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11: 13.18 Relations to Other Functions

… ►Special cases are the error functions … ►

§13.18(v) Orthogonal Polynomials

►Special cases of §13.18(iv) are as follows. … ►

Hermite Polynomials

… ►

Laguerre Polynomials

…

12: 16.18 Special Cases

… ►As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …

13: 15.8 Transformations of Variable

… ►With $m=0,1,2,\mathrm{\dots }$, polynomial cases of (15.8.2)–(15.8.5) are given by …

14: 28.31 Equations of Whittaker–Hill and Ince

… ►

§28.31(ii) Equation of Ince; Ince Polynomials

… ►When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. …and $m=0,1,\mathrm{\dots },n$ in all cases. … ►The normalization is given by …

15: 18.3 Definitions

… ►This table also includes the following special cases of Jacobi polynomials: ultraspherical, Chebyshev, and Legendre. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of $x-1$ for Jacobi polynomials, in powers of $x$ for the other cases). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

16: 18.15 Asymptotic Approximations

… ►These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

17: 18.14 Inequalities

… ►

18.14.3_5 $${\left(\frac{1}{2}(1+x)\right)}^{\beta /2}\left|P_n^{(\alpha ,\beta )}\left(x\right)\right|\le P_n^{(\alpha ,\beta )}\left(1\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\alpha ,\beta \ge 0$.

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Carnicer et al. (2020, Theorem 2)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

Referenced by:

§18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►

18.14.8 $${\mathrm{e}}^{-\frac{1}{2}x}\left|L_n^{(\alpha )}\left(x\right)\right|\le L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ , $\alpha \ge 0$.

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1977, Remark 4.1)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

A&S Ref:

22.14.13

Referenced by:

§18.14(i), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►except that when $\alpha =\beta =-\frac{1}{2}$ (Chebyshev case) $|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|$ is constant. …

18: 31.11 Expansions in Series of Hypergeometric Functions

… ►The case $\alpha =-n$ for nonnegative integer $n$ corresponds to the Heun polynomial ${\mathrm{𝐻𝑝}}_{n,m}\left(z\right)$. … ►In each case $P_j^6$ can be expressed in terms of a Jacobi polynomial (§18.3). …

19: 18.28 Askey–Wilson Class

… ►The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of $q$-Racah polynomials, and cases of these families obtained by specialization of parameters. … ►These systems are the $q$-Racah polynomials and its limit cases. …

20: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. Some properties are included as special cases of properties given in §31.15 below.