# §18.30 Associated OP’s

###### Contents

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 $n\geq 0$ is replaced by

 18.30.1 $A_{n+c}A_{n+c+1}C_{n+c+1}>0,$ $n=0,1,\dots$. ⓘ Symbols: $n$: nonnegative integer, $A_{n}$: coefficient and $C_{n}$: coefficient Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E1 Encodings: TeX, pMML, png See also: Annotations for §18.30 and Ch.18

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 $A_{n+c}B_{n+c}\neq 0,n=0,1,\dots$ Askey and Wimp (1984).

The order $c$ recurrence is initialized as

 18.30.2 $\displaystyle p_{-1}(x;c)$ $\displaystyle=0,$ $\displaystyle p_{0}(x;c)$ $\displaystyle=1,$ ⓘ Symbols: $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.30(vii), §18.30(i), §18.30(iii), §18.30(iv) Permalink: http://dlmf.nist.gov/18.30.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30 and Ch.18

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

 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.

## §18.30(i) Associated Jacobi Polynomials

These are defined by

 18.30.4 $P^{(\alpha,\beta)}_{n}\left(x;c\right)=p_{n}(x;c),$ $n=0,1,\dots$, ⓘ Defines: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Jacobi polynomial Symbols: $p_{n}(x)$: polynomial of degree $n$, $n$: nonnegative integer and $x$: real variable Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E4 Encodings: TeX, pMML, png See also: Annotations for §18.30(i), §18.30 and Ch.18

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 $\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$

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).

## §18.30(ii) Associated Legendre Polynomials

These are defined by

 18.30.6 $P_{n}\left(x;c\right)=P^{(0,0)}_{n}\left(x;c\right),$ $n=0,1,\dots$. ⓘ Defines: $P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Legendre polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.30 Permalink: http://dlmf.nist.gov/18.30.E6 Encodings: TeX, pMML, png See also: Annotations for §18.30(ii), §18.30 and Ch.18

Explicitly,

 18.30.7 $P_{n}\left(x;c\right)=\sum_{\ell=0}^{n}\frac{c}{\ell+c}P_{\ell}\left(x\right)P% _{n-\ell}\left(x\right),$

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).

## §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 $\displaystyle L^{\lambda}_{-1}\left(x;c\right)$ $\displaystyle=0,$ $\displaystyle L^{\lambda}_{0}\left(x;c\right)$ $\displaystyle=1,$ ⓘ Symbols: $L^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Laguerre polynomial and $x$: real variable Referenced by: §18.30, Erratum (V1.2.0) §18.30 Permalink: http://dlmf.nist.gov/18.30.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30(iii), §18.30 and Ch.18

and

 18.30.9 $(n+c+1)L^{\lambda}_{n+1}\left(x;c\right)={(2n+2c+\lambda+1-x)}L^{\lambda}_{n}% \left(x;c\right)-{(n+c+\lambda)}L^{\lambda}_{n-1}\left(x;c\right),$ $n=0,1,\dots$. ⓘ Symbols: $L^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Laguerre polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: (18.30.19) Permalink: http://dlmf.nist.gov/18.30.E9 Encodings: TeX, pMML, png See also: Annotations for §18.30(iii), §18.30 and Ch.18

Orthogonality

 18.30.10 $\int_{0}^{\infty}L^{\lambda}_{n}\left(x;c\right)L^{\lambda}_{m}\left(x;c\right% )w^{\lambda}(x,c)\,\mathrm{d}x=\frac{\Gamma\left(n+c+\lambda+1\right)\Gamma% \left(c+1\right)}{{\left(c+1\right)_{n}}}\delta_{n,m},$ $c+\lambda>-1$, $c\geq 0$, or $c+\lambda\geq 0$, $c>-1$,

with weight function

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

For the confluent hypergeometric function $U$ see §13.2(i). These constraints guarantee that the orthogonality only involves the integral $x\in[0,\infty)$, 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 $\displaystyle H_{-1}\left(x;c\right)$ $\displaystyle=0,$ $\displaystyle H_{0}\left(x;c\right)$ $\displaystyle=1,$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Hermite polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.30.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30(iv), §18.30 and Ch.18

and

 18.30.13 $H_{n+1}\left(x;c\right)={2x}H_{n}\left(x;c\right)-{2(n+c)}H_{n-1}\left(x;c% \right),$ $n=0,1,\dots$. ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Hermite polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: (18.30.20) Permalink: http://dlmf.nist.gov/18.30.E13 Encodings: TeX, pMML, png See also: Annotations for §18.30(iv), §18.30 and Ch.18

