Notebook 4 in connection with the paper
"The Askey scheme as a four-manifold with corners",
Ramanujan J. (2009), DOI 10.1007/s11139-009-9208-7;
arXiv:0909.2822v2 [math.CA],
by Tom H. Koornwinder, email T.H.Koornwinder [at] uva.nl

This gives the computations for section 7.4 (Transformations between the local charts).

Last modified: 15 Sep 2009

Racah:

{α->(t1^(-1)),β->(t2^(-1)) (t1^(-1)),nn->(t2^(-1)) (t4^(-1)),δ->((t2^(-1)) (t3^(-1)) (t4^(-1))+1)(t1^(-1))}

{α->(s2^(-1))((s1^(-1))+1),β->(s2^(-2))((s1^(-1))+1)(s4^(-1)) (s3^(-1)),nn->(s2^(-2))(s4^(-1)),δ->(s2^(-1))((s2^(-1)) (s4^(-1))+1)((s1^(-1))+1)+1}

{α->(u2^(-1)) (1+u1),β->(u2^(-2)) (u1^(-1)) (u3^(-2))(u4^(-1)),nn->(u2^(-2))(u3^(-1))(u4^(-1)),δ->(u2^(-1))+(u2^(-2))(u3^(-1))((u4^(-1))+1)+1}

Solve[{(t1^(-1))==(s2^(-1))((s1^(-1))+1),(t2^(-1)) (t1^(-1))==(s2^(-2))((s1^(-1))+1)(s4^(-1)) (s3^(-1)), (t2^(-1)) (t4^(-1))==(s2^(-2))(s4^(-1)),((t2^(-1)) (t3^(-1)) (t4^(-1))+1)(t1^(-1))==(s2^(-1))((s2^(-1)) (s4^(-1))+1)((s1^(-1))+1)+1},{t1,t2,t3,t4}]

Solve[{(t1^(-1))==(s2^(-1))((s1^(-1))+1),(t2^(-1)) (t1^(-1))==(s2^(-2))((s1^(-1))+1)(s4^(-1)) (s3^(-1)), (t2^(-1)) (t4^(-1))==(s2^(-2))(s4^(-1)),((t2^(-1)) (t3^(-1)) (t4^(-1))+1)(t1^(-1))==(s2^(-1))((s2^(-1)) (s4^(-1))+1)((s1^(-1))+1)+1},{s1,s2,s3,s4}]

Solve[{(u2^(-1)) (1+u1)==(s2^(-1))((s1^(-1))+1),(u2^(-2)) (u1^(-1)) (u3^(-2))(u4^(-1))==(s2^(-2))((s1^(-1))+1)(s4^(-1)) (s3^(-1)), (u2^(-2))(u3^(-1))(u4^(-1))==(s2^(-2))(s4^(-1)),(u2^(-1))+(u2^(-2))(u3^(-1))((u4^(-1))+1)+1==(s2^(-1))((s2^(-1)) (s4^(-1))+1)((s1^(-1))+1)+1},{s1,s2,s3,s4}]

racah1to2={t1->((s1 s2)/(1+s1)),t2->s2 s3 s4,t3->((1+s1)/(s2 (1+s1+s1 (s2^2) s4))),t4->(s2/s3)};
racah2to1={s1->((t1 t3)/(1-t1 t3(1+t2 t4))),s2->(1/t3)-t1 t2 t4,s3->-t1 t2+(1/(t3 t4)),s4->((t2 (t3^2) t4)/(((1-t1 t2 t3 t4))^2))};
racah2to3={s1->(1/(u4(1-u1 u2 u3))),s2->((u2(1+u4-u1 u2 u3 u4))/(1+u1)),
s3->u1 u3 (1+u4-u1 u2 u3 u4),s4->(((((1+u1))^2) u3 u4)/(((1+u4-u1 u2 u3 u4))^2))};
ss=(s4^2)(((1+2s1 +(s1^2)-(s1^2) s2 s3))^2)-4 (s1^2) s3 s4(1+s1);
racah3to2={
u1->(( -2(s1^2)s3+s4(1+s1)(1+2 s1+(s1^2)-(s1^2) s2 s3)-(1+s1)(ss^(1/2)))/(2 (s1^2) s3 (1+s2 s4+s1 s2 s4))),u2->(( s1 (s2^2) s4)/(1+s2 s4+s1 s2 s4))+s2(( s4(1+2 s1+(s1^2)-(s1^2) s2 s3)-(ss^(1/2)))/(2 s1 s3 (1+s2 s4+s1 s2 s4))),u3->(( -2(s1^2)s3+s4(1+s1)(1+2 s1+(s1^2)-(s1^2) s2 s3)+(1+s1)(ss^(1/2)))/(2 s1(1+s1))),u4->(1/s1)+s2((s4(1+2 s1+(s1^2)-(s1^2) s2 s3)-(ss^(1/2)))/(2 s1 (1+s1)))};
racah1to3={t1->(u2/(1+u1)),t2->u1 (1+u1) u2 (u3^2) u4,
t3->((1+u1)/(u2(1+u4(1-u2 u3(u1-u2))))),t4->(u2/(u1 u3(1+u1)))};
tt=(((t2 t3 ))^2)(((t4-(t1^2)))^2)-4 (t1^2)t2 t3(1-t1 t3(1+t2 t4));
racah3to1={u1->((t2 t3 t4(t4-(t1^2))-2 (t1^2)(1-t1 t3-t1 t2 t3 t4)-t4 (tt^(1/2)))/(2 (t1^2)(1+t2 t3 t4-t1 t3-t1 t2 t3 t4))),u2->((t2 t3 t4((t1^2)+t4)-t4 (tt^(1/2)))/(2 t1(1+t2 t3 t4-t1 t3-t1 t2 t3 t4))),u3->((t2 t3 t4(t4-(t1^2))-2 (t1^2)(1-t1 t3-t1 t2 t3 t4)+t4 (tt^(1/2)))/(2 t1 t4(1-t1 t3-t1 t2 t3 t4))),u4->((2(1-t1 t3(1+t2 t4))+t2 t3 (t4-(t1^2) )-(tt^(1/2)))/(2 t1 t3))};

