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QDifferenceEquations[QPochhammer] - q-Pochhammer symbol

QDifferenceEquations[QBinomial] - q-binomial coefficient

QDifferenceEquations[QBrackets] - q-brackets

QDifferenceEquations[QFactorial] - q-factorial

QDifferenceEquations[QGAMMA] - q-Gamma

Calling Sequence

QPochhammer(a, q, infinity)

QPochhammer(a, q, k)

QBinomial(n, k, q)

QBrackets(k, q)

QFactorial(k, q)

QGAMMA(a, q)

Parameters

a

-

algebraic expression

q

-

name used as the parameter q, or an integer power of a name

k

-

symbolic integer value

n

-

symbolic integer value

Description

• 

The QDifferenceEquations package supports five q-hypergeometric terms. They are q-Pochhammer symbol, q-binomial coefficient, q-brackets, q-factorial, and q-Gamma, which correspond to the five functions QPochhammer, QBinomial, QBrackets, QFactorial, and QGAMMA.

• 

These functions are placeholders for the q-objects. The command expand allows expansion of these objects. The command convert...,QPochhammer allows the re-write of QBinomial, QBrackets, QFactorial, and QGAMMA in terms of QPochhammer symbols.

• 

The five q-hypergeometric objects are defined as follows.

QPochhammera,q,=j=01aqj

QPochhammera&comma;q&comma;k&equals;&lcub;j&equals;0k11aqj0<k1k&equals;0j&equals;k11aq&plus;1jk<0

  

Note that QPochhammerseqai&comma;i&equals;1..n&comma;q&comma;k (the compact Gasper and Rahman notation) means i&equals;1nQPochhammerai&comma;q&comma;k.

QBinomialn&comma;k&comma;q&equals;QPochhammerq&comma;q&comma;nQPochhammerq&comma;q&comma;kQPochhammerq&comma;q&comma;nk

QBracketsk&comma;q&equals;qk1q1

QFactorialk&comma;q&equals;QPochhammerq&comma;q&comma;k1qk

QGAMMAz&comma;q=QPochhammerq&comma;q&comma;1q1zQPochhammerqz&comma;q&comma;

• 

The commands QSimpComb and QSimplify are for simplification of expressions involving these q-objects.

• 

This implementation is mainly based on the implementation by H. Boeing, W. Koepf. See the References section.

Examples

withQDifferenceEquations&colon;

expandQPochhammera&comma;q&comma;4

1aaq&plus;1aq2&plus;1aq3&plus;1

(1)

expandQPochhammera&comma;q&comma;4

11aq41aq31aq21aq

(2)

expandQBracketsk&comma;q

qk1q1

(3)

convertQBinomialn&comma;k&comma;q&comma;&apos;QPochhammer&apos;

QDifferenceEquations:-QPochhammerq&comma;q&comma;nQDifferenceEquations:-QPochhammerq&comma;q&comma;kQDifferenceEquations:-QPochhammerq&comma;q&comma;nk

(4)

convertQGAMMAz&comma;q&comma;&apos;QPochhammer&apos;

QDifferenceEquations:-QPochhammerq&comma;q&comma;&infin;1q1zQDifferenceEquations:-QPochhammerqz&comma;q&comma;&infin;

(5)

convertQFactorialk&comma;q&comma;&apos;QPochhammer&apos;

QDifferenceEquations:-QPochhammerq&comma;q&comma;k1qk

(6)

H:=q212q6nQPochhammer1q5&plus;q3&comma;q&comma;nQPochhammer1q4&plus;q2&comma;q&comma;nQPochhammer1q3q21&comma;q&comma;nQPochhammer1q2&comma;q&comma;nQPochhammer1q12q21&comma;q&comma;nQPochhammer1&comma;q&comma;nQPochhammer1q2q21&comma;q&comma;nQPochhammer1q5&comma;q&comma;nQPochhammer1q4&comma;q&comma;n2QPochhammerq4&comma;q&comma;nQPochhammer1q2&plus;1&comma;q&comma;n

H:=q212q6nQDifferenceEquations:-QPochhammer1q5&plus;q3&comma;q&comma;nQDifferenceEquations:-QPochhammer1q4&plus;q2&comma;q&comma;nQDifferenceEquations:-QPochhammerq3q21&comma;q&comma;nQDifferenceEquations:-QPochhammer1q2&comma;q&comma;nQDifferenceEquations:-QPochhammerq12q21&comma;q&comma;nQDifferenceEquations:-QPochhammer1&comma;q&comma;nQDifferenceEquations:-QPochhammerq2q21&comma;q&comma;nQDifferenceEquations:-QPochhammer1q5&comma;q&comma;nQDifferenceEquations:-QPochhammer1q4&comma;q&comma;n2QDifferenceEquations:-QPochhammerq4&comma;q&comma;nQDifferenceEquations:-QPochhammer1q2&plus;1&comma;q&comma;n

(7)

Compute the certificate of H (which is a rational function in qn):

QSimpCombsubsn&equals;n&plus;1&comma;HH

q5q3&plus;qnq2&plus;qn1&plus;qnqnq12&plus;q21qnq3&plus;q21q4q2&plus;qnq2qn&plus;q21q2&plus;qn1q4&plus;qn21&plus;qnq4q5&plus;qn

(8)

See Also

QDifferenceEquations[IsQHypergeometricTerm], QDifferenceEquations[QSimpComb]

References

  

Boeing, H., and Koepf, W. "Algorithms for q-hypergeometric summation in computer algebra." Journal of Symbolic Computation. Vol. 11. (1999): 1-23.


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