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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.