Notebook 2 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 7.2 (From Racah to Hermite along dual Hahn and Charlier).

Last modified: 15 Sep 2009

Off[General::spell, General::spell1]

bnracah=ρ(((n+α+β+1)(n+α+1)(n+β+δ+1)(nn-n))/((2n+α+β+1)(2n+α+β+2))+((n(n+β)(δ-α-n)(n+nn+α+β+1))/((2n+α+β)(2n+α+β+1)))+ σ);
cnracah=(ρ^2)n(n+α)(n+β)(n+α+β)(n+β+δ)((δ-α-n)(n+nn+α+β+1)(nn+1-n))/((2n+α+β-1)(((2n+α+β))^2)(2n+α+β+1));

bndualhahn=ρ(n+γ+1)(nn-n)+ρ n(nn+1+δ-n)+ρ σ;
cndualhahn=(ρ^2)n(n+γ)(nn+1+δ-n)(nn+1-n);

bnmeixner=ρ(((n+(n+β)c)/(1-c))+σ);
cnmeixner=(((ρ^2)n(n+β-1)c)/(((1-c))^2));

bnkrawtchouk=ρ(p(nn-n)+n(1-p)+σ);
cnkrawtchouk=(ρ^2)n p(1-p)(nn+1-n);

bncharlier=ρ(n+a+σ);
cncharlier=(ρ^2)n a;

bnhermite=ρ σ;
cnhermite=(ρ^2) n/2;

subsracah={α->((1+s1)/(s1 s2)),β->((1+s1)/(s1 (s2^2) s3 s4)),nn->(1/((s2^2) s4)),δ->((1+s1+s2 s4(1+s1+s1 s2))/(s1 (s2^2) s4)),ρ->((s1 (s2^(5/2)) s4)/((2^(1/2))(1+s1))),σ->-(((1+s1)(1+s3-(s2^2) s4)+s1 s2)/(s1 (s2^3) s4))};bnracah2=-(2^(-1/2))(s2^(1/2))(2(n^2) (((n+1))^2)(s1^3) (s2^6)(s3^2) (s4^3)(((1+s1))^(-1))+ 4 (n^2)(n+1) (s1^2) (s2^4) s3 (s4^2) (1+s2 s3 s4)+(n^2) s1 (s2^2) s4 (2+2 s1-s3-2 s1 s3-2 s1 (s3^2)+5 s2 s3 s4+5 s1 s2 s3 s4+s2 (s3^2) s4+2 s1 s2 (s3^2) s4+4 s1 s2 (s3^3) s4+3 (s2^2) (s3^2) (s4^2)+3 s1 (s2^2) (s3^2) (s4^2)-2 s1 (s2^3) (s3^2) (s4^2))-n (1+s1+s2 s3 s4+s1 s2 s3 s4+s1 (s2^2) s3 s4) (1+2 s1+2 s1 s3-s2 s4-s1 s2 s4-s2 s3 s4-2 s1 s2 s3 s4-4 s1 s2 (s3^2) s4-(s2^2) s3 (s4^2)-s1 (s2^2) s3 (s4^2)+2 s1 (s2^3) s3 (s4^2))+(1+s1) (1+s2 s3 s4) (-s1 s3-s2 s4-s1 s2 s4+(s3^2) s4+s1 (s3^2) s4+2 s1 s2 (s3^2) s4-(s2^2) s3 (s4^2)-s1 (s2^2) s3 (s4^2)-2 s1 (s2^3) s3 (s4^2)))×((((1+s1)(1+s2 s3 s4)+2 n s1 (s2^2) s3 s4))^(-1)) ((((1+s1)(1+s2 s3 s4)+2(n +1)s1 (s2^2) s3 s4))^(-1));Simplify[bnracah2==(bnracah/.subsracah),{s1>0,s2>0,s3>0,s4>0}]
cnracah2=n (1+s1+n s1 s2) (1+(1-n)(s2^2) s4) (1+s1+(1-n)s1 (s2^2) s4)×((1+s1+n s1 (s2^2) s3 s4) ((1+s1)(1+s2 s3 s4)+s1 s3+(n +1)s1 (s2^2) s3 s4) )/(2 (((1+s1))^2) ((1+s1)(1+s2 s3 s4)+(2 n-1) s1 (s2^2) s3 s4))×(((1+s1)(1+s2 s3 s4)+n s1 (s2^2) s3 s4)((1+s1)(1+s3+s2 s3 s4)+(n+1)s1 (s2^2) s3 s4))/(((((1+s1)(1+s2 s3 s4)+2 n s1 (s2^2) s3 s4))^2) ((1+s1)(1+s2 s3 s4)+(2 n+1) s1 (s2^2) s3 s4));Simplify[cnracah2==(cnracah/.subsracah)]

