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11: 18.1 Notation

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$q$-Hahn Class OP’s

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Askey–Wilson Class OP’s

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Associated OP’s

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Other OP’s

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Classical OP’s in Two Variables

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12: 18.2 General Orthogonal Polynomials

… ►This happens, for example, with the Hahn class OP’s (§18.20(i)). … ►In terms of the monic OP’s $p_n$ define the orthonormal OP’s $q_n$ by … ►are OP’s with orthogonality relation … ► … ►

§18.2(xii) Other Special Constructions Involving General OP’s

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13: 18.19 Hahn Class: Definitions

… ►The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator $\frac{d}{dx}$ in the case of the classical OP’s is played by a suitable difference operator. These eight further families can be grouped in two classes of OP’s: ►

Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

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Wilson class (or quadratic lattice class). These are OP’s $p_n(x)=p_n(\lambda (y))$ ($p_n(x)$ of degree $n$ in $x$, $\lambda (y)$ quadratic in $y$) where the role of the differentiation operator is played by $\frac{{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_y(\lambda (y))}$ or $\frac{{\nabla }_y}{{\nabla }_y(\lambda (y))}$ or $\frac{{\delta }_y}{{\delta }_y(\lambda (y))}$. The Wilson class consists of two discrete and two continuous families.

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Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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14: 18.36 Miscellaneous Polynomials

… ►Similar OP’s can also be constructed for the Laguerre polynomials; see Koornwinder (1984b, (4.8)). … ►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … ►Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. … ►This inequality is violated for $n=1$ if , seemingly precluding such an extension of the Laguerre OP’s into that regime. … ►Hermite EOP’s are defined in terms of classical Hermite OP’s. …

15: 18.39 Applications in the Physical Sciences

… ►These are overviewed in §18.38(iii), and §18.36(vi), and typically involve OP’s or EOP’s. … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►In all of these references these OP’s are simply referred to as the associated Laguerre OP’s. … ►These same solutions are expressed here in terms of Laguerre and Pollaczek OP’s. …

16: 18.40 Methods of Computation

… ►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. For applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. … ►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis. … ►Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. … ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. …

17: 18.37 Classical OP’s in Two or More Variables

§18.37 Classical OP’s in Two or More Variables

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18.37.6 $$R_{m,n}^{(\alpha )}\left(z\right)=\sum _{j=0}^{\min (m,n)}\frac{{(-1)}^j{\left(\alpha +1\right)}_{m+n-j}{\left(-m\right)}_j{\left(-n\right)}_j}{{\left(\alpha +1\right)}_m{\left(\alpha +1\right)}_{n}j!}{z}^{m-j}{\overline{z}}^{n-j}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\overline{z}$: complex conjugate, $R_{m,n}^{(\alpha )}\left(z\right)$: disk polynomial, $!$: factorial (as in $n!$), $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

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http://dlmf.nist.gov/18.37.E6

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See also:

Annotations for §18.37(i), §18.37(i), §18.37 and Ch.18

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§18.37(ii) OP’s on the Triangle

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§18.37(iii) OP’s Associated with Root Systems

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18: 18.3 Definitions

… ►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. There are many ways of characterizing the classical OP’s within the general OP’s $\{p_n(x)\}$, see Al-Salam (1990). … ►

As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_n^{\prime \prime }(x)+B(x)p_n^{\prime }(x)+{\lambda }_n{p}_n(x)=0$, in Table 18.8.1.

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With the property that ${\{p_{n+1}^{\prime }(x)\}}_{n=0}^{\mathrm{\infty }}$ is again a system of OP’s. See §18.9(iii).

… ►Bessel polynomials are often included among the classical OP’s. …

19: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. … ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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OP	$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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►Under certain conditions on their parameters the orthogonality range for the Wilson polynomials and continuous dual Hahn polynomials is $(0,\mathrm{\infty })\cup S$, where $S$ is a specific finite set, e. … ►

Table 18.25.2: Wilson class OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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20: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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