Differential Independence of Solutions of a Class of q-Hypergeometric Difference Equations
Charlotte Hardouin, Michael F. Singer REVISED November 6, 2012
This note is a companion to the paper "Differential Galois Theory of Difference Equations" by the same authors. In a previous version of this note (and in Example 3.14 of this paper) we claimed that for the q-hypergeometric equation with prarameters restricted as below, the associated \317\203\316\264-PV extension has differential transcendence degree 3. We have since founds gaps in the proof of this claim. Nonetheless, we are able to show this claim when we further restrict the paprameters. In the following we show how MAPLE can be used to verify these cases. We shall consider the equations y(q2 x) -[(a+b)x-(1+c/q))/(abx -c/q)]y(qx) + [(x-1)/ 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0YrLUYwNiZGMkY1RjgvRjtRJWF1dG9GJ0YrLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== where a = b, c=q, a\342\210\211 q\342\204\244 , a2 \342\210\210 q\342\204\244 . We shall show in the cases when 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 MAPLE code we give below can be easily altered to test for other values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEiYUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYoLUkjbW5HRiQ2JVEiMkYnRjJGOEYyL0Y2USV0cnVlRicvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0ZCL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUSIuRidGMkY4LyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGW28tRk42LlEifkYnRjJGOEZRRlNGVUZXRllGZW5GZ25GaW5GXG9GXm8tRi82JlEjV2VGJ0YyRjVGOEZeby1GLzYmUShzdXNwZWN0RidGMkY1RjhGXm8tRi82JlEldGhhdEYnRjJGNUY4Rl5vLUYvNiZRJXRoaXNGJ0YyRjVGOEZeby1GLzYmUSdyZXN1bHRGJ0YyRjVGOEZeby1GLzYmUSNpc0YnRjJGNUY4Rl5vRjJGOA== true in general but, as we said above, we do not have a proof. In the paper "Differential Galois Theory of Difference Equations", we showed that for a difference equation \317\203(Y) = AY over C(x) with difference Galois group SL n (C), the differential transcendence degree is less than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JlEibkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYoLUkjbW5HRiQ2JVEiMkYnRjJGOEYyL0Y2USV0cnVlRicvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0ZCL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMkY4 -1 if and only if there exists a matrix B in gln (C(x)) such that \317\203(B) = AB A-1 + \316\264A A-1 , where \316\264=xd/dx. This latter equation is an n 2 x n2 system in the entries of B. In section 1, we shall derive this system for the above family of q-hypergeometric equations. In section 2, we shall derive a scalar equation for one of the entries of B.
1. The equation for B. The matrix equation associated with our scalar equation above is \317\203(Y) = AY. We calculate A below
with(LinearAlgebra):
MM:=simplify(Matrix([[0,1],[ (1-x)/(a*b*x - c/q),((a+b)*x -(1+c/q))/(a*b*x-c/q)]]));
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
M1:=subs(c=q,MM);
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
A:=simplify(subs(b=a,M1));
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
We now calculate the system associated to \317\203(B) = ABA-1 + \316\264A A-1
Bt:=Matrix([[u,v],[w,z]]);
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