Racah -> Krawtchouk: s1->0

bnkrawtchouk2=Simplify[bnracah2/.s1->0];
cnkrawtchouk2=Simplify[cnracah2/.s1->0];
subskrawtchouk={p->((s2 s4 (1+s3+s2 s3 s4))/((1+s2 s4) (1+s2 s3 s4))),nn->(1/((s2^2) s4)),ρ->(2^(-1/2))(s2^(1/2))(1+s2 s4),σ->-((1+s3-(s2^2) s4)/(s2 (1+s2 s4)))};
Simplify[bnkrawtchouk2==(bnkrawtchouk/.subskrawtchouk)]
Simplify[cnkrawtchouk2==(cnkrawtchouk/.subskrawtchouk)]

Racah -> Hermite: s2->0

bnhermite2=Simplify[bnracah2/.s2->0];
cnhermite2=Simplify[cnracah2/.s2->0];
subshermite={ρ->(((((1+s3))^(1/2)) (((1+s1+s1 s3))^(1/2)))/(((1+s1))^(1/2))),σ->0};
Simplify[bnhermite2==(bnhermite/.subshermite)]
Simplify[cnhermite2==(cnhermite/.subshermite)]

Racah -> dual Hahn: s3->0

bndualhahn2=Simplify[bnracah2/.s3->0];
cndualhahn2=Simplify[cnracah2/.s3->0];
subsdualhahn={γ->((1+s1)/(s1 s2)),δ->(1/(s1 (s2^2) s4)),nn->(1/((s2^2) s4)),ρ->((s1 (s2^(5/2)) s4)/((2^(1/2))(1+s1))),σ->-(((1+s1)(1-(s2^2) s4)+s1 s2)/(s1 (s2^3) s4))};
Simplify[bndualhahn2==(bndualhahn/.subsdualhahn)]
Simplify[cndualhahn2==(cndualhahn/.subsdualhahn)]

Racah -> Meixner: s4->0

bnmeixner2=Simplify[bnracah2/.s4->0];
cnmeixner2=Simplify[cnracah2/.s4->0];
subsmeixner={β->((1+s1+s1 s2)/(s1 s2)),c->((s1 (1+s3))/(1+s1+s1 s3)),ρ->((s2^(1/2))/((2^(1/2))(1+s1))),σ->-((1+s1+s3+s1 s2+s1 s3)/s2)};
Simplify[bnmeixner2==(bnmeixner/.subsmeixner)]
Simplify[cnmeixner2==(cnmeixner/.subsmeixner)]

Krawtchouk -> Krawtchouk: s3->0

subskrawtchouk0={s2->(s2/(1+s3+s2 s3 s4)),s4->s4 (((1+s3+s2 s3 s4))^2)};
subskrawtchouk2={p->((s2 s4)/(1+s2 s4)),nn->(1/((s2^2) s4)),ρ->((((s2 )^(1/2))(1+s2 s4))/(2^(1/2))),σ->-((1-(s2^2) s4)/(s2 (1+s2 s4)))};
ρ0=((((1+s3+s2 s3 s4))^(1/2))/(1+s2 s3 s4));
σ0=-(((s2^(1/2))s3 s4(s2+s3))/((2^(1/2))(((1+s3+s2 s3 s4))^(1/2))));
Simplify[(p/.subskrawtchouk)==((p/.subskrawtchouk2)/.subskrawtchouk0)]
Simplify[(nn/.subskrawtchouk)==((nn/.subskrawtchouk2)/.subskrawtchouk0)]
Simplify[((ρ/.subskrawtchouk2)/.subskrawtchouk0)==(((ρ/.subskrawtchouk))/(ρ0)),{s2>0,s3>0,s4>0}]
Simplify[((σ/.subskrawtchouk2)/.subskrawtchouk0)==(σ/.subskrawtchouk)-((ρ0 σ0)/(ρ/.subskrawtchouk)),{s2>0,s3>0,s4>0}]
Simplify[(bnkrawtchouk2/.s3->0)==(bnkrawtchouk/.subskrawtchouk2)]
Simplify[(cnkrawtchouk2/.s3->0)==(cnkrawtchouk/.subskrawtchouk2)]