Simplify[racah1to2/.racah2to1]
Simplify[racah2to1/.racah1to2]

Simplify[racah2to3/.racah3to2]
Simplify[racah3to2/.racah2to3,{u1≥0,u3≥0,u4≥0,u1+u1 u2 u3<=1}]

FullSimplify[racah1to3/.racah3to1]
Simplify[racah3to1/.racah1to3,{u2≥0,u3≥0,u4≥0,u1+u1 u2 u3-1<=0,1+u4+(u2^2) u3 u4-u1 u2 u3 u4>=0}]

Simplify[(ss/.racah2to3)==(((((1+u1))^2) (u3^2) (((-1+u1+u1 u2 u3))^2))/((((-1+u1 u2 u3))^4) (u4^2)))]

Simplify[(-2(s1^2)s3+s4(1+s1)(1+2 s1+(s1^2)-(s1^2) s2 s3)/.racah2to3)==((u3 ((1-u1 u2 u3)+(u1^2)+(u1^2) u2 u3) (1+u4(1-u1 u2 u3)))/((((1-u1 u2 u3))^3) (u4^2)))]

Simplify[ (((-2(s1^2)s3+s4(1+s1)(1+2 s1+(s1^2)-(s1^2) s2 s3)))^2)-(((1+s1))^2)ss==4 (s1^4) (s3^2) (1+s2 s4+s1 s2 s4)]

Simplify[(1-u1-u1 u2 u3/.racah3to2)==(ss^(1/2))((s4(1+2 s1+(s1^2)+(s1^2) s2 s3)-(ss^(1/2)))/(2 (s1^2) (1+s1) s3 s4(1+s2 s4+s1 s2 s4)))]

Simplify[(tt/.racah1to3)==(((u2^2) (u3^2) (((-1+u1+u1 u2 u3))^2) (u4^2))/(((-1-u4+u1 u2 u3 u4-(u2^2) u3 u4))^2))]

Simplify[(((t2 t3 t4(t4-(t1^2))-2 (t1^2)(1-t1 t3-t1 t2 t3 t4)))^2)-(t4^2) tt==4 (t1^4) (1-t1 t3-t1 t2 t3 t4) ((1-t1 t3-t1 t2 t3 t4)+t2 t3 t4)]

Simplify[(((u2^2) (1-u1(1+u2 u3)+u1+(u1^2) (1+u2 u3)) u4)/(u1 (((1+u1))^2) (1+(1-u1 (1+u2 u3)+u1+(u2^2) u3) u4)))==t2 t3 t4(t4-(t1^2))-2 (t1^2)(1-t1 t3-t1 t2 t3 t4)/.racah1to3]

((u2 (1-u1-u1 u2 u3)+2u1 u2)/(u1 u3(((1+u1))^2) ))

Simplify[((((4 (t1^4) (1-t1 t3-t1 t2 t3 t4) (1-t1 t3-t1 t2 t3 t4+t2 t3 t4)+(t4^2)tt))^(1/2))-t4 (tt^(1/2)))/(2 (t1^2)(1+t2 t3 t4-t1 t3-t1 t2 t3 t4))==u1/.racah3to1,{t1>0,t2>0,t3>0,t4>0,1-t1 t3-t1 t2 t3 t4>0,t2 t3 t4(t4-(t1^2))-2 (t1^2)(1-t1 t3-t1 t2 t3 t4)>0}]

Krawtchouk 1<->2:

racah1to2/.s1->0
racah2to1/.t1->0

Meixner 1<->2:

racah1to2/.s4->0
racah2to1/.t2->0

Charlier 1<->2:

racah1to2/.{s1->0,s4->0}
racah2to1/.{t1->0,t2->0}

Hermite 2<->3:

Simplify[racah2to3/.{u2->0}]
Simplify[racah3to2/.{s2->0}]