18 Orthogonal PolynomialsOther Orthogonal Polynomials18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes18.31 Bernstein–Szegő Polynomials

- §18.30(i) Associated Jacobi Polynomials
- §18.30(ii) Associated Legendre Polynomials
- §18.30(iii) Associated Laguerre Polynomials
- §18.30(iv) Associated Hermite Polynomials
- §18.30(v) Associated Meixner–Pollaczek Polynomials
- §18.30(vi) Corecursive Orthogonal Polynomials
- §18.30(vii) Corecursive and Associated Monic Orthogonal Polynomials
- §18.30(viii) Other Associated Polynomials

Assuming equation (18.2.8) with its initialization defines a set of OP’s, ${p}_{n}(x)$, the corresponding *associated orthogonal polynomials* of order $c$ are the ${p}_{n}(x;c)$ 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 ${A}_{n}{A}_{n+1}{C}_{n+1}>0$, for $n\ge 0$ is replaced by

18.30.1 | $${A}_{n+c}{A}_{n+c+1}{C}_{n+c+1}>0,$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

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 $c\in \mathbb{R}$ provided that ${A}_{n+c}{B}_{n+c}\ne 0,n=0,1,\mathrm{\dots}$ Askey and Wimp (1984).

The order $c$ recurrence is initialized as

18.30.2 | ${p}_{-1}(x;c)$ | $=0,$ | ||

${p}_{0}(x;c)$ | $=1,$ | |||

and then for consecutive $n=0,1,2,\mathrm{\dots}$

18.30.3 | $${p}_{n+1}(x;c)=({A}_{n+c}x+{B}_{n+c}){p}_{n}(x;c)-{C}_{n+c}{p}_{n-1}(x;c).$$ | ||

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.

These are defined by

18.30.4 | $${P}_{n}^{(\alpha ,\beta )}(x;c)={p}_{n}(x;c),$$ | ||

$n=0,1,\mathrm{\dots}$, | |||

where ${p}_{n}(x;c)$ is given by (18.30.2) and (18.30.3), with ${A}_{n}$, ${B}_{n}$, and ${C}_{n}$ as in (18.9.2). Explicitly,

18.30.5 | $$\begin{array}{l}\frac{{(-1)}^{n}{\left(\alpha +\beta +c+1\right)}_{n}n!{P}_{n}^{(\alpha ,\beta )}(x;c)}{{\left(\alpha +\beta +2c+1\right)}_{n}{\left(\beta +c+1\right)}_{n}}\\ \phantom{\rule{2em}{0ex}}=\sum _{\mathrm{\ell}=0}^{n}\frac{{\left(-n\right)}_{\mathrm{\ell}}{\left(n+\alpha +\beta +2c+1\right)}_{\mathrm{\ell}}}{{\left(c+1\right)}_{\mathrm{\ell}}{\left(\beta +c+1\right)}_{\mathrm{\ell}}}{\left(\frac{1}{2}x+\frac{1}{2}\right)}^{\mathrm{\ell}}{}_{4}{}^{}F_{3}^{}(\genfrac{}{}{0pt}{}{\mathrm{\ell}-n,n+\mathrm{\ell}+\alpha +\beta +2c+1,\beta +c,c}{\beta +\mathrm{\ell}+c+1,\mathrm{\ell}+c+1,\alpha +\beta +2c};1),\end{array}$$ | ||

where the generalized hypergeometric function ${}_{4}{}^{}F_{3}^{}$ 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).

These are defined by

18.30.6 | $${P}_{n}(x;c)={P}_{n}^{(0,0)}(x;c),$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

Explicitly,

18.30.7 | $${P}_{n}(x;c)=\sum _{\mathrm{\ell}=0}^{n}\frac{c}{\mathrm{\ell}+c}{P}_{\mathrm{\ell}}\left(x\right){P}_{n-\mathrm{\ell}}\left(x\right),$$ | ||

in which ${P}_{n}\left(x\right)$ are the Legendre polynomials of Table 18.3.1.

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

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

18.30.8 | ${L}_{-1}^{\lambda}(x;c)$ | $=0,$ | ||

${L}_{0}^{\lambda}(x;c)$ | $=1,$ | |||

and

18.30.9 | $$(n+c+1){L}_{n+1}^{\lambda}(x;c)=(2n+2c+\lambda +1-x){L}_{n}^{\lambda}(x;c)-(n+c+\lambda ){L}_{n-1}^{\lambda}(x;c),$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

Orthogonality

18.30.10 | $${\int}_{0}^{\mathrm{\infty}}{L}_{n}^{\lambda}(x;c){L}_{m}^{\lambda}(x;c){w}^{\lambda}(x,c)dx=\frac{\mathrm{\Gamma}\left(n+c+\lambda +1\right)\mathrm{\Gamma}\left(c+1\right)}{{\left(c+1\right)}_{n}}{\delta}_{n,m},$$ | ||

$c+\lambda >-1$, $c\ge 0$, or $c+\lambda \ge 0$, $c>-1$, | |||

with weight function

18.30.11 | $${w}^{\lambda}(x,c)=\frac{{x}^{\lambda}{\mathrm{e}}^{-x}}{{\left|U(c,1-\lambda ,x{\mathrm{e}}^{-\mathrm{i}\pi})\right|}^{2}}.$$ | ||

For the confluent hypergeometric function $U$ see §13.2(i). These constraints guarantee that the orthogonality only involves the integral $x\in [0,\mathrm{\infty})$, as above.