Hermite -> Hermite: s1->0

subshermite1=subshermite/.s1->0;
ρ1=((((1+s1+s1 s3))^(1/2))/(((1+s1))^(1/2)));
Simplify[(ρ/.subshermite1)ρ1==(ρ/.subshermite)]
bnhermite12=bnhermite2/.s1->0;
cnhermite12=cnhermite2/.s1->0;
Simplify[bnhermite12==(bnhermite/.subshermite1)]
Simplify[cnhermite12==(cnhermite/.subshermite1)]

Hermite -> Hermite: s3->0

subshermite3=subshermite/.s3->0;
Simplify[(ρ/.subshermite3)==1]
bnhermite23=bnhermite2/.s3->0;
cnhermite23=cnhermite2/.s3->0;
Simplify[bnhermite23==(bnhermite/.subshermite3)]
Simplify[cnhermite23==(cnhermite/.subshermite3)]

Hermite -> Hermite: s4->0

subshermite4=subshermite/.s4->0;
Simplify[(ρ/.subshermite4)==(ρ/.subshermite)]
bnhermite24=bnhermite2/.s4->0;
cnhermite24=cnhermite2/.s4->0;
Simplify[bnhermite24==(bnhermite/.subshermite4)]
Simplify[cnhermite24==(cnhermite/.subshermite4)]

Meixner -> Meixner: s3->0

subsmeixner0={s1->s1(1+s3),s2->((s2(1+s1 + s1 s3))/((1+s1) (1+s3)))};
subsmeixner2={β->((1+s1+s1 s2)/(s1 s2)),c->(s1/(1+s1)),ρ->((s2^(1/2))/((2^(1/2))(1+s1))),σ->-((1+s1+s1 s2)/s2)};
ρ0=(((((1+s3))^(1/2)) (((1+s1+s1 s3))^(1/2)))/(((1+s1))^(1/2)));
σ0=((s1 (s2^(1/2))s3)/((2^(1/2))(((1+s1))^(1/2))(((1+s3))^(1/2)) (((1+s1+s1 s3))^(1/2))));
Simplify[(β/.subsmeixner)==((β/.subsmeixner2)/.subsmeixner0)]
Simplify[(c/.subsmeixner)==((c/.subsmeixner2)/.subsmeixner0)]
Simplify[((ρ/.subsmeixner2)/.subsmeixner0)==(((ρ/.subsmeixner))/(ρ0)),{s1>0,s2>0,s3>0}]
Simplify[((σ/.subsmeixner2)/.subsmeixner0)==(σ/.subsmeixner)-((ρ0 σ0)/(ρ/.subsmeixner)),{s1>0,s2>0,s3>0}]
bnmeixner34=Simplify[bnmeixner2/.s3->0];
cnmeixner34=Simplify[cnmeixner2/.s3->0];
Simplify[bnmeixner34==(bnmeixner/.subsmeixner2)]
Simplify[cnmeixner34==(cnmeixner/.subsmeixner2)]

Meixner -> Hermite: s2->0

Simplify[(bnmeixner2/.s2->0)==bnhermite2]
Simplify[(cnmeixner2/.s2->0)==cnhermite2]

Krawtchouk -> Hermite: s2->0

Simplify[bnhermite12==(bnkrawtchouk2/.s2->0)]
Simplify[cnhermite12==(cnkrawtchouk2/.s2->0)]

Krawtchouk -> Charlier: s4->0

subscharlier={a->((1+s3)/s2),ρ->(2^(-1/2))(s2^(1/2)),σ->-((1+s3)/s2)};
bncharlier4=Simplify[bnkrawtchouk2/.s4->0];
cncharlier4=Simplify[cnkrawtchouk2/.s4->0];
Simplify[bncharlier4==(bncharlier/.subscharlier)]
Simplify[cncharlier4==(cncharlier/.subscharlier)]

