restart: read "qFPS.mpl": with(qFPS):

basic number (or q-brackets), q-factorial and q-binomial [qbrackets(k,q),qfactorial(k,q),qbinomial(n,k,q)]; q-powers and q-Pochhammer symbols [x^k,qpochhammer(x,q,k),qpower(x,a,q,k)]; [qpochhammer(x,q,infinity),qpower(x,a,q,infinity)]; q-exponential functions [qexp(x,q),qExp(x,q),expq(x,q),Expq(x,q)]; q-trigonometric functions [qsin(x,q),qSin(x,q),sinq(x,q),Sinq(x,q)]; [qcos(x,q),qCos(x,q),cosq(x,q),Cosq(x,q)]; classical q-orthogonal polynomials [DiscreteqHermiteII(n,x,q),AlSalamCarlitzII(n,a,x,q),qCharlier(n,a,x,q),qLaguerre(n,alpha,x,q),StieltjesWigert(n,x,q),qMeixner(n,b,c,x,q),QuantumqKrawtchouk(n,p,N,x,q),qKrawtchouk(n,p,N,x,q),LittleqLaguerre(n,a,x,q),AlternativeqCharlier(n,a,x,q),LittleqJacobi(n,a,b,x,q),LittleqLegendre(n,x,q),DiscreteHermiteI(n,x,q),AlSalamCarlitzI(n,a,x,q),BigqLaguerre(n,a,b,x,q),AffineqKrawtchouk(n,p,N,x,q),qHahn(n,a,b,N,x,q),BigqJacobi(n,a,b,c,x,q),BigqLegendre(n,c,x,q)]; generalized q-hypergeometric function qphihypergeom([a,b,c],[d,e],x,q);

q-shifts and q-derivatives of some q-functions [qshift(x^k,x,q),qdiff(x^k,x,q)]; [qshift(qpower(x,a,q,k),x,q),qdiff(qpower(x,a,q,k),x,q)]; [qshift(qexp(x,q),x,q),qdiff(qexp(x,q),x,q)]; [qshift(qsin(x,q),x,q),qdiff(qsin(x,q),x,q)]; [qshift(qLaguerre(n,alpha,x,q),x,q),qdiff(qLaguerre(n,alpha,x,q),x,q)]; [qshift(qphihypergeom([a,b,c],[d,e],x,q),x,q),qdiff(qphihypergeom([a,b,c],[d,e],x,q),x,q)]; q-product rule qdiff(qpochhammer(x,q,k)*qsin(x,q),x,q); q-chain rule qdiff(qsin(a*x^2,q),x,q); qdiff(qpochhammer(a/x,q,k),x,q); higher q-shifts and q-derivatives [qshift(qsin(x,q),[x$2],q),qdiff(qsin(x,q),[x$2],q)]; qdiff(qsin(x*y,q),[x,y],q); application of q-shift and Hahn operator on unknown functions leads to operator notation [qshift(f(x),x,q),qdiff(f(x),x,q)]; [qshift(f(x),[x$2],q),qdiff(f(x),[x$2],q)]; qdiff(Dq[x](f(x)),x,q); with option explicit the definition of the operators is used for unknown functions [qshift(f(x),x,q,explicit),qdiff(f(x),x,q,explicit)]; [qshift(f(x),[x$2],q,explicit),qdiff(f(x),[x$2],q,explicit)]; application w.r.t. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= [qshift(qsin(x,q),x,1/q),qdiff(qsin(x,q),x,1/q)];

q-holonomic recurrence and difference equations of some q-functions qHolonomicRE(qpochhammer(x,q,k),F(x)); qHolonomicDE(qpochhammer(x,q,k),F(x)); qHolonomicRE(qexp(x,q),F(x)); qHolonomicDE(qexp(x,q),F(x)); qHolonomicRE(qsin(x,q),F(x)); qHolonomicDE(qsin(x,q),F(x)); qHolonomicRE(qLaguerre(n,alpha,x,q),F(x)); qHolonomicDE(qLaguerre(n,alpha,x,q),F(x)); qHolonomicRE(qphihypergeom([a,b,c],[d,e],x,q),F(x)); qHolonomicDE(qphihypergeom([a,b,c],[d,e],x,q),F(x)); qHolonomicRE(qexp(x^2,q),F(x)); qHolonomicRE(qsin(x,q)+qcos(x,q),F(x)); qHolonomicRE(qsin(x,q)*qexp(x,q),F(x)); output of q-holonomic recurrence equations without operators qHolonomicRE(qsin(x,q),F(x),explicit); output of q-holonomic recurrence equations with negative q-shifts (WARNING: The operator notation abandons the symbol q for readibility reasons. The operators in the following recurrence equation are negative q-shifts or positive q-shifts w.r.t. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= respectively.) qHolonomicRE(qsin(x,q),F(x),var=1/q); qHolonomicRE(qsin(x,q),F(x),var=1/q,explicit);