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

18.30.12 | ${H}_{-1}(x;c)$ | $=0,$ | ||

${H}_{0}(x;c)$ | $=1,$ | |||

and

18.30.13 | $${H}_{n+1}(x;c)=2x{H}_{n}(x;c)-2(n+c){H}_{n-1}(x;c),$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

Orthogonality

18.30.14 | $${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{H}_{n}(x;c){H}_{m}(x;c)w(x,c)dx={2}^{n}{\pi}^{1/2}\mathrm{\Gamma}\left(n+c+1\right){\delta}_{n,m},$$ | ||

$c>-1$, | |||

with weight function

18.30.15 | $$w(x,c)={\left|U(c-\frac{1}{2},\mathrm{i}x\sqrt{2})\right|}^{-2}.$$ | ||

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

In view of (18.22.8) the associated Meixner–Pollaczek polynomials ${\mathcal{P}}_{n}^{\lambda}(x;\varphi ,c)$ are defined by the recurrence relation

18.30.16 | ${\mathcal{P}}_{-1}^{\lambda}(x;\varphi ,c)$ | $=0,$ | ||

${\mathcal{P}}_{0}^{\lambda}(x;\varphi ,c)$ | $=1,$ | |||

$(n+c+1){\mathcal{P}}_{n+1}^{\lambda}(x;\varphi ,c)$ | $=(2x\mathrm{sin}\varphi +2(n+c+\lambda )\mathrm{cos}\varphi ){\mathcal{P}}_{n}^{\lambda}(x;\varphi ,c)-(n+c+2\lambda -1){\mathcal{P}}_{n-1}^{\lambda}(x;\varphi ,c),$ | |||

$n=0,1,\mathrm{\dots}$. | ||||

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

Orthogonality

18.30.17 | $${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\mathcal{P}}_{n}^{\lambda}(x;\varphi ,c){\mathcal{P}}_{m}^{\lambda}(x;\varphi ,c){w}^{(\lambda )}(x,\varphi ,c)dx=\frac{\mathrm{\Gamma}\left(n+c+2\lambda \right)\mathrm{\Gamma}\left(c+1\right)}{{\left(c+1\right)}_{n}}{\delta}_{n,m},$$ | ||

$$ or $$, | |||

with weight function

18.30.18 | $${w}^{(\lambda )}(x,\varphi ,c)=\frac{{\mathrm{e}}^{(2\varphi -\pi )x}{\left(2\mathrm{sin}\varphi \right)}^{2\lambda}{\left|\mathrm{\Gamma}\left(c+\lambda +\mathrm{i}x\right)\right|}^{2}}{2\pi {\left|F(1-\lambda +\mathrm{i}x,c;c+\lambda +\mathrm{i}x;{\mathrm{e}}^{2\mathrm{i}\varphi})\right|}^{2}}.$$ | ||

For Gauss’ hypergeometric function $F$ see (15.2.1).

The results in the previous two subsections are special limits:

18.30.19 | $${L}_{n}^{\lambda}(x;c)=\underset{\varphi \to 0}{lim}{\mathcal{P}}_{n}^{(\lambda +1)/2}(\frac{-x}{2\mathrm{sin}\varphi};\varphi ,c),$$ | ||

and

18.30.20 | $${H}_{n}(x;c)={\left(c+1\right)}_{n}\underset{\lambda \to \mathrm{\infty}}{lim}{\lambda}^{-n/2}{\mathcal{P}}_{n}^{\lambda}(x{\lambda}^{1/2};\pi /2,c).$$ | ||

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

The *corecursive orthogonal polynomials*, ${p}_{n}^{(0)}(x)$, these being linearly independent solutions of the recurrence for the ${p}_{n}(x)$, are defined as follows:

18.30.21 | ${p}_{0}^{(0)}(x)$ | $=0,$ | ||

${p}_{1}^{(0)}(x)$ | $={A}_{0},$ | |||

and then, as per usual, then, for consecutive $n=1,2,\mathrm{\dots}$,

18.30.22 | $${p}_{n+1}^{(0)}(x)=({A}_{n}x+{B}_{n}){p}_{n}^{(0)}(x)-{C}_{n}{p}_{n-1}^{(0)}(x).$$ | ||

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.

