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21: 18.22 Hahn Class: Recurrence Relations and Differences

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18.22.2 $$-x{p}_n(x)=A_n{p}_{n+1}(x)-\left(A_n+C_n\right)p_n(x)+C_n{p}_{n-1}(x),$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $A_n$: coefficient and $C_n$: coefficient

Referenced by:

§16.4(iii), §18.21(ii), §18.22(i), Table 18.22.1

Permalink:

http://dlmf.nist.gov/18.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

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18.22.5 $$(a+\mathrm{i}x)q_n(x)={\tilde{A}}_n{q}_{n+1}(x)-\left({\tilde{A}}_n+{\tilde{C}}_n\right)q_n(x)+{\tilde{C}}_n{q}_{n-1}(x),$$

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Symbols:

$\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable, ${\tilde{A}}_n$: coefficient and ${\tilde{C}}_n$: coefficient

Permalink:

http://dlmf.nist.gov/18.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

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18.22.10 $$A(x)p_n(x+1)-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-1)-n(n+\alpha +\beta +1)p_n(x)=0,$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $A(x)$: coefficient and $C(x)$: coefficient

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.22.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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18.22.12 $$A(x)p_n(x+1)-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-1)+{\lambda }_n{p}_n(x)=0.$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $C(x)$: coefficient, ${\lambda }_n$: coefficient and $A(x)$: coefficient

Referenced by:

§18.22(ii), Table 18.22.2

Permalink:

http://dlmf.nist.gov/18.22.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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18.22.14 $$A(x)p_n(x+\mathrm{i})-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-\mathrm{i})+n(n+2\mathrm{\Re }\left(a+b\right)-1)p_n(x)=0,$$

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Symbols:

$\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer, $x$: real variable, $A(x)$: coefficient and $C(x)$: coefficient

Referenced by:

§18.22(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

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22: 34.14 Tables

… ►Biedenharn and Louck (1981) give tables of algebraic expressions for Clebsch–Gordan coefficients and $\mathit{6}j$ symbols, together with a bibliography of tables produced prior to 1975. In Varshalovich et al. (1988) algebraic expressions for the Clebsch–Gordan coefficients with all parameters $\le 5$ and numerical values for all parameters $\le 3$ are given on pp. …

23: 5.8 Infinite Products

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5.8.4 $$\sum _{k=1}^m{a}_k=\sum _{k=1}^m{b}_k,$$

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Defines:

$a_k$: coefficient (locally) and $b_k$: coefficient (locally)

Symbols:

$m$: nonnegative integer and $k$: nonnegative integer

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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24: 7.24 Approximations

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ for $x\ge 3$ (15D).

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Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{-1}\mathrm{erf}x$ on $0\le x\le 2$, for $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[2,\mathrm{\infty })$, and for ${\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $[0,\mathrm{\infty })$ (30D).

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Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x){\mathrm{e}}^{x^2}\mathrm{erfc}x$ on $(0,\mathrm{\infty })$ (22D).

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25: 13.5 Continued Fractions

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13.5.1 $$\frac{M(a,b,z)}{M(a+1,b+1,z)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients

Referenced by:

§13.31(ii)

Permalink:

http://dlmf.nist.gov/13.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.5 and Ch.13

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13.5.3 $$\frac{U(a,b,z)}{U(a,b-1,z)}=1+\frac{v_1/z}{1+}\frac{v_2/z}{1+}\mathrm{\cdots },$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $z$: complex variable and $v_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.5 and Ch.13

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26: 13.17 Continued Fractions

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13.17.1 $$\frac{\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)}{M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.17 and Ch.13

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13.17.3 $$\frac{W_{\kappa ,\mu }\left(z\right)}{\sqrt{z}{W}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)}=1+\frac{v_1/z}{1+}\frac{v_2/z}{1+}\mathrm{\cdots },$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $v_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.17 and Ch.13

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27: 18.3 Definitions

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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… ►For explicit power series coefficients up to $n=12$ for these polynomials and for coefficients up to $n=6$ for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). …

28: 33.11 Asymptotic Expansions for Large $\rho$

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k},$$

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Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient and $b$: coefficient

A&S Ref:

14.5.9

Referenced by:

Erratum (V1.0.19) for Equation (33.11.1), Erratum (V1.0.22) for Equation (33.11.1)

Permalink:

http://dlmf.nist.gov/33.11.E1

Encodings:

pMML, png, TeX

Errata (effective with 1.0.22):

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported 2019-01-29 by Gergő Nemes

Errata (effective with 1.0.19):

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.11 and Ch.33

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29: 9.19 Approximations

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Prince (1975) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. The Chebyshev coefficients are given to 10-11D. Fortran programs are included. See also Razaz and Schonfelder (1981).

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Németh (1992, Chapter 8) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, and integrals ${\int }_0^x\mathrm{Ai}\left(t\right)dt$, ${\int }_0^x\mathrm{Bi}\left(t\right)dt$, ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv$, ${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv$ (see also (9.10.20) and (9.10.21)). The Chebyshev coefficients are given to 15D. Chebyshev coefficients are also given for expansions of the second and higher (real) zeros of $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$, again to 15D.

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Razaz and Schonfelder (1980) covers $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$. The Chebyshev coefficients are given to 30D.

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Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\mathrm{Gi}\left(x\right)$ for and $\mathrm{Hi}\left(x\right)$ for . The Chebyshev coefficients are given to 20D.

30: 29.7 Asymptotic Expansions

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29.7.1 $$a_{\nu }^m\left(k^2\right)\sim p\kappa -{\tau }_0-{\tau }_1{\kappa }^{-1}-{\tau }_2{\kappa }^{-2}-\mathrm{\cdots },$$

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Symbols:

$a_{\nu }^n\left(k^2\right)$: eigenvalues of Lamé’s equation, $\sim$: Poincaré asymptotic expansion, $m$: nonnegative integer, $p$: nonnegative integer, $k$: real parameter, $\nu$: real parameter, $\kappa$ and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.3 $${\tau }_0=\frac{1}{2^3}(1+k^2)(1+p^2),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.4 $${\tau }_1=\frac{p}{2^6}({(1+k^2)}^2(p^2+3)-4k^2(p^2+5)).$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

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Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

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