B:=Matrix([[Bt[1,1]],[Bt[1,2]],[Bt[2,1]],[Bt[2,2]]]);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNigtRiM2Ji1JJ210YWJsZUdGJDY4LUkkbXRyR0YkNiYtSSRtdGRHRiQ2KC1GLDYlUSJ1RidGL0YyLyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0Zfby8lK2dyb3VwYWxpZ25HRl9vLyUocm93c3BhbkdRIjFGJy8lK2NvbHVtbnNwYW5HRmZvRl1vRmBvRmJvLUZlbjYmLUZobjYoLUYsNiVRInZGJ0YvRjJGXW9GYG9GYm9GZG9GZ29GXW9GYG9GYm8tRmVuNiYtRmhuNigtRiw2JVEid0YnRi9GMkZdb0Zgb0Zib0Zkb0Znb0Zdb0Zgb0Ziby1GZW42Ji1GaG42KC1GLDYlUSJ6RidGL0YyRl1vRmBvRmJvRmRvRmdvRl1vRmBvRmJvLyUmYWxpZ25HUSVheGlzRicvRl5vUSliYXNlbGluZUYnL0Zhb1EnY2VudGVyRicvRmNvUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGMS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRltyLyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmZyLyUmZnJhbWVHRmZyLyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGPS8lLWVxdWFsY29sdW1uc0dGPS8lLWRpc3BsYXlzdHlsZUdGPS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0Zjci8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5RjkvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZedC8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRicvJSlydGFibGVpZEdRKzQzMjc4MzU3ODRGJ0Zpc0ZcdEY5
Ct:= A.Bt.A^(-1);
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
C:=Matrix([[coeff(Ct[1,1],u),coeff(Ct[1,1],v),coeff(Ct[1,1],w),coeff(Ct[1,1],z)],
[coeff(Ct[1,2],u),coeff(Ct[1,2],v),coeff(Ct[1,2],w),coeff(Ct[1,2],z)],
[coeff(Ct[2,1],u),coeff(Ct[2,1],v),coeff(Ct[2,1],w),coeff(Ct[2,1],z)],
[coeff(Ct[2,2],u),coeff(Ct[2,2],v),coeff(Ct[2,2],w),coeff(Ct[2,2],z)]]);
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
ddA:= map(diff,A,x);
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
XX:=Matrix([[x,0],[0,x]]);
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
dA:= simplify(Multiply(XX,ddA));
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
Et:= dA.A^(-1);
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
E:= simplify(Matrix([[Et[1,1]],[Et[1,2]],[Et[2,1]],[Et[2,2]]]));
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
The associated system for B (now written as a colum vector as above) is \317\203(B) = CB+E, where C and E are as above.
2. The scalar equation for v. To compute a scalar equation for the component v of the vector B, we proceed as follows. We will compute matrices Ci and Ei such that \317\203 i (B) = Ci B + Ei . Let ci denote the entry in second row of Ci and ei denote the entry of the second row of Ei . We will find elements y i such that \316\243 y i c i = 0. We then will have the scalar equation \316\243 y i \317\203 i (v) = \316\243 y i e i . Note that if e = (0,1,0,0), then for any matrix M, eM is the second row of M.
e:=Matrix([[0,1,0,0]]);
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
s1C:=subs(x=q*x,C);
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
s2C:=subs(x=q*x,s1C);
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
s3C:=subs(x=q*x,s2C);
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
s1E:=subs(x=q*x,E);
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
s2E:=subs(x=q*x,s1E);
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
Since C0 = I, we have
c0:=e:
Since C1 = C, we have
c1:= Multiply(e,C):
Since C2 = \317\203(C)C, we have
c2:=simplify(Multiply(e,Multiply(s1C,C))):
Since C2 = \317\203 2 (C)\317\203(C)C, we have
c3:= simplify(Multiply(e,Multiply(s2C,Multiply(s1C,C)))):
A priori, there will be a dependence among e0,e1,e2,e3,e4 but in this case there is actually a dependence among e0,e1,e2,e3. To find such a dependence let
MM:=Transpose(Matrix([[c0],[c1],[c2],[c3]])):
The dependence will be given by the following element of the nullspace of this matrix.
VV:=(Transpose(op(1,NullSpace(MM))));
