Notebook 3 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 7.3 (From Racah to Hermite along dual Hahn and Laguerre).

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));

bnlaguerre=ρ(2n+α+1+σ);
cnlaguerre=(ρ^2)n(n+α);

bnhermite=ρ σ;
cnhermite=(ρ^2) n/2;

subsracah={α->((1+u1)/u2),β->(1/(u1 (u2^2) (u3^2) u4)),nn->(1/((u2^2) u3 u4)),δ->((1+u4+u2 u3 u4+(u2^2) u3 u4)/((u2^2) u3 u4)),ρ->(2^(-1/2))(u2^(5/2)) u3 u4,σ->-(((1+u1) (1+u1 u3+u1 u3 u4))/((u2^3) u3 u4))};
bnracah0=-(2^(-1/2))(u2^(1/2)) (2 (n^2)(((n+1))^2) (u1^2) (u2^6) (u3^5) (u4^3)+4 (n^2) (n+1)u1 (u2^4) (u3^3) (u4^2) (1+u1 u2 (u3^2) u4+(u1^2) u2 (u3^2) u4)+(n^2) (u2^2) u3 u4 (2-2 u1 u3-2 (u1^2) (u3^2)-u1 u3 u4-2 (u1^2) (u3^2) u4+5 u1 u2 (u3^2) u4+6 (u1^2) u2 (u3^2) u4+2 (u1^2) u2 (u3^3) u4+4 (u1^3) u2 (u3^3) u4-4 (u1^2) (u2^2) (u3^3) u4+4 (u1^3) u2 (u3^4) u4+4 (u1^4) u2 (u3^4) u4+(u1^2) u2 (u3^3) (u4^2)+(u1^3) u2 (u3^3) (u4^2)+4 (u1^3) u2 (u3^4) (u4^2)+4 (u1^4) u2 (u3^4) (u4^2)+3 (u1^2) (u2^2) (u3^4) (u4^2)+5 (u1^3) (u2^2) (u3^4) (u4^2)+2 (u1^4) (u2^2) (u3^4) (u4^2)+2 (u1^2) (u2^3) (u3^4) (u4^2)+2 (u1^3) (u2^3) (u3^4) (u4^2))+n (1+u1 u2 (u3^2) u4+(u1^2) u2 (u3^2) u4+u1 (u2^2) (u3^2) u4) (-2-2 u1 u3-u4-2 u1 u3 u4+u2 u3 u4+2 u1 u2 u3 u4+2 u1 u2 (u3^2) u4+4 (u1^2) u2 (u3^2) u4-4 u1 (u2^2) (u3^2) u4+4 (u1^2) u2 (u3^3) u4+4 (u1^3) u2 (u3^3) u4+u1 u2 (u3^2) (u4^2)+(u1^2) u2 (u3^2) (u4^2)+4 (u1^2) u2 (u3^3) (u4^2)+4 (u1^3) u2 (u3^3) (u4^2)+u1 (u2^2) (u3^3) (u4^2)+(u1^2) (u2^2) (u3^3) (u4^2)+2 u1 (u2^3) (u3^3) (u4^2)+2 (u1^2) (u2^3) (u3^3) (u4^2))+(1+u1 u2 (u3^2) u4+(u1^2) u2 (u3^2) u4) (-1-u1 u3-u1 u3 u4+(u1^2) (u3^2) u4+(u1^3) (u3^2) u4-u1 u2 (u3^2) u4-2 u1 (u2^2) (u3^2) u4+(u1^2) (u3^3) u4+2 (u1^3) (u3^3) u4+(u1^4) (u3^3) u4+2 (u1^2) u2 (u3^3) u4+2 (u1^3) u2 (u3^3) u4+(u1^2) (u3^3) (u4^2)+2 (u1^3) (u3^3) (u4^2)+(u1^4) (u3^3) (u4^2)+2 (u1^2) u2 (u3^3) (u4^2)+2 (u1^3) u2 (u3^3) (u4^2)))×(((1+u1 u2 (u3^2) u4(1+u1)+2 n u1 (u2^2) (u3^2) u4))^(-1)) (((1+u1 u2 (u3^2) u4(1+u1)+2(n+1) u1 (u2^2) (u3^2) u4))^(-1));
Simplify[bnracah0==(bnracah/.subsracah),{u1>0,u2>0,u3>0,u4>0}]
cnracah0=(1/2)n(1+u1+n u2) (1+(1-n)(u2^2) u3 u4)×
 (1+u4-u1 u2 u3 u4+(1-n)(u2^2) u3 u4)×
((1+u1 u2 (u3^2) u4(1+u1)+n u1 (u2^2) (u3^2) u4)/(1+u1 u2 (u3^2) u4(1+u1)+(2 n-1) u1 (u2^2) (u3^2) u4))×
(1+u1 u3(1+u4+u2 u3 u4)+(n+1)u1 (u2^2) (u3^2) u4)/(((1+u1 u2 (u3^2) u4(1+u1)+2 n u1 (u2^2) (u3^2) u4))^2) ×  (1+n u1 (u2^2) (u3^2) u4)(1+u1 u3(1+u2 u3 u4+u1 u2 u3 u4)+(n+1)u1 (u2^2) (u3^2) u4)/(1+u1 u2 (u3^2) u4(1+u1)+(2 n+1) u1 (u2^2) (u3^2) u4);Simplify[cnracah0==cnracah/.subsracah]