Orthogonality

 18.30.14 $\int_{-\infty}^{\infty}H_{n}\left(x;c\right)H_{m}\left(x;c\right)w(x,c)\,% \mathrm{d}x=2^{n}{\pi}^{\ifrac{1}{2}}\Gamma\left(n+c+1\right)\delta_{n,m},$ $c>-1$,

with weight function

 18.30.15 $w(x,c)={\left|U\left(c-\tfrac{1}{2},\mathrm{i}x\sqrt{2}\right)\right|}^{-2}.$ ⓘ Symbols: $\mathrm{i}$: imaginary unit, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $w(x)$: weight function and $x$: real variable Permalink: http://dlmf.nist.gov/18.30.E15 Encodings: TeX, pMML, png See also: Annotations for §18.30(iv), §18.30 and Ch.18

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 ${\mathscr{P}}^{\lambda}_{n}\left(x;\phi,c\right)$ are defined by the recurrence relation

 18.30.16 $\displaystyle{\mathscr{P}}^{\lambda}_{-1}\left(x;\phi,c\right)$ $\displaystyle=0,$ $\displaystyle{\mathscr{P}}^{\lambda}_{0}\left(x;\phi,c\right)$ $\displaystyle=1,$ $\displaystyle(n+c+1){\mathscr{P}}^{\lambda}_{n+1}\left(x;\phi,c\right)$ $\displaystyle=(2x\sin\phi+2(n+c+\lambda)\cos\phi){\mathscr{P}}^{\lambda}_{n}% \left(x;\phi,c\right)-{(n+c+2\lambda-1)}{\mathscr{P}}^{\lambda}_{n-1}\left(x;% \phi,c\right),$ $n=0,1,\dots$. ⓘ Symbols: ${\mathscr{P}}^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{\phi},\NVar{c}\right)$: associated Meixner–Pollaczek polynomial, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $x$: real variable Source: Askey and Wimp (1984, (1.8)) Referenced by: (18.30.19), (18.30.20) Permalink: http://dlmf.nist.gov/18.30.E16 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.30(v), §18.30 and Ch.18

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 $\int_{-\infty}^{\infty}{\mathscr{P}}^{\lambda}_{n}\left(x;\phi,c\right){% \mathscr{P}}^{\lambda}_{m}\left(x;\phi,c\right)w^{(\lambda)}(x,\phi,c)\,% \mathrm{d}x=\frac{\Gamma\left(n+c+2\lambda\right)\Gamma\left(c+1\right)}{{% \left(c+1\right)_{n}}}\,\delta_{n,m},$ $0<\phi<\pi,c+2\lambda>0,c\geq 0$ or $0<\phi<\pi,c+2\lambda\geq 1,c>-1$,

with weight function

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

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

The results in the previous two subsections are special limits:

 18.30.19 $L^{\lambda}_{n}\left(x;c\right)=\lim_{\phi\to 0}{\mathscr{P}}^{(\lambda+1)/2}_% {n}\left(\frac{-x}{2\sin\phi};\phi,c\right),$ ⓘ Symbols: $L^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Laguerre polynomial, ${\mathscr{P}}^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{\phi},\NVar{c}\right)$: associated Meixner–Pollaczek polynomial, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $x$: real variable Proof sketch: Using the limit will convert (18.30.16) into (18.30.9). See also Askey and Wimp (1984, §2). Permalink: http://dlmf.nist.gov/18.30.E19 Encodings: TeX, pMML, png See also: Annotations for §18.30(v), §18.30 and Ch.18

and

 18.30.20 $H_{n}\left(x;c\right)={\left(c+1\right)_{n}}\lim_{\lambda\to\infty}\lambda^{-n% /2}{\mathscr{P}}^{\lambda}_{n}\left(x{\lambda}^{1/2};\pi/2,c\right).$ ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $H_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Hermite polynomial, ${\mathscr{P}}^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{\phi},\NVar{c}\right)$: associated Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $x$: real variable Proof sketch: Using the limit will convert (18.30.16) into (18.30.13). See also Askey and Wimp (1984, §4). Permalink: http://dlmf.nist.gov/18.30.E20 Encodings: TeX, pMML, png See also: Annotations for §18.30(v), §18.30 and Ch.18

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, $p_{n}^{(0)}(x)$, these being linearly independent solutions of the recurrence for the $p_{n}(x)$, are defined as follows:

 18.30.21 $\displaystyle p_{0}^{(0)}(x)$ $\displaystyle=0,$ $\displaystyle p_{1}^{(0)}(x)$ $\displaystyle=A_{0},$ ⓘ Symbols: $p_{n}(x)$: polynomial of degree $n$, $x$: real variable and $A_{n}$: coefficient Permalink: http://dlmf.nist.gov/18.30.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30(vi), §18.30 and Ch.18

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

 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.

