<Text-field style="Heading 1" layout="Heading 1">Example 1</Text-field>

restart;

read "qFPS.mpl";

with(qFPS):

qshift(qsin(x,q),x,q);

qdiff(qLaguerre(n,alpha,x,q),x,1/q);

<Text-field style="Heading 1" layout="Heading 1">Example 2</Text-field>

qRE1:=qHolonomicRE(qcos(x,q),F(x));

qRE2:=qHolonomicRE(qsin(x,1/q),F(x));

qRE:=qProductRE(qRE1,qRE2,F(x));

<Text-field style="Heading 1" layout="Heading 1">Example 3</Text-field>

qDE1:=qProductDE(qHolonomicDE(qsin(x,q),F(x)),qHolonomicDE(qSin(x,q),F(x)),F(x));

qOrder(qDE1,F(x));

qDE1:=qProductDE(qHolonomicDE(qcos(x,q),F(x)),qHolonomicDE(qCos(x,q),F(x)),F(x));

qOrder(qDE2,F(x));

qSumDE(qDE1,qDE2,F(x));

f:=qsin(x,q)*qSin(x,q): g:=qcos(x,q)*qCos(x,q):

qHolonomicDE(f+g,F(x));

<Text-field style="Heading 1" layout="Heading 1">Example 4</Text-field>

RE:=qREtoRE(qRE,F(x),c(j),base=qpower,expansionpt=0);

<Text-field style="Heading 1" layout="Heading 1">Example 5</Text-field>

qHypergeomSolveRE(RE,c(j));

<Text-field style="Heading 1" layout="Heading 1">Example 6</Text-field>

convert(qcos(x,q)*qsin(x,1/q),qFPS);

sums with finite support

PS:=convert(x^n,qFPS,x,expansionpt=1);

convert(PS,qbinomial);

convert(qLaguerre(n,alpha,x,q),qFPS,x,base=qpochhammer,expansionpt=-1);

q-hypergeometric series

convert(qexp(x,q),qFPS,base=qpochhammer,expansionpt=a,var=1/q);

convert(qsin(x,q)+qSin(x,1/q),qFPS);

linear combinations of q-hypergeometric series

convert(qsin(x,q),qFPS,expansionpt=a);

convert(sinq(x,q)*qExp(x,q),qFPS);

convert((a*x+b)*qcos(x,q)+(c*x+d)*qsin(x,q),qFPS,x);

<Text-field style="Heading 1" layout="Heading 1">Example 7</Text-field>

term:=qpochhammer(c/b,q,n)/qpochhammer(c,q,n)*b^n;

PS:=convert(term,qFPS,b,base=qpochhammer,expansionpt=1);

convert(PS,qphihypergeom);

<Text-field style="Heading 1" layout="Heading 1">Appendix</Text-field>

Jackson's formula

PS:=convert(qpochhammer(q/b,q,infinity)/qpochhammer(q/b/q^n,q,infinity)*qphihypergeom([1/q^n,c/a],[c],a*q/b,q),qFPS,a,base=qpochhammer,expansionpt=1,initialcheck=false);

convert(subs(b=1/b,qpochhammer(q/b,q,infinity)*qphihypergeom([1/q^n,c],[c],q/b,q)/qpochhammer(q/q^n/b,q,infinity)),qFPS,b);

convert(subs(qphihypergeom([q^(-n),c],[c],q/b,q)=qpochhammer(q/b/q^n,q,infinity)/qpochhammer(q/b,q,infinity),PS),qphihypergeom);

q-Hypergeometric series representations of Taylor coefficients of generating functions (q-Laguerre)

PS1:=convert(qphihypergeom([-x],[0],q^(alpha+1)*t,q),qFPS,x);

PS2:=convert(qphihypergeom([0],[0],q^(alpha+1)*t,q)/qpochhammer(q^(alpha+1)*t,q,k)/qpochhammer(t,q,infinity),qFPS,t,j);

as double sum

S:=Sum(Sum(simplify(op(1,PS1)/qphihypergeom([0],[0],q^(alpha+1)*t,q)*qpochhammer(q^(alpha+1)*t,q,k)*subs(j=n-k,op(1,PS2)),power),k=0..n),n=0..infinity);

inner sum

PS:=Sum(op([1,1],S)/t^n,k=0..infinity);

PS:=subs(qpochhammer(q,q,n-k)=qpochhammer(q,q,n)/qpochhammer(q^(-n),q,k)*(-1)^k*q^(binomial(k,2)-n*k),qpochhammer(q^(alpha+1+k),q,n-k)=qpochhammer(q^(alpha+1),q,n)/qpochhammer(q^(alpha+1),q,k),PS);

convert(PS,qphihypergeom);

PS:=convert(qLaguerre(n,alpha,x,q),qFPS,x);

convert(PS,qphihypergeom);

Identites for the small q-exponential function

convert(qexp(x,q),qFPS);

convert(expq(x/(1-q),q),qFPS);

convert(1/qpochhammer(x,q,infinity),qFPS);

convert(qphihypergeom([0],[],x,q),qFPS);

convert(convert(qexp(x,q),qFPS,expansionpt=a),qphihypergeom);

convert(convert(qexp(x,q),qFPS,base=qpochhammer,expansionpt=a),qphihypergeom,var=1/q);

convert(convert(qexp(x,q),qFPS,base=qpochhammer,expansionpt=a),qphihypergeom);

convert(convert(qexp(x,q),qFPS,expansionpt=a),qphihypergeom,var=1/q);

q-Addition theorems

convert(qsin(x,q)*qSin(x,q)+qcos(x,q)*qCos(x,q),qFPS);

convert(qsin(x,q)*qCos(x,q)-qSin(x,q)*qcos(x,q),qFPS);