RE1:=qHolonomicRE(qexp(x,q),F(x)); RE2:=qHolonomicRE(qsin(x,q),F(x)); q-holonomic recurrence equation, which is valid for the product of the solutions of RE1 and RE2 qProductRE(RE1,RE2,F(x)); q-holonomic recurrence equation, which is valid for the sum of the solutions of RE1 and RE2 qSumRE(RE1,RE2,F(x)); q-holonomic recurrence equation, which is valid for the composition of the solution of RE1 with the power LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEjYXhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1JI21uR0YkNiRRIjJGJy9GNlEnbm9ybWFsRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y+ qCompositionRE(RE1,F(x),a*x^2); comparison with qHolonomicRE qHolonomicRE(qexp(a*x^2,q),F(x)); For q-holonomic difference equations we use the corresponding DE-procedures. To convert a recurrence equation into a difference equation (and vice versa) we apply the following procedures DE1:=qREtoqDE(RE1,F(x)); DE2:=qREtoqDE(RE2,F(x)); test evalb(RE1=qDEtoqRE(DE1,F(x))); qProductDE(DE1,DE2,F(x)); qSumDE(DE1,DE2,F(x)); qCompositionDE(DE1,F(x),a*x^2);

RE1:=(q-1)*(q^3*x-2*x*q^2+q*x-1)*F(x)+(6*x^2*q^3+q^5*x^2-4*q^4*x^2+q*x^2-4*x^2*q^2-1)*Sq[x](F(x))+q*x*(q-1)*(x*q^2-2*q*x-1+x)*Sq[x, x](F(x)); RE2:=F(x)+(q-1)*x*Sq[x](F(x)); right division with quotient polynomial and remainder polynomial qDivideRE(RE1,RE2,F(x),polynomial=quo); qDivideRE(RE1,RE2,F(x)); left division with quotient polynomial and remainder polynomial qDivideRE(RE1,RE2,F(x),direction=left,polynomial=quo); qDivideRE(RE1,RE2,F(x),direction=left); right multiplication of q-recurrence operators qMultiplyRE(RE1,RE2,F(x)); left multiplication of q-recurrence operators qMultiplyRE(RE1,RE2,F(x),direction=left); least common left multiple qLCM(RE1,RE2,F(x)); least common right multiple qLCM(RE1,RE2,F(x),direction=right); greatest common right divisor qGCD(RE1,RE2,F(x)); greatest common left divisor qGCD(RE1,RE2,F(x),direction=left); q-adjoint operator qAdjoint(RE1,F(x));

qRE1:=qHolonomicRE(qsin(x,q),F(x)); determination of a q-holonomic recurrence equation for the coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRImpGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== of 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 the q-power basis. For LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= we obtain qREtoRE(qRE1,F(x),c(j)); qREtoRE(qRE1,F(x),c(j),expansionpt=a); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZGVub3RlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSS1JJW1zdXBHRiQ2JS1JJW1zdWJHRiQ2JS1GLDYlUSJQRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JC1GLDYlUSJqRidGVUZXRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GIzYkLUYsNiVRImFGJ0ZVRldGMi8lMXN1cGVyc2NyaXB0c2hpZnRHRmpuLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnRlVGV0YyRjJGMg== the q-Pochhammer basis RE:=qREtoRE(qRE1,F(x),c(j),base=qpochhammer,expansionpt=a); the reverse transformation is done by the following procedure qRE2:=REtoqRE(RE,c(j),F(x),base=qpochhammer,expansionpt=a); qDivideRE(qRE2,qRE1,F(x)); qNormal(subs(a=0,qRE2),F(x)); procedures for q-holonomic difference equations qDE:=qREtoqDE(qRE1,F(x)); RE:=qDEtoRE(qDE,F(x),c(j)); REtoqDE(RE,c(j),F(x));

