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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: 18.34 Bessel Polynomials

… ►Because the coefficients $C_n$ in (18.34.4) are not all positive, the polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function. … ►

18.34.6 $$\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}{z}^{a-2}{y}_n(z;a)y_m(z;a){\mathrm{e}}^{-2/z}dz=\frac{{(-1)}^{n+a-1}n!{\mathrm{\hspace{0.17em}2}}^{a-1}}{(n+a-2)!(2n+a-1)}{\delta }_{n,m},$$ $a=1,2,\mathrm{\dots }$,

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Symbols:

$y_n(x;a)$: Bessel polynomial, ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

… ►

18.34.7 $$x^2{y}_n^{\prime \prime }(x;a)+(ax+2)y_n^{\prime }(x;a)-n(n+a-1)y_n(x;a)=0,$$

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Symbols:

$y_n(x;a)$: Bessel polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.34.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

… ►

18.34.8 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,a-\alpha -2)}\left(1+\alpha x\right)}{P_n^{(\alpha ,a-\alpha -2)}\left(1\right)}=y_n(x;a).$$

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Symbols:

$y_n(x;a)$: Bessel polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.34.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

►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). …

4: 10.19 Asymptotic Expansions for Large Order

… ►

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

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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

… ►

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

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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: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

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

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Defines:

${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: 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 $0,1,a$. …

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, $x$: real variable, $a$: parameter and $n$: nonnegative integer

Permalink:

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

Encodings:

pMML, png, TeX

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, $n$: nonnegative integer, $N$: positive integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

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

Encodings:

pMML, png, TeX

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: 35.4 Partitions and Zonal Polynomials

… ►

35.4.2

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Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $\mathbf{I}$: $m\times m$ identity matrix, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer, $\kappa$: vector of nonnegative integers and $\mathrm{\ell }(\kappa )$: number of nonzeros

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

… ►Therefore $Z_{\kappa }\left(\mathbf{T}\right)$ is a symmetric polynomial in the eigenvalues of $\mathbf{T}$. … ►

35.4.6 $$\sum _{|\kappa |=k}{Z}_{\kappa }\left(\mathbf{T}\right)={(tr\mathbf{T})}^k.$$

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Symbols:

$tr\mathbf{A}$: trace of matrix, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $\mathbf{T}$: real symmetric $m\times m$ matrix, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.8 $${\int }_{\mathbf{\Omega }}etr\left(-\mathbf{T}\mathbf{X}\right){\left|\mathbf{X}\right|}^{a-\frac{1}{2}(m+1)}{Z}_{\kappa }\left(\mathbf{X}\right)\mathrm{d}\mathbf{X}={\mathrm{\Gamma }}_m\left(a+\kappa \right){\left|\mathbf{T}\right|}^{-a}{Z}_{\kappa }\left({\mathbf{T}}^{-1}\right),$$

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Symbols:

$etr\left(\mathbf{X}\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $a$: complex variable, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$ , $\mathrm{d}$: differential of matrix, $\mathbf{\Omega }$: space of positive-definite real symmetric $m\times m$ matrices, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

►

35.4.9

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(\mathbf{T}\right)$: zonal polynomial, $a$: complex variable, $\mathbf{I}$: $m\times m$ identity matrix, $\mathbf{T}$: real symmetric $m\times m$ matrix, $\mathbf{X}$: real symmetric $m\times m$ matrix, $\left|\mathbf{X}\right|$: determinant of $\mathbf{X}$ , $\mathrm{d}$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer, $\mathbf{0}$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35