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEjVlZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWFjdGlvbkdGJDYlLUkobWZlbmNlZEdGJDYoLUknbXRhYmxlR0YkNjUtSSRtdHJHRiQ2KS1JJG10ZEdGJDYoLUYjNictRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl5vLUkmbWZyYWNHRiQ2KC1GIzYqLUZTNiQtRiM2J0Zqbi1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGXW9GX28tRiw2JVEieEYnRi9GMkY5RjktRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZncC1GUzYkLUYjNiYtRiM2KC1JJW1zdXBHRiQ2JS1GLDYlUSJhRidGL0YyLUZqbzYkUSIyRidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGY3BGYHBGY3AtRmBxNiUtRiw2JVEicUYnRi9GMkZlcUZocUY5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGXW9GX29GaW9GOUY5RmNwLUZTNiQtRiM2Ji1GIzYoRmBwRmNwRl1yRmNwRmJxRjlGYHJGaW9GOUY5LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy8lKXJlYWRvbmx5R0Y9RjktRiM2Ki1GUzYkLUYjNiYtRiM2JkZgcEZjcEZicUY5RmByRmlvRjlGOUZjcC1GUzYkLUYjNiYtRiM2KEZfcUZjcEZgcEZjcEZdckY5RmByRmlvRjlGOUZjcC1GUzYkLUYjNidGam5GaW9GXXAtRiM2JkZgcEZjcEZbckY5RjlGOUZpckZcc0Y5LyUubGluZXRoaWNrbmVzc0dGXHAvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZnQvJSliZXZlbGxlZEdGPUZpckZcc0Y5LyUpcm93YWxpZ25HUSFGJy8lLGNvbHVtbmFsaWduR0ZddS8lK2dyb3VwYWxpZ25HRl11LyUocm93c3BhbkdGXHAvJStjb2x1bW5zcGFuR0ZccC1GZm42KC1GYW82KC1GIzYoLUZTNiQtRiM2Li1GIzYmRl9xRmNwRmBwRjlGXXAtRmpvNiRRIjNGJ0Y5Rl1wLUYjNipGYnZGY3AtRmBxNiVGYHBGZXFGaHFGY3BGXXJGY3BGX3FGOUZdcC1GIzYmRmBwRmNwRl1yRjlGYHItRiM2Ki1Gam82JFEiNEYnRjlGY3BGYHBGY3BGXXJGY3BGYnFGOUZgci1GIzYoRl13RmNwRmBwRmNwRmJxRjlGOUY5RmNwRmlwRmlyRlxzRjktRiM2Ki1GUzYkLUYjNiZGYHZGYHJGaW9GOUY5RmNwRmZzRmNwRlx0RmlyRlxzRjlGYnRGZHRGZ3RGaXRGW3VGXnVGYHVGYnVGZHUtRmZuNigtRiM2J0Zqbi1GYW82KC1GIzYqRmlwRmNwRmNyRmNwRlx2RmlyRlxzRjktRiM2KkZcdEZjcC1GYHE2JUZmc0ZlcUZocUZjcEZgc0ZpckZcc0Y5RmJ0RmR0Rmd0Rml0RmlyRlxzRjlGW3VGXnVGYHVGYnVGZHUtRmZuNihGaW9GW3VGXnVGYHVGYnVGZHVGW3VGXnVGYHUvJSZhbGlnbkdRJWF4aXNGJy9GXHVRKWJhc2VsaW5lRicvRl91RmZ0L0ZhdVEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZieS8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zdei8lJmZyYW1lR0Zdei8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj0vJS1lcXVhbGNvbHVtbnNHRj0vJS1kaXNwbGF5c3R5bGVHRj0vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGanlGOS9JK21zZW1hbnRpY3NHRiRRKlJvd1ZlY3RvckYnLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmBbbC8lK2FjdGlvbnR5cGVHUS5ydGFibGVhZGRyZXNzRicvJSlydGFibGVpZEdRKzQzNzg3OTE5NjBGJ0ZpckZcc0Y5
Letting yi-1 be the ith entry of this vector, we have that \317\203 3 (v ) + y2 \317\203 2 (v ) + y1 \317\203 (v ) +y0 v = e3 + y2 e2 + y1 e1 +y0 e0 . Clearly e0 =e1 = 0. A calculation shows that E2 =\317\203(C)E + \317\203(E) and E2 =\317\203 2 (C)\317\203(C)E + \317\203 2 (C)\317\203(E) + \317\203 2 (E) so
e0:=0;
IiIh
e1:=(e.E)[1,1];
IiIh
e2:=(Multiply(e,Multiply(s1C,E)+s1E))[1,1];
LCQqMCIiIyIiIiwmRiUhIiIqJkkieEc2IkYlSSJxR0YqRiVGJUYnLCYqKClJImFHRipGJEYlRilGJUYrRiVGJUYlRidGJSwmRiVGJ0YvRiVGJUYpRiUsJiomRi5GJUYpRiVGJUYlRidGJywmRiVGJ0YpRiVGJ0Yl
e3:=Multiply(e,(Multiply(s2C,Multiply(s1C,E))+Multiply(s2C,s1E)+s2E))[1,1];
LCYqKCwmIiIiISIiKiZJInhHNiJGJSlJInFHRikiIiNGJUYlRiYsJiooKUkiYUdGKUYsRiVGKEYlRipGJUYlRiVGJkYlLCYqMiIiKUYlKSwmKihGKEYlRitGJUYwRiVGJUYlRiZGLEYlLCYqKEYvRiVGKEYlRitGJUYlRiVGJkYmLCZGJUYmKiZGKEYlRitGJUYlRiYsJkYlRiZGMEYlRiVGKEYlLCYqJkYvRiVGKEYlRiVGJUYmRiYsJkYlRiZGKEYlRiZGJiowRixGJUY1RiVGN0YmRihGJSwmRiVGJiokRi9GJUYlRiVGPEYmRj5GJkYlRiVGJioyRixGJUYkRiZGLUYlRjtGJUYoRiVGK0YlRjdGJkY5RiZGJQ==
EE:=Matrix([[e0],[e1],[e2],[e3]]);
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
The following WWW will be the left hand side of \317\203 3 (v ) + y2 \317\203 2 (v ) + y1 \317\203 (v ) +y0 v = e3 + y2 e2 + y1 e1 +y0 e0 and the entries of VV will be the coefficients of the the right hand side.