### Numerator and Denominator Polynomials

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, $z\in\mathbb{C}$,

 18.30.23 $F(z)=\cfrac{A_{0}}{A_{0}z+B_{0}-\cfrac{C_{1}}{A_{1}z+B_{1}-\cfrac{C_{2}}{A_{2}% z+B_{2}-\cdots}}}$ ⓘ Symbols: $z$: complex variable, $A_{n}$: coefficient, $B_{n}$: coefficient and $C_{n}$: coefficient Referenced by: §18.30(vi), §18.30(vi) Permalink: http://dlmf.nist.gov/18.30.E23 Encodings: TeX, pMML, png See also: Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

is given by

 18.30.24 $F_{n}(z)=p_{n}^{(0)}(z)/p_{n}(z)=\cfrac{A_{0}}{A_{0}z+B_{0}-\cfrac{C_{1}}{A_{1% }z+B_{1}-\cdots}}\frac{C_{n-1}}{A_{n-1}z+B_{n-1}}.$

$F(z)$ 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.

### Markov’s Theorem

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 $\lim_{n\to\infty}F_{n}(x)=\lim_{n\to\infty}p_{n}^{(0)}(z)/p_{n}(z)=\frac{1}{% \mu_{0}}\int_{a}^{b}\frac{\,\mathrm{d}\mu(x)}{z-x},$ $z\in\mathbb{C}\backslash[a,b]$.

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).

## §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 $p_{n}(x)$, namely, $x^{n}$, is unity; for a note on this standardization, see §18.2(iii). The notations $\hat{p}_{n}(x;c)$ and $\hat{p}_{n}^{(0)}(x)$ 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.

### Associated Monic OP’s

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

 18.30.26 $\displaystyle\hat{p}_{0}(x;c)$ $\displaystyle=1,$ $\displaystyle\hat{p}_{1}(x;c)$ $\displaystyle=x-\alpha_{c},$ ⓘ Symbols: $x$: real variable Referenced by: §18.30(vii) Permalink: http://dlmf.nist.gov/18.30.E26 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

and employing the recurrence

 18.30.27 $x\hat{p}_{n}(x;c)=\hat{p}_{n+1}(x;c)+\alpha_{n+c}\hat{p}_{n}(x;c)+\beta_{n+c}% \hat{p}_{n-1}(x;c),$ $n=1,2,\dots$. ⓘ Symbols: $n$: nonnegative integer and $x$: real variable Referenced by: §18.30(vii), §18.30(vii), §18.30(vii) Permalink: http://dlmf.nist.gov/18.30.E27 Encodings: TeX, pMML, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

### The “Zeroth” Corecursive Monic OP

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

 18.30.28 $\displaystyle\hat{p}_{0}^{(0)}(x)$ $\displaystyle=0,$ $\displaystyle\hat{p}_{1}^{(0)}(x)$ $\displaystyle=1,$ ⓘ Symbols: $x$: real variable Referenced by: §18.30(vii), §18.30(vii) Permalink: http://dlmf.nist.gov/18.30.E28 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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 $\hat{p}_{2}^{(0)}(x)=\hat{p}_{1}(x;1)=x-\alpha_{1}$, and then

 18.30.29 $\hat{p}_{n}^{(0)}(x)=\hat{p}_{n-1}(x;1)$ ⓘ Symbols: $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.30.E29 Encodings: TeX, pMML, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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 $\hat{p}_{n}(x;1)$, are referred to as the first associated such polynomials in §18.2(x), is now evident. The ratio $\hat{p}_{n-1}(x;1)/\hat{p}_{n}(x)$ 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 $\hat{p}_{n}^{(k)}(x)=\hat{p}_{n-1}(x;k+1).$ ⓘ Symbols: $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.30.E30 Encodings: TeX, pMML, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

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 $F_{n}(x,k)=\hat{p}_{n}^{(k)}(x)/\hat{p}_{n}(x;k).$ ⓘ Symbols: $(\NVar{a},\NVar{b})$: open interval, $k$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.30, Erratum (V1.2.0) §18.30 Permalink: http://dlmf.nist.gov/18.30.E31 Encodings: TeX, pMML, png See also: Annotations for §18.30(vii), §18.30(vii), §18.30 and Ch.18

## §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 $C_{n}^{(\lambda)}(x;c)=P^{{(\lambda)}}_{n}\left(x;0,0,c\right)$.