Polynomial Solutions RE:=Sq[x,x](F(x))-(q+1)*Sq[x](F(x))+q*F(x)=0; qPolynomialSolveRE(RE,F(x)); qPolynomialSolveRE(RE,F(x),output=onesol); Rational Solutions RE:=q^2*Sq[x,x](F(x))-(q+1)*q*Sq[x](F(x))+q*F(x)=0; qRationalSolveRE(RE,F(x)); qRationalSolveRE(RE,F(x),output=onesol); Solutions of q-Hypergeometric Type RE:=Sq[x,x](F(x))+(1-q)*x*F(x)=0; qHypergeomTypeSolveRE(RE,F(x)); qHypergeomTypeSolveRE(RE,F(x),certificate); transformation of the recurrence from multiplicative in additive form RE:=qREtoqRE(RE,F(x),A(k)); qHypergeomTypeSolveRE(RE,A(k)); qHypergeomTypeSolveRE(RE,A(k),certificate); q-Hypergeometric Solutions RE:=(q^2*x-1)*Sq[x,x](F(x))+x*q*(q^2*x+q*x-1)*Sq[x](F(x))+q^2*x^3*F(x)=0; qHypergeomSolveRE(RE,F(x)); qHypergeomSolveRE(RE,F(x),certificate); transformation of the recurrence from multiplicative in additive form RE:=qREtoqRE(RE,F(x),A(k)); qHypergeomSolveRE(RE,A(k)); qHypergeomSolveRE(RE,A(k),certificate); choice of method (the first method is preset) qHypergeomSolveRE(RE,A(k),method=modqPetkovsek); qHypergeomSolveRE(RE,A(k),method=qPetkovsek); qHypergeomSolveRE(RE,A(k),method=qVanHoeij);

qFPS Algorithm the standard basis is the q-power basis convert(qsin(x,q),qFPS); PS1:=convert(qexp(x,q),qFPS); PS2:=convert(qLaguerre(n,alpha,x,q),qFPS,x); PS3:=convert(qphihypergeom([a,b,c],[d,e],x,q),qFPS,x); convert(qsin(x,q)+qSin(x,1/q),qFPS); convert(qsin(x,q)+x*qcos(x,q),qFPS); specification of the expansion point convert(qexp(x,q),qFPS,expansionpt=a); convert(qsin(x,q),qFPS,expansionpt=a); q-series expansion of the q-Charlier polynomials w.r.t. the q-Pochhammer basis with expansion point 1 convert(qCharlier(n,a,x,q),qFPS,x,base=qpochhammer,expansionpt=1); linear combinations of q-hypergeometric series convert(qcos(x,q)*qsin(x,1/q),qFPS,x); convert(sinq(x,q)*qExp(x,q),qFPS); with option termwise the input will be expanded, then the qFPS algorithm will be applied to every single term and the result will be combined to one (if possible). convert((a*x^2+b*x+c)*qexp(x,q),qFPS,x,termwise); additional information about the qFPS algorithm infolevel[qFPS]:=4: convert(1/x*qexp(x^2,q),qFPS,expansionpt=a); infolevel[qFPS]:=0: conversion w.r.t. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= convert(qsin(x,q),qFPS,x,var=1/q); Conversion into the Notation of a Generalized q-Hypergeometric Function convert(PS1,qphihypergeom); convert(PS2,qphihypergeom); convert(PS3,qphihypergeom); convert(qphihypergeom([a,b,c],[d],x,1/q),qFPS,x); convert(%,qphihypergeom,var=1/q);

We consider the following example. Let myfunction be a function, which is defined by the following term term:=expq(x,q)*qExp(x,q); qshift(term,x,q); Of course, there exists no q-shift rule for myfunction. qshift(myfunction(x,q),x,q); addqshift('myfunction', proc(y,p,x,q) if x=y and p=q then (x*(q-1)+1)/(x+1)*myfunction(x,q) end if end proc ); For adding a q-derivative rule or a q-recurrence equation respectively one uses the procedures addqdiff or addqrec respectively. Now, qFPS 'knows' the function qshift(myfunction(x,q),x,q); We set the function value of myfunction in 0. myfunction(0,q):=1; convert(myfunction(x,q),qFPS);

