Differential Independence of Solutions of a Class of q-Hypergeometric Difference Equations
Charlotte Hardouin, Michael F. Singer
This note is a companion to the paper "Differential Galois Theory of Difference Equations" by the same authors. In particular, we ashow how simple Maple commands can be used to verify a claim in Example 3.14 of that paper. We will show that for a certain class of q-hypergeometric difference equations, the associated \317\203\316\264-PV extension has differential transcendence degree 3. In particular 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 . 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 n 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 . This latter equation is an n2 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 and show that this scalar equation has no rational solution. We note the maple commands can be easily modified to carry out the other claims in Example 3.13 and 3.14 of our paper.
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],[((a+b)*x -(1+c/q))/(a*b*x-c/q), (1-x)/(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]]]);
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
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 second row of Ci and ei denote 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);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEkczFDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1hY3Rpb25HRiQ2JS1JKG1mZW5jZWRHRiQ2KC1GIzYmLUknbXRhYmxlR0YkNjgtSSRtdHJHRiQ2KS1JJG10ZEdGJDYoLUkjbW5HRiQ2JFEiMEYnRjkvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmBvLyUrZ3JvdXBhbGlnbkdGYG8vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGZ29GZ24tRmhuNigtRiM2KC1JJm1mcmFjR0YkNigtRltvNiRRIjFGJ0Y5LUZbbzYkUSIyRidGOS8lLmxpbmV0aGlja25lc3NHRmdvLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRltxLyUpYmV2ZWxsZWRHRj0tRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZkcS1GX3A2KC1GIzYoLUZTNiQtRiM2KS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYnJGYXAtRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSEZhckZjci1GIzYoLUYsNiVRInhGJ0YvRjJGYHEtRiw2JVEicUYnRi9GMi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGPUY5Rl9zRmJzRjlGOUZgcS1GUzYkLUYjNigtRiM2Ki1JJW1zdXBHRiQ2JS1GLDYlUSJhRidGL0YyRmRwLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZgcUZpckZgcUZcc0Zfc0Zic0Y5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkhGYXJGY3JGYXBGX3NGYnNGOUY5Rl9zRmJzRjktRiM2KC1GUzYkLUYjNihGaHNGY3QtRiM2Ji1GW3Q2JUZcc0ZkcEZgdEZfc0Zic0Y5Rl9zRmJzRjlGOUZgcS1GUzYkLUYjNigtRiM2KkZpckZgcUZcc0ZgcUZddEZfc0Zic0Y5RmN0RmFwRl9zRmJzRjlGOUZfc0Zic0Y5RmdwRmlwRlxxRl5xRl9zRmJzRjlGXm9GYW9GY29GZW9GaG8tRmhuNihGYXBGXm9GYW9GY29GZW9GaG9GXm9GYW9GY28tRmVuNilGZ25GZ24tRmhuNigtRiM2KEZecEZgcS1GX3A2KC1GIzYmRmZzRl9zRmJzRjktRiM2JkZidUZfc0Zic0Y5RmdwRmlwRlxxRl5xRl9zRmJzRjlGXm9GYW9GY29GZW9GaG9GZ25GXm9GYW9GY28tRmVuNiktRmhuNigtRl9wNigtRiM2JkZcckZfc0Zic0Y5LUYjNiZGanRGX3NGYnNGOUZncEZpcEZccUZecUZeb0Zhb0Zjb0Zlb0Zoby1GaG42KC1GX3A2KC1GIzYoRmRwRmBxRmB1Rl9zRmJzRjlGYHZGZ3BGaXBGXHFGXnFGXm9GYW9GY29GZW9GaG8tRmhuNigtRiM2J0Zeci1GIzYoRl5wRmBxLUZfcDYoLUYjNigtRlt0NiVGanFGZHBGYHRGYHFGZHNGX3NGYnNGOS1GIzYoLUZbdDYlRmh0RmRwRmB0RmBxRmB1Rl9zRmJzRjlGZ3BGaXBGXHFGXnFGX3NGYnNGOUZfc0Zic0Y5Rl5vRmFvRmNvRmVvRmhvLUZobjYoLUYjNidGXnJGaHZGX3NGYnNGOUZeb0Zhb0Zjb0Zlb0Zob0Zeb0Zhb0Zjby1GZW42KUZmdUZnbi1GaG42KC1GIzYnRl5yRlxwRl9zRmJzRjlGXm9GYW9GY29GZW9GaG9GZ25GXm9GYW9GY28vJSZhbGlnbkdRJWF4aXNGJy9GX29RKWJhc2VsaW5lRicvRmJvRltxL0Zkb1EnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0ZqeS8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0Zlei8lJmZyYW1lR0Zlei8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj0vJS1lcXVhbGNvbHVtbnNHRj0vJS1kaXNwbGF5c3R5bGVHRj0vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGYnpGX3NGYnNGOUY5L0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGaFtsLyUrYWN0aW9udHlwZUdRLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EoMzE3NDQ0OEYnRl9zRmJzRjk=