Racah -> dual Hahn: u1->0

bndualhahn1=bnracah0/.u1->0;
cndualhahn1=cnracah0/.u1->0;
subsdualhahn={γ-> (1/u2) ,δ->(1/((u2^2) u3)),nn->(1/((u2^2) u3 u4)),ρ->(2^(-1/2))(u2^(5/2)) u3 u4,σ->-(1/((u2^3) u3 u4))};
Simplify[bndualhahn1==bndualhahn/.subsdualhahn]
Simplify[cndualhahn1==cndualhahn/.subsdualhahn]

Racah -> Hermite: u2->0

bnhermite2=bnracah0/.u2->0;
cnhermite2=cnracah0/.u2->0;
subshermite={ρ->(((1+u1))^(1/2))  (((1+u4))^(1/2)) (((1+u1 u3))^(1/2))(((1+u1 u3 (1+u4)))^(1/2)),σ->0};
Simplify[bnhermite2==(bnhermite/.subshermite)]
Simplify[cnhermite2==(cnhermite/.subshermite)]

Racah -> Meixner: u3->0

bnmeixner3=bnracah0/.u3->0;
cnmeixner3=cnracah0/.u3->0;
subsmeixner={β-> ((1+u1+u2)/u2),c->(1/(1+u4)),ρ->(2^(-1/2))(u2^(1/2)) u4,σ->-((1+u1)/(u2 u4))};
Simplify[bnmeixner3==(bnmeixner/.subsmeixner)]
Simplify[cnmeixner3==(cnmeixner/.subsmeixner)]

Racah -> Laguerre: u4->0

bnlaguerre4=bnracah0/.u4->0;
cnlaguerre4=cnracah0/.u4->0;
subslaguerre={α->((1+u1)/u2),ρ->(2^(-1/2))(u2^(1/2))(1+u1 u3),σ->-((1+u1)/u2)};

Simplify[bnlaguerre4==(bnlaguerre/.subslaguerre)]
Simplify[cnlaguerre4==(cnlaguerre/.subslaguerre)]

Meixner -> Meixner: u1->0

subsmeixner0={u2->(u2/(1+u1))};
subsmeixner1=subsmeixner/.u1->0;
ρ0=(((1+u1))^(1/2));
Simplify[(subsmeixner1/.subsmeixner0){1,1,0,1}==subsmeixner{1,1,0,1}]
Simplify[((ρ/.subsmeixner1)/.subsmeixner0)ρ0==(ρ/.subsmeixner),{u1>0,u2>0}]
bnmeixner13=(bnmeixner3/.u1->0);
cnmeixner13=(cnmeixner3/.u1->0);
Simplify[bnmeixner13==(bnmeixner/.subsmeixner1)]
Simplify[cnmeixner13==(cnmeixner/.subsmeixner1)]

