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1: 18.31 Bernstein–Szegő Polynomials

… ►Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell }$ and positive when $-1\le x\le 1$. …

2: 15.13 Zeros

… ►If $a$, $b$, $c$, $c-a$, or $c-b\in \{0,-1,-2,\mathrm{\dots }\}$, then $F(a,b;c;z)$ is not defined, or reduces to a polynomial, or reduces to ${(1-z)}^{c-a-b}$ times a polynomial. …

3: 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 $0,1,a$. …

4: 10.19 Asymptotic Expansions for Large Order

… ►

10.19.9 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim \frac{2^{\frac{4}{3}}}{{\nu }^{\frac{1}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{P_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{5}{3}}}{\nu }{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{Q_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $P_k(a)$: polynomial coefficient and $Q_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.0.27):

Originally the polynomials $P_k(a)$, $Q_k(a)$ were incorrectly described as Legendre polynomials and integer-degree, zero-order associated Legendre functions of the second kind respectively.

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

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10.19.13 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\end{array}\}\sim -\frac{2^{\frac{5}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{e}}^{\pm \pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{4}{3}}}{{\nu }^{\frac{4}{3}}}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\mp \pi \mathrm{i}/3}{2}^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\nu$: complex parameter, $R_k(a)$: polynomial coefficient and $S_k(a)$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.19.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.19(iii), §10.19 and Ch.10

…

5: 18.34 Bessel Polynomials

… ►With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by $y_n(x;a-2)$. …where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … Sometimes the polynomials ${\theta }_n(x;a,b)$ are called reverse Bessel polynomials. … ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : … ►For uniform asymptotic expansions of $y_n(x;a)$ as $n\to \mathrm{\infty }$ in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). …

6: 17.17 Physical Applications

… ►In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. … ►They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. See Kassel (1995). ►A substantial literature on $q$-deformed quantum-mechanical Schrödinger equations has developed recently. It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

7: 8.7 Series Expansions

… ►

8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

…

8: 18.21 Hahn Class: Interrelations

… ►

$C_n(x;a)=C_x(n;a),$ $n,x=0,1,2,\mathrm{\dots }$.

… ►

18.21.6 $$\underset{N\to \mathrm{\infty }}{lim}{K}_n(x;N^{-1}a,N)=C_n(x;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E6

Encodings:

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See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.7 $$\underset{\beta \to \mathrm{\infty }}{lim}{M}_n(x;\beta ,a{(a+\beta )}^{-1})=C_n(x;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.21.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.9 $$\underset{a\to \mathrm{\infty }}{lim}{(2a)}^{\frac{1}{2}n}{C}_n({(2a)}^{\frac{1}{2}}x+a;a)={(-1)}^n{H}_n\left(x\right).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.11 $$p_n(x;a,a+\frac{1}{2},a,a+\frac{1}{2})=2^{-2n}{\left(4a+n\right)}_n{P}_n^{(2a)}(2x;\frac{1}{2}\pi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

…

9: 1.11 Zeros of Polynomials

… ►A polynomial of degree $n$ with real or complex coefficients has exactly $n$ real or complex zeros counting multiplicity. …A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative. … ►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. … ►The discriminant of $f(z)$ is defined by …The elementary symmetric functions of the zeros are (with $a_n\ne 0$) …

10: 18.35 Pollaczek Polynomials

… ►The type 2 polynomials reduce for $a=b=0$ to ultraspherical polynomials, see (18.35.8). ►The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8)) … ►More generally, the $P_n^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)). … ►See Bo and Wong (1996) for an asymptotic expansion of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$ as $n\to \mathrm{\infty }$, with $a$ and $b$ fixed. …Also included is an asymptotic approximation for the zeros of $P_n^{(\frac{1}{2})}(\cos \left(n^{-\frac{1}{2}}\theta \right);a,b)$. …