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§18.35 Pollaczek Polynomials

  1. §18.35(i) Definition and Hypergeometric Representation
  2. §18.35(ii) Orthogonality
  3. §18.35(iii) Other Properties

§18.35(i) Definition and Hypergeometric Representation

There are 3 types of Pollaczek polynomials:

18.35.0_5 Pn(12)(x;a,b),
Pn(λ)(x;a,b) =Pn(λ)(x;a,b,0),

Thus type 3 with c=0 reduces to type 2, and type 3 with c=0 and λ=12 reduces to type 1, also in subsequent formulas. The three types of Pollaczek polynomials were successively introduced in Pollaczek (1949a, b, 1950), see also Erdélyi et al. (1953b, p.219) and, for type 1 and 2, Szegö (1950) and Askey (1982b). The type 2 polynomials reduce for a=b=0 to ultraspherical polynomials, see (18.35.8).

The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8))

18.35.1 P1(λ)(x;a,b,c) =0,
P0(λ)(x;a,b,c) =1,
18.35.2 Pn+1(λ)(x;a,b,c) =2(n+c+λ+a)x+2bn+c+1Pn(λ)(x;a,b,c)n+c+2λ1n+c+1Pn1(λ)(x;a,b,c),

or, equivalently in second form (18.2.10),

18.35.2_1 xPn(λ)(x;a,b,c)=n+c+12(n+c+λ+a)Pn+1(λ)(x;a,b,c)bn+c+λ+aPn(λ)(x;a,b,c)+n+c+2λ12(n+c+λ+a)Pn1(λ)(x;a,b,c),

For the monic polynomials

18.35.2_2 Qn(λ)(x;a,b,c)=(c+1)n2n(c+λ+a)nPn(λ)(x;a,b,c)

the recurrence relation of form (18.2.11_5) becomes

18.35.2_3 Q1(λ)(x;a,b,c) =0,
Q0(λ)(x;a,b,c) =1,
18.35.2_4 xQn(λ)(x;a,b,c) =Qn+1(λ)(x;a,b,c)bn+c+λ+aQn(λ)(x;a,b,c)+(n+c)(n+c+2λ1)4(n+c+λ+a1)(n+c+λ+a)Qn1(λ)(x;a,b,c),

There is the symmetry

18.35.2_5 Pn(λ)(x;a,b,c)=(1)nPn(λ)(x;a,b,c).

As in the coefficients of the above recurrence relations n and c only occur in the form n+c, the type 3 Pollaczek polynomials may also be called the associated type 2 Pollaczek polynomials by using the terminology of §18.30.

For type 2, with notation

18.35.3 τa,b(θ)=acosθ+bsinθ,

we have the explicit representations

18.35.4 Pn(λ)(cosθ;a,b)=(λiτa,b(θ))nn!einθF(n,λ+iτa,b(θ)nλ+1+iτa,b(θ);e2iθ)==0n(λ+iτa,b(θ))!(λiτa,b(θ))n(n)!ei(n2)θ,
18.35.4_5 Pn(λ)(cosθ;a,b)=(2λ)nn!einθF(n,λ+iτa,b(θ)2λ;1e2iθ).

For type 1 take λ=12 and for Gauss’ hypergeometric function F see (15.2.1).

§18.35(ii) Orthogonality

First consider type 2.

18.35.5 11Pn(λ)(x;a,b)Pm(λ)(x;a,b)w(λ)(x;a,b)dx=Γ(2λ+n)n!(λ+a+n)δn,m,
aba, λ>0,


18.35.6 w(λ)(cosθ;a,b)=π1e(2θπ)τa,b(θ)(2sinθ)2λ1|Γ(λ+iτa,b(θ))|2,

Note that

18.35.6_1 ln(w(λ)(cosθ;a,b))={2π(a+b)θ1+(2λ1)ln(a+b)+λln4+2(a+b)+O(θ),θ0+,2π(ba)(πθ)1+(2λ1)ln(ab)+λln4+2(ab)+O(πθ),θπ,

indicating the presence of essential singularities. Hence, only in the case a=b=0 does ln(w(λ)(x;a,b)) satisfy the condition (18.2.39) for the Szegő class 𝒢.

