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§18.30 Associated OP’s

  1. §18.30(i) Associated Jacobi Polynomials
  2. §18.30(ii) Associated Legendre Polynomials
  3. §18.30(iii) Associated Laguerre Polynomials
  4. §18.30(iv) Associated Hermite Polynomials
  5. §18.30(v) Associated Meixner–Pollaczek Polynomials
  6. §18.30(vi) Corecursive Orthogonal Polynomials
  7. §18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials
  8. §18.30(viii) Other Associated Polynomials

Assuming equation (18.2.8) with its initialization defines a set of OP’s, pn(x), the corresponding associated orthogonal polynomials of order c are the pn(x;c) as defined by shifting the index n in the recurrence coefficients by adding a constant c, functions of n, say f(n), being replaced by f(n+c). The inequality AnAn+1Cn+1>0, for n0 is replaced by

18.30.1 An+cAn+c+1Cn+c+1>0,

The constant c is usually taken as a positive integer. However, if the recurrence coefficients are polynomial, or rational, functions of n, polynomials of degree n may be well defined for c provided that An+cBn+c0,n=0,1, Askey and Wimp (1984).

The order c recurrence is initialized as

18.30.2 p1(x;c) =0,
p0(x;c) =1,

and then for consecutive n=0,1,2,

18.30.3 pn+1(x;c)=(An+cx+Bn+c)pn(x;c)Cn+cpn1(x;c).

Associated polynomials and the related corecursive polynomials appear in Ismail (2009, §§2.3, 2.6, and 2.10), where the relationship of OP’s to continued fractions is made evident. The lowest order monic versions of both of these appear in §18.2(x), (18.2.31) defining the c=1 associated monic polynomials, and (18.2.32) their closely related cousins the c=0 corecursive polynomials.

§18.30(i) Associated Jacobi Polynomials

These are defined by

18.30.4 Pn(α,β)(x;c)=pn(x;c),

where pn(x;c) is given by (18.30.2) and (18.30.3), with An, Bn, and Cn as in (18.9.2). Explicitly,

18.30.5 (1)n(α+β+c+1)nn!Pn(α,β)(x;c)(α+β+2c+1)n(β+c+1)n==0n(n)(n+α+β+2c+1)(c+1)(β+c+1)(12x+12)F34(n,n++α+β+2c+1,β+c,cβ++c+1,+c+1,α+β+2c;1),

where the generalized hypergeometric function F34 is defined by (16.2.1).

For corresponding corecursive associated Jacobi polynomials, corecursive associated polynomials being discussed in §18.30(vii), see Letessier (1995). For other results for associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991).

§18.30(ii) Associated Legendre Polynomials

These are defined by

18.30.6 Pn(x;c)=Pn(0,0)(x;c),


18.30.7 Pn(x;c)==0nc+cP(x)Pn(x),

in which Pn(x) are the Legendre polynomials of Table 18.3.1.

For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12).

§18.30(iii) Associated Laguerre Polynomials

The recursion relation for the associated Laguerre polynomials, see (18.30.2), (18.30.3) is

18.30.8 L1λ(x;c) =0,
L0λ(x;c) =1,


18.30.9 (n+c+1)Ln+1λ(x;c)=(2n+2c+λ+1x)Lnλ(x;c)(n+c+λ)Ln1λ(x;c),


18.30.10 0Lnλ(x;c)Lmλ(x;c)wλ(x,c)dx=Γ(n+c+λ+1)Γ(c+1)(c+1)nδn,m,
c+λ>1, c0, or c+λ0, c>1,

with weight function

18.30.11 wλ(x,c)=xλex|U(c,1λ,xeiπ)|2.

For the confluent hypergeometric function U see §13.2(i). These constraints guarantee that the orthogonality only involves the integral x[0,), as above.

For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real x-axis each multiplied by the polynomial product evaluated at the corresponding values of x, as in (18.2.3).

§18.30(iv) Associated Hermite Polynomials

The recursion relation for the associated Hermite polynomials, see (18.30.2), and (18.30.3), is

18.30.12 H1(x;c) =0,
H0(x;c) =1,


18.30.13 Hn+1(x;c)=2xHn(x;c)2(n+c)Hn1(x;c),


18.30.14 Hn(x;c)Hm(x;c)w(x,c)dx=2nπ1/2Γ(n+c+1)δn,m,

with weight function

18.30.15 w(x,c)=|U(c12,ix2)|2.

For the parabolic cylinder function U see §12.2(i).

§18.30(v) Associated Meixner–Pollaczek Polynomials

In view of (18.22.8) the associated Meixner–Pollaczek polynomials 𝒫nλ(x;ϕ,c) are defined by the recurrence relation

18.30.16 𝒫1λ(x;ϕ,c) =0,
𝒫0λ(x;ϕ,c) =1,
(n+c+1)𝒫n+1λ(x;ϕ,c) =(2xsinϕ+2(n+c+λ)cosϕ)𝒫nλ(x;ϕ,c)(n+c+2λ1)𝒫n1λ(x;ϕ,c),

They can be expressed in terms of type 3 Pollaczek polynomials (which are also associated type 2 Pollaczek polynomials) by (18.35.10).