Conversion of q-Functions convert(qbinomial(n,k,q),qpochhammer); convert(q^(n*k-k*(k-1)/2)*qpochhammer(1/q^n,q,k)/qpochhammer(q,q,k)/(-1)^k,qbinomial); convert(qpochhammer(q,q,k)/(q-1)^k/(-1)^k,qfactorial); convert(x^n,qFPS,x,expansionpt=1); convert(%,qbinomial); q-Normal Form q-normal form for q-holonomic recurrence equations RE:=1/(x-1)/q*F(x)+q*(q-1)/x/(x-1)*Sq[x](F(x))=0; qNormal(RE,F(x)); q-Content content of a q-holonomic recurrence equation RE:=q*x*(x-1)*F(x)+q^3*(q-1)*(x-1)*Sq[x](F(x))=0; qContent(RE,F(x)); the recurrence equation divided by its content qContent(RE,F(x),eq); q-Order order of a q-holonomic recurrence equation RE:=q*x*(x-1)*F(x)+q^3*(q-1)*(x-1)*Sq[x,x](F(x))=0; qOrder(RE,F(x)); order of a q-holonomic difference equation qOrder(qREtoqDE(RE,F(x)),F(x)); q-Shifting of Recurrence Equations RE:=F(x)*x+q^2*(q-1)*Sq[x](F(x))=0; input of a q-holonomic recurrence equation (considered as recurrence with negative q-shifts), which is converted into a recurrence equation with only positive q-shifts qShiftRE(RE,F(x)); input of a q-holonomic recurrence equation (considered as recurrence with positive q-shifts), which is converted into a recurrence equation with only negative q-shifts qShiftRE(RE,F(x),var=1/q); q-Dispersion Set determination of the q-dispersion set p:=mul(q^k*x+modp(k,3),k=1..10); q-dispersion qDispersion(qshift(p,[x$2],q),p,x); q-dispersion set qDispersion(qshift(p,[x$2],q),p,x,set); q-Newton Polygon RE:=qshift(F(x),[x$5],q)+(q^2*x^2-1)*qshift(F(x),[x$4],q)+x*qshift(F(x),[x$3],q)-(x^4+q*x+1)*qshift(F(x),[x$2],q)+q^2*x^4*qshift(F(x),x,q)-(x^3+q)*F(x)=0; vertices of the q-Newton polygon qNewtonPolygon(RE,F(x),q); charakteristic equations qNewtonPolygon(RE,F(x),q,eq); graphical representation of the q-Newton polygon qPlotNewtonPolygon(RE,F(x),q); selection of the evaluation qNewtonPolygon(RE,F(x),q,v=((a,x)->ldegree(a,x))); Local Type of a q-Hypergeometric Term local type of a q-hypergeometric term term:=qpochhammer(x,q,k); through entering the q-hypergeometric term qLocalTypes(term,x,q); cert:=qCertificate(term,x,q); through entering the corresponding q-certificate qLocalTypes(cert,x,q,certificate); qLocalTypes(term,x,q,ltype=infinity); qLocalTypes(term,x,q,ltype=0); qLocalTypes(term,x,q,ltype=points); candidates of local types for q-hypergeometric solutions of a q-holonomic recurrence equation RE:=(q*x-1)*Sq[x,x](F(x))+(1-q)*x*F(x)=0; qLocalTypesCandidates(RE,F(x),q); Application of the Sq Operators or Dq Operators respectively RE:=qHolonomicRE(qsin(x,q),F(x)); term:=subs(F(x)=qsin(x,q),lhs(RE)); expand(Sqtoqshift(term,q)); DE:=qHolonomicDE(qsin(x,q),F(x)); term:=subs(F(x)=qsin(x,q),lhs(DE)); Dqtoqdiff(term,q); Conversion from Sq Operator in Dq Operator and vice versa DqtoSq(Dq[x,x,x](F(x)),q); SqtoDq(Sq[x,x,x](F(x)),q); q-Shift and q-Contiguity Relations determination of a q-shift relation of a generalized q-hypergeometric function qShiftRelation(qphihypergeom([a,b,c],[d,e],x,q),F(d)); determination of a q-contiguity relation of a generalized q-hypergeometric function qContiguityRelation(qphihypergeom([a,b,c],[d,e],x,q),F(d)); determination of an operator, which increments (with q multiplied) or decrements (divided by q) respectively one of the upper or lower parameter respectively qRelation(qphihypergeom([a,b,c],[d,e],x,q),F(x),e,direction=up); qRelation(qphihypergeom([a,b,c],[d,e],x,q),F(x),e,direction=down);