Notebook 1 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 7.1 (From Racah to Hermite along Hahn).

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)));

bnhahn=ρ((((n+α+β+1)(n+α+1)(nn-n))/((2n+α+β+1)(2n+α+β+2)))+((n(n+α+β+nn+1)(n+β))/((2n+α+β)(2n+α+β+1)))+σ);
cnhahn=(ρ^2)(n(n+α)(n+β)(n+α+β)(n+α+β+nn+1)(nn-n+1))/((2n+α+β-1)(((2n+α+β))^2)(2n+α+β+1));

bnjacobi=ρ((((β^2)-(α^2))/((2n+α+β)(2n+α+β+2)))+σ);
cnjacobi=(ρ^2) ((4n(n+α)(n+β)(n+α+β))/((2n+α+β-1)(((2n+α+β))^2)(2n+α+β+1)));

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);

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

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

bnhermite=ρ σ;
cnhermite=(ρ^2) n/2;

subsracah={α->(1/t1),β-> (1/(t1 t2)),nn->(1/(t2 t4)),δ->((1+t2 t3 t4)/(t1 t2 t3 t4)),ρ->((t1 t2 (((1+t2))^(3/2)) t3 (t4^2))/((((t1+t4+t2 t4))^(1/2)) (((1+(1+t2) t3 t4))^(1/2)))),σ->-(((1+t1) (1+(1+t2+t1 t2)t3 t4))/(t1 t2 (1+t2+2 t1 t2) t3 (t4^2)))};
bnracah2=-n (((1+t2))^(3/2)) (1+t2+(n+1)t1 t2)/((1+t2+2 t1 t2) (1+t2+2 n t1 t2) (1+t2+2(n+1) t1 t2) (((1+t3 t4+t2 t3 t4))^(1/2)))×(2 n t1 t2  t3 (t4^2)(1+t2+(n+1)t1 t2)(1+t2+2 t1 t2)+t2 t3 (t4^2)(1+t1)(1+t2)(1+t2+2 t1 t2)-t4(1-(t2^2))(1+t1 t3)-2 t1(1-t2)(1+ t2 t4))/(((t1+t4+t2 t4))^(1/2));
Simplify[(bnracah/.subsracah)==bnracah2]
cnracah2=((1+t2+n t1 t2) (1+(1-n)t2 t4)(1-n t1 t2 t3 t4) (1+t3 t4+t2 t3 t4+n t1 t2 t3 t4))/((1+t2+(2n-1)t1 t2)(((1+t2+2 n t1 t2))^2)  (1+t2+(2n+1)t1 t2))×n (1+n t1)(1+n t1 t2) ((((1+t2))^3)/(1+t3 t4+t2 t3 t4))(1+(n+1)((t1 t2 t4)/(t1+t4+t2 t4)));
Simplify[(cnracah/.subsracah)==cnracah2]

Racah -> Krawtchouk: t1->0

bnkrawtchouk2=Simplify[bnracah2/.t1->0];
cnkrawtchouk2=Simplify[cnracah2/.t1->0];
subskrawtchouk={p->((t2 (1+t3 t4+t2 t3 t4))/((1+t2) (1+t2 t3 t4))),nn->(1/(t2 t4)),ρ->(((t4^(1/2))(1+t2)(1+t2 t3 t4))/(((1+t3 t4+t2 t3 t4))^(1/2))),σ->-((1+t3 t4+t2 t3 t4)/((1+t2) t4 (1+t2 t3 t4)))};
Simplify[bnkrawtchouk2==(bnkrawtchouk/.subskrawtchouk),{t2>0,t3>0,t4>0}]
Simplify[cnkrawtchouk2==(cnkrawtchouk/.subskrawtchouk)]

Racah -> Meixner: t2->0

bnmeixner2=Simplify[bnracah2/.t2->0];
cnmeixner2=Simplify[cnracah2/.t2->0];
subsmeixner={β->((1+t1)/t1),c->((t1 (1+t3 t4))/(t1+t4)),ρ->(((1-t1 t3) t4)/((((t1+t4))^(1/2)) (((1+t3 t4))^(1/2)))),σ->-(((1+t1) (1+t3 t4))/((1-t1 t3) t4))};
Simplify[bnmeixner2==(bnmeixner/.subsmeixner)]
Simplify[cnmeixner2==(cnmeixner/.subsmeixner)]

Racah -> Hahn: t3->0

bnhahn2=Simplify[bnracah2/.t3->0];
cnhahn2=Simplify[cnracah2/.t3->0];
subshahn={α->(1/t1),β-> (1/(t1 t2)),nn->(1/(t2 t4)),ρ-> (((((1+t2))^(3/2)) t4)/(((t1+t4+t2 t4))^(1/2))) ,σ->-((1+t1)/((1+t2+2 t1 t2) t4))};
Simplify[bnhahn2==(bnhahn/.subshahn)]
Simplify[cnhahn2==(cnhahn/.subshahn)]

Racah -> Jacobi: t4->0

bnjacobi2=bnracah2/.t4->0;
cnjacobi2=cnracah2/.t4->0;
subsjacobi={α->(1/t1),β-> (1/(t1 t2)),ρ-> -((((1+t2))^(3/2))/(2(t1^(1/2)) t2)),σ-> ((-1+t2)/(1+t2+2 t1 t2))};
Simplify[bnjacobi2==(bnjacobi/.subsjacobi)]
Simplify[cnjacobi2==(cnjacobi/.subsjacobi)]