18.30.17 𝒫nλ(x;ϕ,c)𝒫mλ(x;ϕ,c)w(λ)(x,ϕ,c)dx=Γ(n+c+2λ)Γ(c+1)(c+1)nδn,m,
0<ϕ<π,c+2λ>0,c0 or 0<ϕ<π,c+2λ1,c>1,

with weight function

18.30.18 w(λ)(x,ϕ,c)=e(2ϕπ)x(2sinϕ)2λ|Γ(c+λ+ix)|22π|F(1λ+ix,c;c+λ+ix;e2iϕ)|2.

For Gauss’ hypergeometric function F see (15.2.1).

The results in the previous two subsections are special limits:

18.30.19 Lnλ(x;c)=limϕ0𝒫n(λ+1)/2(x2sinϕ;ϕ,c),


18.30.20 Hn(x;c)=(c+1)nlimλλn/2𝒫nλ(xλ1/2;π/2,c).

The corresponding results for c=0 appear as (18.21.12) and (18.21.13), respectively.

§18.30(vi) Corecursive Orthogonal Polynomials

The corecursive orthogonal polynomials, pn(0)(x), these being linearly independent solutions of the recurrence for the pn(x), are defined as follows:

18.30.21 p0(0)(x) =0,
p1(0)(x) =A0,

and then, as per usual, then, for consecutive n=1,2,,

18.30.22 pn+1(0)(x)=(Anx+Bn)pn(0)(x)Cnpn1(0)(x).

Note that this is the same recurrence as in (18.2.8) for the traditional OP’s, but with a different initialization. Ismail (2009, §2.3) discusses the meaning of linearly independent in this situation.

Numerator and Denominator Polynomials

The pn(0)(x) are also referred to as the numerator polynomials, the pn(x) then being the denominator polynomials, in that the n-th approximant of the continued fraction, z,

18.30.23 F(z)=A0A0z+B0C1A1z+B1C2A2z+B2

is given by

18.30.24 Fn(z)=pn(0)(z)/pn(z)=A0A0z+B0C1A1z+B1Cn1An1z+Bn1.

F(z) and Fn(z) of (18.30.23) and (18.30.24) are, also, precisely those of (18.2.34) and (18.2.35), now expressed via the traditional, An, Bn, Cn coefficients, rather than the monic, αn, βn, recursion coefficients.

Markov’s Theorem

The ratio pn(0)(z)/pn(z), as defined here, thus provides the same statement of Markov’s Theorem, as in (18.2.9_5), but now in terms of differently obtained numerator and denominator polynomials. Namely, if the interval [a,b] is bounded, then

18.30.25 limnFn(x)=limnpn(0)(z)/pn(z)=1μ0abdμ(x)zx,

Ismail (2009, §2.6) discusses this in a different Nn/Dn notation; also note the assumption that μ0=1, made throughout that reference, Ismail (2009, p. 16).

§18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials

Defining associated orthogonal polynomials and their relationship to their corecursive counterparts is particularly simple via use of the recursion relations for the monic, rather than via those for the traditional polynomials. The simplicity of the relationship follows from the fact that the monic polynomials have been rescaled so that the coefficient of the highest power of x in pn(x), namely, xn, is unity; for a note on this standardization, see §18.2(iii). The notations p^n(x;c) and p^n(0)(x) are used here to distinguish the two sets of monic polynomials from the (traditional) polynomials pn(x;c) and pn(0)(x) of the preceding subsection.

Associated Monic OP’s

In the monic case, the monic associated polynomials p^n(x;c) of order c with respect to the p^n(x) are obtained by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to

18.30.26 p^0(x;c) =1,
p^1(x;c) =xαc,

and employing the recurrence

18.30.27 xp^n(x;c)=p^n+1(x;c)+αn+cp^n(x;c)+βn+cp^n1(x;c),

The “Zeroth” Corecursive Monic OP

The zeroth order corecursive monic polynomials p^n(0)(x) follow directly from the alternate initialization

18.30.28 p^0(0)(x) =0,
p^1(0)(x) =1,

followed by use of the c=0 recursion of (18.30.27).

Relationship of Monic Corecursive and Monic Associated OP’s

It is easily seen that p^2(0)(x)=p^1(x;1)=xα1, and then

18.30.29 p^n(0)(x)=p^n1(x;1)

follows by induction on n. This being the relationship established in §18.2(x) following (18.2.32). The usage of §18.2(x), where the monic associated polynomials, there denoted pn(1)(x), instead of p^n(x;1), are referred to as the first associated such polynomials in §18.2(x), is now evident. The ratio p^n1(x;1)/p^n(x) is then the Fn(x) of (18.2.35), leading to Markov’s theorem as stated in (18.30.25).

More generally, the kth corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the c=k recurrence of (18.30.27)) are related to the (k+1)st monic associated polynomials by

18.30.30 p^n(k)(x)=p^n1(x;k+1).

See Ismail (2009, p. 46 ), where the kth corecursive polynomial is also related to an appropriate continued fraction, given here as its nth convergent,

18.30.31 Fn(x,k)=p^n(k)(x)/p^n(x;k).

§18.30(viii) Other Associated Polynomials

For associated Askey–Wilson polynomials see Rahman (2001). The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. The relationship (18.35.8) implies the definition for the associated ultraspherical polynomials Cn(λ)(x;c)=Pn(λ)(x;0,0,c).