Example 1

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(1.4) (1.4) restart;

read "qFPS.mpl":

with(qFPS):

Example 1

Computation of 2-fold q-hypergeometric solutions of a linear q-recurrence equation m:=2;

m:= 2 r:=(q+1)*q^7*(x-1)/(q^2-2)/(x-q)/x^4;

r:= qC1 q⁷ xK1 q²K2 xKq x⁴

RE1:=denom(r)*qshift(f(x),[x$m],q)-numer(r)*f(x)=0;

RE1:= q²K2 KxCq x⁴ Sq_x,_x f x C qC1 q⁷ xK1 f x = 0

RE2:=x^2*qshift(f(x),[x$2],q)-(q-1)*qshift(f(x),x,q)+q*(x-2)*f(x)

=0;

RE2:=x² Sq_x,_x f x K qK1 Sq_x f x Cq xK2 f x = 0 RE:=qNormal(qMultiplyRE(RE2,RE1,f(x)),f(x)):

qLocalTypesCandidates(RE,f(x),q,mhypersol=m);

q⁷ qC1

q²K2 , 4 , K1

q⁷ , 1 , q⁶ qC1

q²K2 ,K4 , xK2, 0, 1 , xK1, 0, 1 , xKq, K1, 0

st:=time():

qHypergeomSolveRE(RE,f(x),method=qPetkovsek,mhypersol=m);

time()-st;

K qC1 q⁷ xK1 q²K2 KxCq x⁴

0.265 st:=time():

qHypergeomSolveRE(RE,f(x),method=modqPetkovsek,mhypersol=m);

time()-st;

0.048 st:=time():

qHypergeomSolveRE(RE,f(x),method=qVanHoeij,mhypersol=m);

time()-st;

0.754

Example 2

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(3.2) (3.2) (2.6) (2.6) (2.1) (2.1) Computation of 3-fold q-hypergeometric solutions of a linear q-recurrence equation

m:=3;

m:= 3 r:=sqrt(2)*q*(q-1)*x/(x-1);

r:= 2 q qK1 x xK1

RE1:=denom(r)*qshift(f(x),[x$m],q)-numer(r)*f(x)=0;

RE1:= xK1 Sq_x,_x,_x f x K 2 q qK1 x f x = 0

RE2:=(3*x-1)*x*qshift(f(x),[x$2],q)-(q-1)*x*qshift(f(x),x,q)-q^2*

(x-q)*f(x)=0;

RE2:= 3 xK1 x Sq_x,_x f x K qK1 x Sq_x f x Kq² xKq f x = 0 RE:=qNormal(qMultiplyRE(RE2,RE1,f(x)),f(x)):

2 qK1 q, 0 , K 2 qK1 q, 1 , xK1, K1, 0 , xKq, 0, 1 , x K 1

3 q², K1, 0 st:=time():

time()-st;

2 q qK1 x xK1 0.139 st:=time():

time()-st;

2 q qK1 x xK1 0.034 st:=time():

time()-st;

2 q qK1 x xK1 8.588

Example 3

Computation of 2-fold q-hypergeometric solutions of a linear q-recurrence equation with two solutions

m:=2;

m:= 2 r1:=(x+2*q)^2/(x-q);

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r1:= xC2 q ² xKq

RE1:=denom(r1)*qshift(f(x),[x$m],q)-numer(r1)*f(x)=0;

RE1:= KxCq Sq_x,_x f x C xC2 q ² f x = 0 r2:=(x-q)^2/(x-2);

r2:= xKq ² xK2

RE2:=denom(r2)*qshift(f(x),[x$m],q)-numer(r2)*f(x)=0;

RE2:= xK2 Sq_x,_x f x K KxCq ² f x = 0 RE:=qNormal(qLCM(RE2,RE1,f(x)),f(x)):

1,K1 , 1

q² ,K1 , K4 q, 0 , K1

2 q², 0 , q⁷Cq⁴ xK8 q⁶C7 q x²K8 q³ xK2 x²

7 qK2 , K1, 1 , xK2, K1, 0 , xKq,

K1, 2 , xC2 q, 0, 2 st:=time():

time()-st;

