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18 Orthogonal PolynomialsAskey Scheme

§18.22 Hahn Class: Recurrence Relations and Differences

Contents

§18.22(i) Recurrence Relations in n

Hahn

With

18.22.1 pn(x)=Qn(x;α,β,N),
18.22.2 -xpn(x)=Anpn+1(x)-(An+Cn)pn(x)+Cnpn-1(x),

where

18.22.3 An =(n+α+β+1)(n+α+1)(N-n)(2n+α+β+1)(2n+α+β+2),
Cn =n(n+α+β+N+1)(n+β)(2n+α+β)(2n+α+β+1).

Krawtchouk, Meixner, and Charlier

These polynomials satisfy (18.22.2) with pn(x), An, and Cn as in Table 18.22.1.

Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.
pn(x) An Cn
Kn(x;p,N) p(N-n) n(1-p)
Mn(x;β,c) c(n+β)1-c n1-c
Cn(x,a) a n

Continuous Hahn

With

18.22.4 qn(x)=pn(x;a,b,a¯,b¯)/pn(ia;a,b,a¯,b¯),
18.22.5 (a+ix)qn(x)=A~nqn+1(x)-(A~n+C~n)qn(x)+C~nqn-1(x),

where

18.22.6 A~n =-(n+2(a+b)-1)(n+a+a¯)(n+a+b¯)(2n+2(a+b)-1)(2n+2(a+b)),
C~n =n(n+b+a¯-1)(n+b+b¯-1)(2n+2(a+b)-2)(2n+2(a+b)-1).

Meixner–Pollaczek

With

18.22.7 pn(x)=Pn(λ)(x;ϕ),
18.22.8 (n+1)pn+1(x)=2(xsinϕ+(n+λ)cosϕ)pn(x)-(n+2λ-1)pn-1(x).

§18.22(ii) Difference Equations in x

Hahn

With

18.22.9 pn(x)=Qn(x;α,β,N),
18.22.10 A(x)pn(x+1)-(A(x)+C(x))pn(x)+C(x)pn(x-1)-n(n+α+β+1)pn(x)=0,

where

18.22.11 A(x) =(x+α+1)(x-N),
C(x) =x(x-β-N-1).

Krawtchouk, Meixner, and Charlier

18.22.12 A(x)pn(x+1)-(A(x)+C(x))pn(x)+C(x)pn(x-1)+λnpn(x)=0.

For A(x), C(x), and λn in (18.22.12) see Table 18.22.2.

Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.
pn(x) A(x) C(x) λn
Kn(x;p,N) p(x-N) (p-1)x -n
Mn(x;β,c) c(x+β) x n(1-c)
Cn(x,a) a x n

Continuous Hahn

With

18.22.13 pn(x)=pn(x;a,b,a¯,b¯),
18.22.14 A(x)pn(x+i)-(A(x)+C(x))pn(x)+C(x)pn(x-i)+n(n+2(a+b)-1)pn(x)=0,

where

18.22.15 A(x) =(x+ia¯)(x+ib¯),
C(x) =(x-ia)(x-ib).

Meixner–Pollaczek

With

18.22.16 pn(x)=Pn(λ)(x;ϕ),
18.22.17 A(x)pn(x+i)-(A(x)+C(x))pn(x)+C(x)pn(x-i)+2nsinϕpn(x)=0,

where

18.22.18 A(x) =eiϕ(x+iλ),
C(x) =e-iϕ(x-iλ).

§18.22(iii) x-Differences

Hahn

18.22.19 ΔxQn(x;α,β,N) =-n(n+α+β+1)(α+1)NQn-1(x;α+1,β+1,N-1),
18.22.20 x((α+1)x(β+1)N-xx!(N-x)!Qn(x;α,β,N)) =N+1β(α)x(β)N+1-xx!(N+1-x)!Qn+1(x;α-1,β-1,N+1).

Krawtchouk

18.22.21 ΔxKn(x;p,N) =-npNKn-1(x;p,N-1),
18.22.22 x((Nx)px(1-p)N-xKn(x;p,N)) =(N+1x)px(1-p)N-xKn+1(x;p,N+1).

Meixner

18.22.23 ΔxMn(x;β,c)=-n(1-c)βcMn-1(x;β+1,c),
18.22.24 x((β)xcxx!Mn(x;β,c))=(β-1)xcxx!Mn+1(x;β-1,c).

Charlier

18.22.25 ΔxCn(x,a) =-naCn-1(x,a),
18.22.26 x(axx!Cn(x,a)) =axx!Cn+1(x,a).

Continuous Hahn

18.22.27 δx(pn(x;a,b,a¯,b¯))=(n+2(a+b)-1)pn-1(x;a+12,b+12,a¯+12,b¯+12),
18.22.28 δx(w(x;a+12,b+12,a¯+12,b¯+12)pn(x;a+12,b+12,a¯+12,b¯+12))=-(n+1)w(x;a,b,a¯,b¯)pn+1(x;a,b,a¯,b¯).

Meixner–Pollaczek

18.22.29 δx(Pn(λ)(x;ϕ))=2sinϕPn-1(λ+12)(x;ϕ),
18.22.30 δx(w(λ+12)(x;ϕ)Pn(λ+12)(x;ϕ))=-(n+1)w(λ)(x;ϕ)Pn+1(λ)(x;ϕ).