The ${p}_{n}^{(0)}(x)$ are also referred to as the *numerator polynomials*, the ${p}_{n}(x)$ then being the *denominator* polynomials,
in that the $n$-th approximant of the continued fraction, $z\in \u2102$,

18.30.23 | $$F(z)=\frac{{A}_{0}}{{A}_{0}z+{B}_{0}-}\frac{{C}_{1}}{{A}_{1}z+{B}_{1}-}\frac{{C}_{2}}{{A}_{2}z+{B}_{2}-}\mathrm{\cdots}$$ | ||

is given by

18.30.24 | $${F}_{n}(z)={p}_{n}^{(0)}(z)/{p}_{n}(z)=\frac{{A}_{0}}{{A}_{0}z+{B}_{0}-}\frac{{C}_{1}}{{A}_{1}z+{B}_{1}-}\mathrm{\cdots}\frac{{C}_{n-1}}{{A}_{n-1}z+{B}_{n-1}}.$$ | ||

$F(z)$ and ${F}_{n}(z)$ 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, ${A}_{n}$, ${B}_{n}$, ${C}_{n}$ coefficients, rather than the monic, ${\alpha}_{n}$, ${\beta}_{n}$, recursion coefficients.

The ratio ${p}_{n}^{(0)}(z)/{p}_{n}(z)$, 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 | $$\underset{n\to \mathrm{\infty}}{lim}{F}_{n}(x)=\underset{n\to \mathrm{\infty}}{lim}{p}_{n}^{(0)}(z)/{p}_{n}(z)=\frac{1}{{\mu}_{0}}{\int}_{a}^{b}\frac{d\mu (x)}{z-x},$$ | ||

$z\in \u2102\backslash [a,b]$. | |||

Ismail (2009, §2.6) discusses this in a different ${N}_{n}/{D}_{n}$ notation; also note the assumption that ${\mu}_{0}=1$, made throughout that reference, Ismail (2009, p. 16).

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 ${p}_{n}(x)$, namely, ${x}^{n}$, is unity; for a note on this
standardization, see §18.2(iii). The notations ${\widehat{p}}_{n}(x;c)$ and ${\widehat{p}}_{n}^{(0)}(x)$ are used here to distinguish the two sets of
*monic* polynomials from the (traditional) polynomials ${p}_{n}(x;c)$ and ${p}_{n}^{(0)}(x)$ of the preceding subsection.

In the monic case, the *monic associated polynomials* ${\widehat{p}}_{n}(x;c)$ of order $c$ with respect to the ${\widehat{p}}_{n}(x)$ are obtained
by simply changing the initialization and recursions, respectively, of (18.30.2) and (18.30.3) to

18.30.26 | ${\widehat{p}}_{0}(x;c)$ | $=1,$ | ||

${\widehat{p}}_{1}(x;c)$ | $=x-{\alpha}_{c},$ | |||

and employing the recurrence

18.30.27 | $$x{\widehat{p}}_{n}(x;c)={\widehat{p}}_{n+1}(x;c)+{\alpha}_{n+c}{\widehat{p}}_{n}(x;c)+{\beta}_{n+c}{\widehat{p}}_{n-1}(x;c),$$ | ||

$n=1,2,\mathrm{\dots}$. | |||

The *zeroth order corecursive monic polynomials* ${\widehat{p}}_{n}^{(0)}(x)$ follow directly from the alternate initialization

18.30.28 | ${\widehat{p}}_{0}^{(0)}(x)$ | $=0,$ | ||

${\widehat{p}}_{1}^{(0)}(x)$ | $=1,$ | |||

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

It is easily seen that ${\widehat{p}}_{2}^{(0)}(x)={\widehat{p}}_{1}(x;1)=x-{\alpha}_{1}$, and then

18.30.29 | $${\widehat{p}}_{n}^{(0)}(x)={\widehat{p}}_{n-1}(x;1)$$ | ||

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 ${p}_{n}^{(1)}(x)$, instead of ${\widehat{p}}_{n}(x;1)$, are referred
to as the *first associated* such polynomials in §18.2(x), is now evident.
The ratio ${\widehat{p}}_{n-1}(x;1)/{\widehat{p}}_{n}(x)$ is then the ${F}_{n}(x)$ of (18.2.35), leading to Markov’s theorem as stated
in (18.30.25).

More generally, the $k$th 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 | $${\widehat{p}}_{n}^{(k)}(x)={\widehat{p}}_{n-1}(x;k+1).$$ | ||

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

18.30.31 | $${F}_{n}(x,k)={\widehat{p}}_{n}^{(k)}(x)/{\widehat{p}}_{n}(x;k).$$ | ||

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 ${C}_{n}^{(\lambda )}(x;c)={P}_{n}^{(\lambda )}(x;0,0,c)$.