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 an element of the nullspace of this matrix
VV:=(Transpose(op(1,NullSpace(MM))));
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
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];
LCQqMiMiIiIiIiNGJSwmKihJInhHNiJGJUkicUdGKkYlSSJhR0YqRiVGJUYlISIiRi0sJiooKUYsRiZGJUYpRiVGK0YlRiVGJUYtRiVGKUYlLDYqKEYmRiUpRiwiIiRGJUYpRiVGLSomRixGJSlGK0YmRiVGJSomKUYpRiZGJUYzRiVGJSomRjBGJUY2RiVGLSomRjhGJSlGLCIiJUYlRi0qKkYmRiVGKUYlRjBGJUY2RiVGJSooRjhGJUYzRiVGNkYlRi0qJEY2RiVGLSomKUYsIiImRiVGOEYlRiUqJEYwRiVGJUYlLCYqJkYwRiVGKUYlRiVGJUYtRi0pLCZGRUYlRj9GLUYmRi0sJiomRilGJUYsRiVGJUYlRi1GLUYt
e3:=Multiply(e,(Multiply(s2C,Multiply(s1C,E))+Multiply(s2C,s1E)+s2E))[1,1];
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
EE:=Matrix([[e0],[e1],[e2],[e3]]);
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
WW:=simplify(VV.EE);
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
VV;
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
The entry of WW is the right 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 while the entries of VV are y0 ,y1 , y2 , and y3 =1 respectively. We claim that under our assumpions, x=1/qa is a pole of WW. To see this we must show that 1/qa is not a zero of the numerator of WW. Substituting, we have
factor(subs(x=1/(q*a), numer(WW[1])));
Ki4pLCYiIiIhIiJJImFHNiJGJSIiI0YlKSwmKiZGJ0YlSSJxR0YoRiVGJUYlRiZGKUYlLCZGJ0YlKiQpRi1GKUYlRiZGJSwmRidGJSokKUYtIiIkRiVGJkYlRjNGJilGJ0YpRiY=
Since a\342\210\211 q\342\204\244 , this latter expression is not 0. Note that our assumptions also imply that 1/qa is not a pole of any of the yi . Therefore 1/qa must be a pole of \317\203 3 ( v ) ,\317\203 2 (v ) ,\317\203 ( v ) or v for any putative solution of \317\203 3 ( v ) + y2 \317\203 2 (v ) + y1 \317\203 ( v ) +y0 v = e3 + y2 e2 + y1 e1 +y0 e0 . Before we proceed, we make two definitions. If R(x) is a rational function, we say that \316\261 is a maximal pole of R(x) if \316\261 is a pole of R but \316\261/q n is not a pole of R for any n>0 and \316\262 is a minimal pole of R(x) if \316\262 is a pole but \316\262q n is not a pole of R for any n>0. We shall now show that the assumption that 1/qa is a pole of some \317\203 i ( v ) for i=0,1,2,3, leads to a contradiction. 1/qa is a pole of v: In this case, v has a maximal pole of the form 1/(qn a) for some n \342\211\245 1. This implies that \317\203 3 ( v ) has a maximal pole of the form 1/(qn+3 a) and this can be seen to not cancel any pole in any other term of \317\203 3 ( v ) + y2 \317\203 2 (v ) + y1 \317\203 ( v ) +y0 v = e3 + y2 e2 + y1 e1 +y0 e0 , a contradiction. 1/qa is a pole of \317\203 ( v ) : In this case, \317\203 ( v ) has a maximal pole of the form 1/(qn a) for some n \342\211\245 1. This implies that \317\203 3 ( v ) has a maximal pole of the form 1/(qn+2 a,) which again leads to a contradiction. 1/qa is a pole of \317\203 2 ( v ) : In this case, \317\203 2 ( v ) has a minimal pole of the form qn / a for some n \342\211\245 -1 , so v has a minimal pole of the form qn+2 /a for some n\342\211\245-1. From the formula for y0 , one sees that y0 v also has a pole of this form and this can be seen to not cancel with a pole of any other term in \317\203 3 ( v ) + y2 \317\203 2 (v ) + y1 \317\203 ( v ) +y0 v = e3 + y2 e2 + y1 e1 +y0 e 0 . 1/qa is a pole of \317\203 3 ( v ) : In this case has a minimal poe of the form \317\203 3 ( v ) qn / a for some n \342\211\245 -1 , so v has a minimal pole of the form qn+3 /a for some n\342\211\245-1 and this again leads toa contradiction as above.