WWW:=simplify(VV.EE);
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
VV;
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
We now lcear the denominators and produce the third order equation EQN for v(x).
W:=denom(WWW[1])*(a^2*x*q-1)*WWW;
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
V:= denom(WWW[1])*(a^2*x*q-1)*VV;
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
y:= Matrix([[v(x),v(q*x),v(q^2*x),v(q^3*x)]]);
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
EQN:= (V.Transpose(y))[1] = W[1];
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
We wish to show that this equation has no nonzero rational solutions. In order to use MAPLE taking into account that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y/L0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== is a power of q. To do this we will replace q by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Jy1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y/L0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGNQ== and consider a sixth order equation. This avoids haing to deal with the square root of q when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2KC1JI21uR0YkNiRRIjJGJ0Y1L0YzUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y/LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnL0Y2USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGRUY1 is an odd power of q. We then apply the MAPLE command RationalSolution. This command will return an empty set when there are no rational solutions.
EQN0:=subs(q=t,EQN);
LywqKjAsJiIiIiEiIiomSSJ4RzYiRiZJInRHRipGJkYmRiYsJiooKUkiYUdGKiIiI0YmRilGJkYrRiZGJkYmRidGJiwmKiZGLkYmRilGJkYmRiZGJ0YmLCZGJkYnRilGJkYmLCYqKEYuRiZGKUYmKUYrRjBGJkYmRiZGJ0YmLCYqKEYpRiZGK0YmRi9GJkYmRiZGJ0YmLUkidkdGKjYjRilGJkYnKi5GJUYmRixGJiwmKiZGKUYmRi9GJkYmRiZGJ0YmLC5GMkYmIiIkRiYqKkZARiYpRilGMEYmRitGJkYuRiZGJkYoRiYqKiIiJUYmRilGJkYrRiZGL0YmRicqKEZERiZGKUYmRi9GJkYnRiZGNEYmLUY6NiNGKEYmRiYqLkYlRiZGMUYmRjRGJkY3RiZGP0YmLUY6NiMqJkYpRiZGNkYmRiZGJyouRiVGJilGLEYwRiZGMUYmLCZGJkYnRktGJkYmRj1GJi1GOjYjKiZGKUYmKUYrRkBGJkYmRiYsJCosRjBGJkYsRiYsMiooRkJGJkY2RiZGLkYmRiYqKEZCRiZGK0YmKUYvRkRGJkYmKihGQkYmRjZGJilGL0ZARiZGJyooRkJGJkYrRiZGWkYmRidGL0YmKiRGLkYmRidGK0YnKiZGK0YmRi9GJkYmRiZGNEYmRilGJkYm
EQN1:=subs(a=q^1,t=q^2,EQN0);
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
with(QDifferenceEquations):
The MAPLE command RationalSolution will yield a basis of the rational solutions of a q-difference equation. When the output is empty, we can conclude that there are no nonzero solutions.
sol1 := RationalSolution(EQN1, v(x), {}, output = basis[_K]);
NiI=
EQN3:=subs(a=q^3,t=q^2,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQN3, v(x), {}, output = basis[_K]);
NiI=
EQN5:=subs(a=q^5,t=q^2,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQN5, v(x), {}, output = basis[_K]);
NiI=
EQNminus1:=subs(a=q^(-1),t=q^2,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQNminus1, v(x), {}, output = basis[_K]);
NiI=
EQNminus3:=subs(a=q^(-3),t=q^2,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQNminus3, v(x), {}, output = basis[_K]);
NiI=
EQNminus5:=subs(a=q^(-5),t=q^2,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQNminus5, v(x), {}, output = basis[_K]);
NiI=
EQN2:=subs(a=-q,t=q,EQN0):
with(QDifferenceEquations):
sol1 := RationalSolution(EQN2, v(x), {}, output = basis[_K]);
NiI=