More generally, the Pn(λ)(x;a,b) are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)).

18.35.6_2 (i)λ>0 and a+λ>0,(ii)12<λ<0 and 1<a+λ<0,(iii)λ=0 and a=b=0.


18.35.6_3 11Pn(λ)(x;a,b)Pm(λ)(x;a,b)w(λ)(x;a,b)dx+ζDPn(λ)(ζ;a,b)Pm(λ)(ζ;a,b)wζ(λ)(a,b)=Γ(2λ+n)n!(λ+a+n)δn,m,

where, depending on a,b,λ, D is a discrete subset of and the wζ(λ)(a,b) are certain weights. See Ismail (2009, §5.5). In particular, if a>b>a and condition (ii) of (18.35.6_2) holds then |D|=2 (see Ismail (2009, Theorem 5.5.1)). Also, if b>ab, λ+a>0 then

18.35.6_4 D ={xk=(λ+k)Δaba2(λ+k)2}k=0,
wxk(λ)(a,b) =ρ2k1(1ρ2)2λ+1Γ(2λ+k)2Δk!,
Δ =(λ+k)2+b2a2,
ρ =Δbλ+ka,

and similarly if ba>b, λ+a>0 by application of (18.35.2_5).

For type 3 orthogonality (18.35.5) generalizes to

18.35.6_5 11Pn(λ)(x;a,b,c)Pm(λ)(x;a,b,c)w(λ)(x;a,b,c)dx=Γ(c+1)Γ(2λ+c+n)(c+1)n(λ+a+c+n)δn,m,


18.35.6_6 w(λ)(cosθ;a,b,c)=e(2θπ)τa,b(θ)(2sinθ)2λ1|Γ(c+λ+iτa,b(θ))|2π|F(1λ+iτa,b(θ),cc+λ+iτa,b(θ);e2iθ)|2,

with two possible constraints: a>b>a, 2λ+c>0, c0, or a>b>a, 2λ+c1, c>1. For Gauss’ hypergeometric function F see (15.2.1).

§18.35(iii) Other Properties

18.35.7 (1zeiθ)λ+iτa,b(θ)(1zeiθ)λiτa,b(θ)=n=0Pn(λ)(cosθ;a,b)zn,
|z|<1, 0<θ<π.
18.35.8 Pn(λ)(x;0,0)=Cn(λ)(x),
18.35.9 Pn(λ)(x;ϕ) =Pn(λ)(cosϕ;0,xsinϕ),
Pn(λ)(cosθ;a,b) =Pn(λ)(τa,b(θ);θ),
18.35.10 𝒫nλ(x;ϕ,c)=Pn(λ)(cosϕ;0,xsinϕ,c).

For the ultraspherical polynomials Cn(λ)(x), the Meixner–Pollaczek polynomials Pn(λ)(x;ϕ) and the associated Meixner–Pollaczek polynomials 𝒫nλ(x;ϕ,c) see §§18.3, 18.19 and 18.30(v), respectively.

See Bo and Wong (1996) for an asymptotic expansion of Pn(12)(cos(n12θ);a,b) as n, with a and b fixed. This expansion is in terms of the Airy function Ai(x) and its derivative (§9.2), and is uniform in any compact θ-interval in (0,). Also included is an asymptotic approximation for the zeros of Pn(12)(cos(n12θ);a,b).

See Szegő (1975, Appendix, §§ 1–5), Askey (1982b), and Ismail (2009, §§ 5.4–5.5) for further results on type 2 Pollaczek polynomials. These polynomials also occur in connection with the Coulomb problem, see §18.39(iv).