Meixner -> Charlier: s1->0

Simplify[bncharlier4==(bnmeixner2/.s1->0)]
Simplify[cncharlier4==(cnmeixner2/.s1->0)]

dual Hahn -> Krawtchouk: s1->0

Simplify[(bndualhahn2/.s1->0)==(bnkrawtchouk/.subskrawtchouk2)]
Simplify[(cndualhahn2/.s1->0)==(cnkrawtchouk/.subskrawtchouk2)]

dual Hahn -> Meixner: s4->0

Simplify[(bndualhahn2/.s4->0)==(bnmeixner/.subsmeixner2)]
Simplify[(cndualhahn2/.s4->0)==(cnmeixner/.subsmeixner2)]

Dual Hahn -> Hermite: s2->0,

Simplify[bnhermite23==(bndualhahn2/.s2->0)]
Simplify[cnhermite23==(cndualhahn2/.s2->0)]

Charlier -> Charlier: s3->0

subscharlier2={a->(1/s2),ρ->(((s2 )^(1/2))/(2^(1/2))),σ->-(1/s2)};
subscharlier0={s2->(s2/(1+s3 ))};
ρ0=(((1+s3))^(1/2));
Simplify[(a/.subscharlier)==((a/.subscharlier2)/.subscharlier0)]
Simplify[((ρ/.subscharlier2)/.subscharlier0)==(((ρ/.subscharlier))/(ρ0)),{s2>0,s3>0}]
Simplify[((σ/.subscharlier2)/.subscharlier0)==(σ/.subscharlier),{s2>0,s3>0}]
bncharlier34=Simplify[bncharlier4/.s3->0];
cncharlier34=Simplify[cncharlier4/.s3->0];
Simplify[bncharlier34==(bncharlier/.subscharlier2)]
Simplify[cncharlier34==(cncharlier/.subscharlier2)]

Hermite -> Hermite: s1->0, s3->0

subshermite13=subshermite1/.s3->0;
Simplify[(ρ/.subshermite13)==1]
bnhermite123=bnhermite12/.s3->0;
cnhermite123=cnhermite12/.s3->0;
Simplify[bnhermite123==(bnhermite/.subshermite13)]
Simplify[cnhermite123==(cnhermite/.subshermite13)]
Simplify[bnhermite123==bnhermite23/.s1->0]
Simplify[cnhermite123==cnhermite23/.s1->0]

Hermite -> Hermite: s1->0, s4->0

subshermite14=subshermite1/.s4->0;
Simplify[(ρ/.subshermite14)ρ1==(ρ/.subshermite)]
bnhermite124=bnhermite12/.s4->0;
cnhermite124=cnhermite12/.s4->0;
Simplify[bnhermite124==(bnhermite/.subshermite14)]
Simplify[cnhermite124==(cnhermite/.subshermite14)]
Simplify[bnhermite124==bnhermite24/.s1->0]
Simplify[cnhermite124==cnhermite24/.s1->0]

Hermite -> Hermite: s3->0, s4->0

subshermite34=subshermite3/.s4->0;
Simplify[(ρ/.subshermite34)==1]
bnhermite234=bnhermite23/.s4->0;
cnhermite234=cnhermite23/.s4->0;
Simplify[bnhermite234==(bnhermite/.subshermite34)]
Simplify[cnhermite234==(cnhermite/.subshermite34)]
Simplify[bnhermite234==bnhermite24/.s3->0]
Simplify[cnhermite234==cnhermite24/.s3->0]

Meixner -> Charlier: s1->0

Simplify[bncharlier34==(bnmeixner34/.s1->0)]
Simplify[cncharlier34==(cnmeixner34/.s1->0)]

Charlier -> Hermite:  s2->0

Simplify[bnhermite124==(bncharlier4/.s2->0)]
Simplify[cnhermite124==(cncharlier4/.s2->0)]

Hermite -> Hermite: s1->0, s3->0, s4->0

subshermite134=subshermite13/.s4->0;
Simplify[(ρ/.subshermite134)==1]
bnhermite1234=bnhermite123/.s4->0;
cnhermite1234=cnhermite123/.u4->0;
Simplify[bnhermite1234==(bnhermite/.subshermite134)]
Simplify[cnhermite1234==(cnhermite/.subshermite134)]
Simplify[bnhermite1234==bnhermite124/.s3->0]
Simplify[cnhermite1234==cnhermite124/.s3->0]
Simplify[bnhermite1234==bnhermite234/.s1->0]
Simplify[cnhermite1234==cnhermite234/.s1->0]