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18 Orthogonal PolynomialsNotation

§18.1 Notation


§18.1(i) Special Notation

(For other notation see Notation for the Special Functions.)

x,y real variables.
z(=x+iy) complex variable.
q real variable such that 0<q<1, unless stated otherwise.
,m nonnegative integers.
n nonnegative integer, except in §18.30.
N positive integer.
δ(x-a) Dirac delta (§1.17).
δ arbitrary small positive constant.
pn(x) polynomial in x of degree n.
p-1(x) 0.
w(x) weight function (0) on an open interval (a,b).
wx weights (>0) at points xX of a finite or countably infinite subset of .
OP’s orthogonal polynomials.


Forward differences:

Δx(f(x)) =f(x+1)-f(x),
Δxn+1(f(x)) =Δx(Δxn(f(x))).

Backward differences:

x(f(x)) =f(x)-f(x-1),
xn+1(f(x)) =x(xn(f(x))).

Central differences in imaginary direction:

δx(f(x)) =(f(x+12i)-f(x-12i))/i,
δxn+1(f(x)) =δx(δxn(f(x))).

q-Pochhammer Symbol

(z;q)0 =1,
(z;q)n =(1-z)(1-zq)(1-zqn-1),

Infinite q-Product

(z;q) =j=0(1-zqj),
(z1,,zk;q) =(z1;q)(zk;q).

§18.1(ii) Main Functions

The main functions treated in this chapter are:

Classical OP’s

  • Jacobi: Pn(α,β)(x).

  • Ultraspherical (or Gegenbauer): Cn(λ)(x).

  • Chebyshev of first, second, third, and fourth kinds: Tn(x), Un(x), Vn(x), Wn(x).

  • Shifted Chebyshev of first and second kinds: Tn*(x), Un*(x).

  • Legendre: Pn(x).

  • Shifted Legendre: Pn*(x).

  • Laguerre: Ln(α)(x) and Ln(x)=Ln(0)(x). (Ln(α)(x) with α0 is also called Generalized Laguerre.)

  • Hermite: Hn(x), Hen(x).

Hahn Class OP’s

  • Hahn: Qn(x;α,β,N).

  • Krawtchouk: Kn(x;p,N).

  • Meixner: Mn(x;β,c).

  • Charlier: Cn(x;a).

  • Continuous Hahn: pn(x;a,b,a¯,b¯).

  • Meixner–Pollaczek: Pn(λ)(x;ϕ).

Wilson Class OP’s

  • Wilson: Wn(x;a,b,c,d).

  • Racah: Rn(x;α,β,γ,δ).

  • Continuous Dual Hahn: Sn(x;a,b,c).

  • Dual Hahn: Rn(x;γ,δ,N).

q-Hahn Class OP’s

  • q-Hahn: Qn(x;α,β,N;q).

  • Big q-Jacobi: Pn(x;a,b,c;q).

  • Little q-Jacobi: pn(x;a,b;q).

  • q-Laguerre: Ln(α)(x;q).

  • Stieltjes–Wigert: Sn(x;q).

  • Discrete q-Hermite I: hn(x;q).

  • Discrete q-Hermite II: h~n(x;q).

Askey–Wilson Class OP’s

  • Askey–Wilson: pn(x;a,b,c,d|q).

  • Al-Salam–Chihara: Qn(x;a,b|q).

  • Continuous q-Ultraspherical: Cn(x;β|q).

  • Continuous q-Hermite: Hn(x|q).

  • Continuous q-1-Hermite: hn(x|q)

  • q-Racah: Rn(x;α,β,γ,δ|q).

Other OP’s

  • Bessel: yn(x;a).

  • Pollaczek: Pn(λ)(x;a,b).

Classical OP’s in Two Variables

  • Disk: Rm,n(α)(z).

  • Triangle: Pm,nα,β,γ(x,y).

§18.1(iii) Other Notations

In Szegö (1975, §4.7) the ultraspherical polynomials Cn(λ)(x) are denoted by Pn(λ)(x). The ultraspherical polynomials will not be considered for λ=0. They are defined in the literature by C0(0)(x)=1 and

18.1.1 Cn(0)(x)=2nTn(x)=2(n-1)!(12)nPn(-12,-12)(x),

Nor do we consider the shifted Jacobi polynomials:

18.1.2 Gn(p,q,x)=n!(n+p)nPn(p-q,q-1)(2x-1),

or the dilated Chebyshev polynomials of the first and second kinds:

18.1.3 Cn(x) =2Tn(12x),
Sn(x) =Un(12x).

In Koekoek et al. (2010) δx denotes the operator iδx.

In Mason and Handscomb (2003), the definitions of the Chebyshev polynomials of the third and fourth kinds Vn(x) and Wn(x) are the converse of the definitions in this chapter.