KxCq ²

xK2 ,K xC2 q ² KxCq 1.964

st:=time():

time()-st;

KxCq ²

st:=time():

time()-st;

KxCq ²

Example 4

Computation of all m-fold q-hypergeometric solutions of a linear q-recurrence equation m1:=2;

m1:= 2 r1:=sqrt(2)*q*(q-1)*x/(x-1);

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r1:= 2 q qK1 x xK1

RE1:=denom(r1)*qshift(f(x),[x$m1],q)-numer(r1)*f(x)=0;

RE1:= xK1 Sq_x,_x f x K 2 q qK1 x f x = 0 m2:=3;

m2:= 3 r2:=(q-1)*(x-3)^2/(x-q);

r2:= qK1 xK3 ² xKq

RE2:=denom(r2)*qshift(f(x),[x$m2],q)-numer(r2)*f(x)=0;

RE2:= KxCq Sq_x,_x,_x f x C qK1 xK3 ² f x = 0 RE:=qNormal(qLCM(RE2,RE1,f(x)),f(x)):

st:=time():

qHypergeomSolveRE(RE,f(x),mhypersol=0);

time()-st;

2, 2 q qK1 x

xK1 , 3, K qK1 xK3 ² KxCq 57.867

Example 5

Computation of q-hypergeometric solutions of a linear q-recurrence equation showing additionally Newton polygon

r1:=q*(q-1)*x/(x-1);

r1:= q qK1 x xK1

L1:=denom(r1)*qshift(f(x),x,q)-numer(r1)*f(x);

L1:= xK1 Sq_x f x Kq qK1 x f x

r2:=q/(x-q)^2;

r2:= q xKq ²

L2:=denom(r2)*qshift(f(x),x,q)-numer(r2)*f(x);

L2:= KxCq ² Sq_x f x Kq f x L:=qNormal(qLCM(L1,L2,f(x)),f(x));

L:=q qK1 x K2 q⁴ x²Cq⁴ xCq⁴ x³C2 q³ x²Kq³ xKq³ x³Kq xC1 f x K x K1 1K3 q⁵ x³Kq xCq⁵ x⁵K2 q⁴ x⁵Cq³ x⁵C3 q⁵ x⁴Cx³ q⁷Kq⁷ x²

C2 q⁴ x³K2 q⁶ x⁴Kq⁵ x²C2 x² q⁶Kx⁴ q³ Sq_x f x C xK1 ² q q xK1 1 KxKq² xCq³ xC2 q x²K2 q² x²Kx³Cx³ q Sq_x,_x f x = 0

qPlotNewtonPolygon(L,f(x),q,v=((a,x)->ldegree(a,x)));

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0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1

qPlotNewtonPolygon(L,f(x),q,v=((a,x)->-degree(a,x)));

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0 0.5 1 1.5 2

K6 K5.5 K5 K4.5 K4

st:=time():

qHypergeomSolveRE(L,f(x),method=qPetkovsek);

time()-st;

q

KxCq ² , q qK1 x xK1 0.237 st:=time():

qHypergeomSolveRE(L,f(x),method=qVanHoeij);

time()-st;

q

KxCq ² , q qK1 x xK1 0.257 st:=time():

qHypergeomSolveRE(L,f(x),method=modqPetkovsek);

time()-st;

q

KxCq ² , q qK1 x xK1 0.055

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Example 6 (Timings)

J:=8:

Example Operator (3.2)

L1:=mul(x+i*q^i,i=1..floor((J-1)/2))*qshift(F(x),[x$2],q)+mul (x-i*q^i,i=1..floor(J/2))*F(x):

L1:=qNormal(qMultiplyRE(L1,qshift(F(x),[x],q)-x*F(x),F(x)),F (x));

L1:= xCq xC2 q² xC3 q³ Sq_x,_x,_x F x K xCq xC2 q² x C3 q³ q² x Sq_x,_x F x C KxCq KxC2 q² KxC3 q³ Kx

C4 q⁴ Sq_x F x K KxCq KxC2 q² KxC3 q³ KxC4 q⁴ x F x

= 0

st:=time():

qHypergeomSolveRE(L1,F(x),method=qPetkovsek);

time()-st;

x 2.573 st:=time():

qHypergeomSolveRE(L1,F(x),method=qVanHoeij);

time()-st;

x 1.106 st:=time():

qHypergeomSolveRE(L1,F(x),method=modqPetkovsek);

time()-st;

x 0.128

Example Operator (3.3)