Krawtchouk -> Krawtchouk: t3->0

subskrawtchouk2={p->(t2/(1+t2)),nn->(1/(t2 t4)),ρ->(1+t2) (t4^(1/2)),σ->-(1/((1+t2) t4))};
Simplify[subskrawtchouk==(subskrawtchouk2/.{t2->t2(1+t3 t4+t2 t3 t4),t4->(t4/(1+t3 t4+t2 t3 t4))}),{t2>0,t3>0,t4>0}]
Simplify[(bnkrawtchouk2/.t3->0)==(bnkrawtchouk/.subskrawtchouk2),{t2>0,t4>0}]
Simplify[(cnkrawtchouk2/.t3->0)==(cnkrawtchouk/.subskrawtchouk2)]

Meixner -> Meixner: t3->0

subsmeixner2={β->((1+t1)/t1),c->(t1/(t1+t4)),ρ->(t4/(((t1+t4))^(1/2))),σ->-((1+t1)/t4)};
Simplify[subsmeixner==(subsmeixner2/.{t4->((t4(1-t1 t3))/(1+t3 t4))}),{t1>0,t3>0,t4>0}]
Simplify[(bnmeixner2/.t3->0)==(bnmeixner/.subsmeixner2)]
Simplify[(cnmeixner2/.t3->0)==(cnmeixner/.subsmeixner2)]

Hahn -> Krawtchouk: t1->0

Simplify[(bnhahn2/.t1->0)==(bnkrawtchouk/.subskrawtchouk2),{t2>0,t4>0}]
Simplify[(cnhahn2/.t1->0)==(cnkrawtchouk/.subskrawtchouk2)]

Hahn -> Meixner: t2->0

Simplify[(bnhahn2/.t2->0)==(bnmeixner/.subsmeixner2)]
Simplify[(cnhahn2/.t2->0)==(cnmeixner/.subsmeixner2)]

Hahn -> Jacobi: t4->0

Simplify[(bnhahn2/.t4->0)==bnjacobi2]
Simplify[(cnhahn2/.t4->0)==cnjacobi2]

Jacobi -> Laguerre: t2->0

bnlaguerre2=bnjacobi2/.t2->0;
cnlaguerre2=cnjacobi2/.t2->0;
subslaguerre={α-> (1/t1),ρ->(t1^(1/2)),σ->-((1+t1)/t1)};
Simplify[bnlaguerre2==(bnlaguerre/.subslaguerre)]
Simplify[cnlaguerre2==(cnlaguerre/.subslaguerre)]

Meixner -> Laguerre: t4->0

Simplify[(bnmeixner2/.t4->0)==bnlaguerre2]
Simplify[(cnmeixner2/.t4->0)==cnlaguerre2]

Meixner -> Charlier: t1->0

bncharlier2=bnmeixner2/.t1->0;
cncharlier2=cnmeixner2/.t1->0;
subscharlier={a->((1+t3 t4)/t4),ρ->((t4^(1/2))/(((1+t3 t4))^(1/2))),σ->- ((1+t3 t4)/t4)};
Simplify[bncharlier2==bncharlier/.subscharlier]
Simplify[cncharlier2==cncharlier/.subscharlier]

Krawtchouk -> Charlier: t2->0

Simplify[(bnkrawtchouk2/.t2->0)==bncharlier2]
Simplify[(cnkrawtchouk2/.t2->0)==cncharlier2]

Krawtchouk -> Hermite: t4->0

bnhermite4=bnkrawtchouk2/.t4->0;
cnhermite4=cnkrawtchouk2/.t4->0;
subshermite={ρ->(2^(1/2)),σ->0};
Simplify[bnhermite4==bnhermite/.subshermite]
Simplify[cnhermite4==cnhermite/.subshermite]

Jacobi -> Hermite: t1->0

(bnjacobi2/.t1->0)==bnhermite4
(cnjacobi2/.t1->0)==cnhermite4

Charlier -> Charlier: t3->0

subscharlier2={a->(1/t4),ρ->(t4^(1/2)),σ->- (1/t4)};
Simplify[subscharlier==(subscharlier2/.{t4->(t4/(1+t3 t4))}),{t3>0,t4>0}]
Simplify[(bncharlier2/.t3->0)==(bncharlier/.subscharlier2)]
Simplify[(cncharlier2/.t3->0)==(cncharlier/.subscharlier2)]

Hermite -> Hermite: t2->0

subshermite2=subshermite/.t2->0;
Simplify[(ρ/.subshermite2)==(ρ/.subshermite)]
bnhermite24=bnhermite4/.t2->0;
cnhermite24=cnhermite4/.t2->0;
Simplify[bnhermite24==(bnhermite/.subshermite2)]
Simplify[cnhermite24==(cnhermite/.subshermite2)]

Hermite -> Hermite: t3->0

subshermite3=subshermite/.t3->0;
Simplify[(ρ/.subshermite3)==(ρ/.subshermite)]
bnhermite34=bnhermite4/.t3->0;
cnhermite34=cnhermite4/.t3->0;
Simplify[bnhermite34==(bnhermite/.subshermite3)]
Simplify[cnhermite34==(cnhermite/.subshermite3)]

Charlier -> Hermite: t4->0

(bncharlier2/.t4->0)==bnhermite24
(cncharlier2/.t4->0)==cnhermite24

Laguerre -> Hermite: t1->0

(bnlaguerre2/.t1->0)==bnhermite24
(cnlaguerre2/.t1->0)==cnhermite24