Laguerre -> Laguerre: u1->0

subslaguerre0={u2->(u2/(1+u1))};
subslaguerre1=subslaguerre/.u1->0;
ρ0=(((1+u1))^(1/2))(1+u1 u3);
Simplify[(subslaguerre1/.subslaguerre0){1,0,1}==subslaguerre{1,0,1}]
Simplify[((ρ/.subslaguerre1)/.subslaguerre0)ρ0==(ρ/.subslaguerre),{u1>0,u2>0}]
bnlaguerre14=(bnlaguerre4/.u1->0);
cnlaguerre14=(cnlaguerre4/.u1->0);
Simplify[bnlaguerre14==(bnlaguerre/.subslaguerre1)]
Simplify[cnlaguerre14==(cnlaguerre/.subslaguerre1)]

Laguerre -> Laguerre: u3->0

subslaguerre3=subslaguerre/.u3->0;
ρ0=1+u1 u3;
Simplify[subslaguerre3{1,0,1}==subslaguerre{1,0,1}]
Simplify[(ρ/.subslaguerre3)ρ0==(ρ/.subslaguerre)]
bnlaguerre34=(bnlaguerre4/.u3->0);
cnlaguerre34=(cnlaguerre4/.u3->0);
Simplify[bnlaguerre34==(bnlaguerre/.subslaguerre3)]
Simplify[cnlaguerre34==(cnlaguerre/.subslaguerre3)]

Hermite -> Hermite: u1->0

subshermite1=subshermite/.u1->0;
ρ1=(((1+u1))^(1/2)) (((1+u1 u3))^(1/2))(((1+u1 u3 (1+u4)))^(1/2));
Simplify[(ρ/.subshermite1)ρ1==(ρ/.subshermite),{u1>0,u3>0,u4>0}]
bnhermite12=bnhermite2/.u1->0;
cnhermite12=cnhermite2/.u1->0;
Simplify[bnhermite12==(bnhermite/.subshermite1)]
Simplify[cnhermite12==(cnhermite/.subshermite1)]

Hermite -> Hermite: u3->0

subshermite3=subshermite/.u3->0;
ρ3= (((1+u1 u3))^(1/2))(((1+u1 u3 (1+u4)))^(1/2));
Simplify[(ρ/.subshermite3)ρ3==(ρ/.subshermite)]
bnhermite23=bnhermite2/.u3->0;
cnhermite23=cnhermite2/.u3->0;
Simplify[bnhermite23==(bnhermite/.subshermite3)]
Simplify[cnhermite23==(cnhermite/.subshermite3)]

Hermite -> Hermite: u4->0

subshermite4=subshermite/.u4->0;
ρ4=(((1+u1 u3))^(-1/2)) (((1+u4))^(1/2)) (((1+u1 u3 (1+u4)))^(1/2));
Simplify[(ρ/.subshermite4)ρ4==(ρ/.subshermite),{u1>0,u3>0,u4>0}]
bnhermite24=bnhermite2/.u4->0;
cnhermite24=cnhermite2/.u4->0;
Simplify[bnhermite24==(bnhermite/.subshermite4)]
Simplify[cnhermite24==(cnhermite/.subshermite4)]

Dual Hahn -> Hermite: u2->0

Simplify[(bndualhahn1/.u2->0)==bnhermite12]
Simplify[(cndualhahn1/.u2->0)==cnhermite12]

dual Hahn -> Meixner: u3->0

Simplify[(bndualhahn1/.u3->0)==bnmeixner13]
Simplify[(cndualhahn1/.u3->0)==cnmeixner13]

Dual Hahn -> Laguerre: u4->0

Simplify[(bndualhahn1/.u4->0)==bnlaguerre14]
Simplify[(cndualhahn1/.u4->0)==cnlaguerre14]

Meixner -> Laguerre: u4->0

Simplify[(bnmeixner3/.u4->0)==bnlaguerre34]
Simplify[(cnmeixner3/.u4->0)==cnlaguerre34]

Meixner -> Hermite: u2->0

Simplify[(bnmeixner3/.u2->0)==bnhermite23]
Simplify[(cnmeixner3/.u2->0)==cnhermite23]

Laguerre -> Hermite: u2->0

Simplify[(bnlaguerre4/.u2->0)==bnhermite24]
Simplify[(cnlaguerre4/.u2->0)==cnhermite24]

