-

expression; specified summand

k

-

name

m, n

-

expressions or integers

Description

• 

The pFqToStandardFunctions(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k by first converting the specified sum into hypergeometric functions. If possible, the output is then converted to standard functions.

• 

If _EnvFormal is assigned true, the command computes the result in the sense of analytic continuation. Otherwise, the command computes the closed form in the domain of convergence (see hypergeom).

Examples

withSumToolsDefiniteSum:

F22kGAMMAknGAMMAk+nzkPi12GAMMA2k+1

F22kΓknΓk+nzkπΓ2k+1

(1)

pFqToStandardFunctionsF,k=0..∞

FAIL

(2)

k=0∞F=pFqToStandardFunctionsF,k=0..∞assumingz1

k=022kΓknΓk+nzkπΓ2k+1=πcos2narcsinzcscπnn

(3)

_EnvFormaltrue:

k=0∞F=pFqToStandardFunctionsF,k=0..∞

k=022kΓknΓk+nzkπΓ2k+1=πcos2narcsinzcscπnn

(4)

References

  

Petkovsek, M.; Wilf, H.; and Zeilberger, D. A=B, Ch. 3. Wellesley, Massachusetts: A K Peters, Ltd., 1996.

  

Prudnikov, A. P.; Brychkov, Yu.; and Marichev, O. Integrals and Series. Gordon and Breach Science Publishers, 1990. Vol. 3: More Special Functions.

  

Roach, K. "Hypergeometric Function Representations." Proceedings ISSAC 1996, pp. 301-308. New York: ACM Press, 1996.

See Also

hypergeom

SumTools[DefiniteSum]

SumTools[DefiniteSummation]

SumTools[IndefiniteSummation]

SumTools[Summation]