L2:=mul(x-q^i,i=1..floor((J-1)/2))*qshift(F(x),[x$2],q)+mul(x- i*q^i,i=1..floor(J/2))*F(x):

L2:=qNormal(qMultiplyRE(L2,qshift(F(x),[x],q)-x*F(x),F(x)),F (x));

L2:=x KxC3 q³ KxC4 q⁴ KxC2 q² F x K KxC3 q³ KxC4 q⁴ Kx C2 q² Sq_x F x Kq² x KxCq² KxCq³ Sq_x,_x F x C KxCq² Kx

Cq³ Sq_x,_x,_x F x = 0 st:=time():

qHypergeomSolveRE(L2,F(x),method=qPetkovsek);

time()-st;

x 0.583 st:=time():

qHypergeomSolveRE(L2,F(x),method=qVanHoeij);

time()-st;

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x 0.314 st:=time():

qHypergeomSolveRE(L2,F(x),method=modqPetkovsek);

time()-st;

x 0.054

Example 7 (Example of 2 and 3-fold q-hypergeometric solutions)

m1:=2;

m1:= 2 cert1:=q*(q-1)*x/(x-1);

cert1:= q qK1 x xK1

RE1:=denom(cert1)*qshift(f(x),[x$m1],q)-numer(cert1)*f(x)=0;

RE1:= xK1 Sq_x,_x f x Kq qK1 x f x = 0 m2:=3;

m2:= 3 cert2:=(q-1)*(x-3)^2/(x-q);

cert2:= qK1 xK3 ² xKq

RE2:=denom(cert2)*qshift(f(x),[x$m2],q)-numer(cert2)*f(x)=0;

RE2:= KxCq Sq_x,_x,_x f x C qK1 xK3 ² f x = 0 RE:=qNormal(qLCM(RE2,RE1,f(x)),f(x)):

qOrder(RE,f(x));

5 st:=time():

qHypergeomSolveRE(RE,f(x),method=qVanHoeij,mhypersol=2);

time()-st;

q qK1 x xK1 9.878 st:=time():

qHypergeomSolveRE(RE,f(x),method=qPetkovsek,mhypersol=2);

time()-st;

q qK1 x xK1 2.128 st:=time():

qHypergeomSolveRE(RE,f(x),method=modqPetkovsek,mhypersol=2);

time()-st;

q qK1 x xK1

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(3.2) (3.2)

0.241 st:=time():

qHypergeomSolveRE(RE,f(x),method=qVanHoeij,mhypersol=3);

time()-st;

K xK3 ² qK1 KxCq 17.025 st:=time():

qHypergeomSolveRE(RE,f(x),method=qPetkovsek,mhypersol=3);

time()-st;

qHypergeomSolveRE(RE,f(x),method=modqPetkovsek,mhypersol=3);

time()-st;

qHypergeomSolveRE(RE,f(x),mhypersol=0);

time()-st;

2, q qK1 x

xK1 , 3, K xK3 ² qK1 KxCq 2.020

Example 8 (Example of computation of 2-fold q-

hypergeometric series expansion of small q-sine function)

Computation of series cofficients of small q-sine function qDE:=qHolonomicDE(qsin(y,q),F(y));

qDE:=F y C qK1 ² Dq_y,_y F y = 0

RE:=qDEtoRE(qDE,F(y),c(j),base=qpower,expansionpt=a);

RE:=c j Cq a q^j qC1 c jC1 C Kq^j qKq^j q²C q^j ² q³Cq⁴ q^j ² a² C1 c jC2 Cq a q^j qC1 q^j q³K1 q^j q²K1 c jC3

Cq⁴ q^j ² a² q^j q⁴K1 q^j q³K1 c jC4 = 0 qHypergeomSolveRE(RE,c(j),mhypersol=2);

A25₁ K1 ^j

qpochhammer q,q, 2 j , A25₂ qK1 K1 ^j

q^j ² qK1 qpochhammer q,q, 2 j