Laguerre -> Laguerre: u1->0, u3->0

subslaguerre0={u2->(u2/(1+u1))};subslaguerre13=subslaguerre3/.u1->0;
ρ0=(((1+u1))^(1/2))(1+u1 u3);
Simplify[(subslaguerre13/.subslaguerre0){1,0,1}==subslaguerre{1,0,1}]
Simplify[((ρ/.subslaguerre13)/.subslaguerre0) ρ0==(ρ/.subslaguerre),{u1>0,u2>0}]
bnlaguerre134=(bnlaguerre34/.u1->0);
cnlaguerre134=(cnlaguerre34/.u1->0);
Simplify[bnlaguerre134==(bnlaguerre/.subslaguerre13)]
Simplify[cnlaguerre134==(cnlaguerre/.subslaguerre13)]
Simplify[bnlaguerre134==(bnlaguerre14/.u3->0)]
Simplify[cnlaguerre134==(cnlaguerre14/.u3->0)]

Hermite -> Hermite: u1->0, u3->0

subshermite13=subshermite1/.u3->0;
Simplify[(ρ/.subshermite13)ρ1==(ρ/.subshermite)]
bnhermite123=bnhermite12/.u3->0;
cnhermite123=cnhermite12/.u3->0;
Simplify[bnhermite123==(bnhermite/.subshermite13)]
Simplify[cnhermite123==(cnhermite/.subshermite13)]
Simplify[bnhermite123==bnhermite23/.u1->0]
Simplify[cnhermite123==cnhermite23/.u1->0]

Hermite -> Hermite: u1->0, u4->0

subshermite14=subshermite1/.u4->0;
Simplify[(ρ/.subshermite14)==1]
bnhermite124=bnhermite12/.u4->0;
cnhermite124=cnhermite12/.u4->0;
Simplify[bnhermite124==(bnhermite/.subshermite14)]
Simplify[cnhermite124==(cnhermite/.subshermite14)]
Simplify[bnhermite124==bnhermite24/.u1->0]
Simplify[cnhermite124==cnhermite24/.u1->0]

Hermite -> Hermite: u3->0, u4->0

subshermite34=subshermite3/.u4->0;
ρ34=(((1+u4))^(1/2))(((1+u1 u3))^(1/2)) (((1+u1 u3 (1+u4)))^(1/2));
Simplify[(ρ/.subshermite34)ρ34==(ρ/.subshermite)]
bnhermite234=bnhermite23/.u4->0;
cnhermite234=cnhermite23/.u4->0;
Simplify[bnhermite234==(bnhermite/.subshermite34)]
Simplify[cnhermite234==(cnhermite/.subshermite34)]
Simplify[bnhermite234==bnhermite24/.u3->0]
Simplify[cnhermite234==cnhermite24/.u3->0]

Meixner -> Hermite: u2->0

Simplify[(bnmeixner13/.u2->0)==bnhermite123]
Simplify[(cnmeixner13/.u2->0)==cnhermite123]

Laguerre -> Hermite: u2->0

Simplify[(bnlaguerre14/.u2->0)==bnhermite124]
Simplify[(cnlaguerre14/.u2->0)==cnhermite124]
Simplify[(bnlaguerre34/.u2->0)==bnhermite234]
Simplify[(cnlaguerre34/.u2->0)==cnhermite234]

Hermite -> Hermite: u1->0, u3->0, u4->0

subshermite134=subshermite13/.u4->0;
Simplify[(ρ/.subshermite134)==1]
bnhermite1234=bnhermite123/.u4->0;
cnhermite1234=cnhermite123/.u4->0;
Simplify[bnhermite1234==(bnhermite/.subshermite134)]
Simplify[cnhermite1234==(cnhermite/.subshermite134)]
Simplify[bnhermite1234==bnhermite124/.u3->0]
Simplify[cnhermite1234==cnhermite124/.u3->0]
Simplify[bnhermite1234==bnhermite234/.u1->0]
Simplify[cnhermite1234==cnhermite234/.u1->0]

Laguerre -> Hermite: u2->0

Simplify[(bnlaguerre134/.u2->0)==bnhermite1234]
Simplify[(cnlaguerre134/.u2->0)==cnhermite1234]