§18.7 Interrelations and Limit Relations

Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.7

§18.7(i) Linear Transformations
§18.7(ii) Quadratic Transformations
§18.7(iii) Limit Relations

§18.7(i) Linear Transformations

Notes:: For (18.7.1) and (18.7.2) see Szegö (1975, (4.7.1)). For (18.7.3) and (18.7.4) see Szegö (1975, (4.1.7)). For (18.7.5) and (18.7.6) see Szegö (1975, (4.1.8)). (18.7.7)–(18.7.12) follow from the definitions given by Table 18.3.1.
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.7.i

¶ Ultraspherical and Jacobi

18.7.1 $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(2\lambda\right)_{{n}}}{\left(\lambda+\frac{1}{2}\right)_{{n}}}\mathop{P^{{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.27
Referenced by:: §18.15(ii), §18.7(i), §18.7(ii), §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E1
Encodings:: TeX, pMML, png

18.7.2 $\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(\alpha+1\right)_{{n}}}{\left(2\alpha+1\right)_{{n}}}\mathop{C^{{(\alpha+\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.20
Referenced by:: §18.6(i), §18.7(i)
Permalink:: http://dlmf.nist.gov/18.7.E2
Encodings:: TeX, pMML, png

¶ Chebyshev, Ultraspherical, and Jacobi

Keywords:: Chebyshev polynomials, Jacobi polynomials

18.7.3 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{\mathop{P^{{(-\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(-\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)},$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.15, 22.5.31
Referenced by:: §18.17(viii), ¶ ‣ §18.18(i), ¶ ‣ §18.5(iii), §18.7(i), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E3
Encodings:: TeX, pMML, png

18.7.4 $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)=\mathop{C^{{(1)}}_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{(n+1)\mathop{P^{{(\frac{1}{2},\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\frac{1}{2},\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)},$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.3.16, 22.5.32
Referenced by:: §18.12, ¶ ‣ §18.3, §18.7(i), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E4
Encodings:: TeX, pMML, png

18.7.5 $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{(2n+1)\mathop{P^{{(\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)},$

Symbols:: $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the third kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii), §18.5(i), §18.7(i), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E5
Encodings:: TeX, pMML, png

18.7.6 $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)=\ifrac{\mathop{P^{{(-\frac{1}{2},\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(-\frac{1}{2},\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)}.$

Symbols:: $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: ¶ ‣ §18.18(i), ¶ ‣ §18.3, ¶ ‣ §18.5(iii), §18.5(i), §18.7(i), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E6
Encodings:: TeX, pMML, png

18.7.7 $\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{T_{{n}}\/}\nolimits\!\left(2x-1\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the first kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.14
Referenced by:: §18.7(i)
Permalink:: http://dlmf.nist.gov/18.7.E7
Encodings:: TeX, pMML, png

18.7.8 $\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{U_{{n}}\/}\nolimits\!\left(2x-1\right).$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.15
Permalink:: http://dlmf.nist.gov/18.7.E8
Encodings:: TeX, pMML, png

¶ Legendre, Ultraspherical, and Jacobi

Keywords:: Legendre polynomials

18.7.9 $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)=\mathop{C^{{(\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P^{{(0,0)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.35, 22.5.36
Referenced by:: §10.60(iii), ¶ ‣ §18.18(ii), ¶ ‣ §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.7.E9
Encodings:: TeX, pMML, png

18.7.10 $\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P_{{n}}\/}\nolimits\!\left(2x-1\right).$

Symbols:: $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Legendre polynomial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.7.E10
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: classical orthogonal polynomials

18.7.11 $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)=2^{{-\frac{1}{2}n}}\mathop{H_{{n}}\/}\nolimits\!\left(2^{{-\frac{1}{2}}}x\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.18
Referenced by:: ¶ ‣ §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.7.E11
Encodings:: TeX, pMML, png

18.7.12 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=2^{{\frac{1}{2}n}}\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(2^{{\frac{1}{2}}}x\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.19
Referenced by:: §18.7(i)
Permalink:: http://dlmf.nist.gov/18.7.E12
Encodings:: TeX, pMML, png

§18.7(ii) Quadratic Transformations

Notes:: For (18.7.13) and (18.7.14) see Szegö (1975, (4.1.5)). (18.7.15) and (18.7.16) follow from (18.7.13) and (18.7.14), combined with (18.7.1). (18.7.17) follows from (18.7.4), (18.7.5), and (18.7.13). (18.7.18) follows from (18.7.3), (18.7.6), and (18.7.14). For (18.7.19) and (18.7.20) see Szegö (1975, (5.6.1)).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.7.ii

18.7.13 $\frac{\mathop{P^{{(\alpha,\alpha)}}_{{2n}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\alpha,\alpha)}}_{{2n}}\/}\nolimits\!\left(1\right)}=\frac{\mathop{P^{{(\alpha,-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(2x^{2}-1\right)}{\mathop{P^{{(\alpha,-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)},$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E13
Encodings:: TeX, pMML, png

18.7.14 $\frac{\mathop{P^{{(\alpha,\alpha)}}_{{2n+1}}\/}\nolimits\!\left(x\right)}{\mathop{P^{{(\alpha,\alpha)}}_{{2n+1}}\/}\nolimits\!\left(1\right)}=\frac{x\mathop{P^{{(\alpha,\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(2x^{2}-1\right)}{\mathop{P^{{(\alpha,\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(1\right)}.$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E14
Encodings:: TeX, pMML, png

18.7.15 $\mathop{C^{{(\lambda)}}_{{2n}}\/}\nolimits\!\left(x\right)=\frac{\left(\lambda\right)_{{n}}}{\left(\tfrac{1}{2}\right)_{{n}}}\mathop{P^{{(\lambda-\frac{1}{2},-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(2x^{2}-1\right),$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.25
Referenced by:: §18.7(ii), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.7.E15
Encodings:: TeX, pMML, png

18.7.16 $\mathop{C^{{(\lambda)}}_{{2n+1}}\/}\nolimits\!\left(x\right)=\frac{\left(\lambda\right)_{{n+1}}}{\left(\frac{1}{2}\right)_{{n+1}}}x\mathop{P^{{(\lambda-\frac{1}{2},\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(2x^{2}-1\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.26
Referenced by:: §18.7(ii), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.7.E16
Encodings:: TeX, pMML, png

18.7.17 $\mathop{U_{{2n}}\/}\nolimits\!\left(x\right)=\mathop{V_{{n}}\/}\nolimits\!\left(2x^{2}-1\right),$

Symbols:: $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{V_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.30
Referenced by:: §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E17
Encodings:: TeX, pMML, png

18.7.18 $\mathop{T_{{2n+1}}\/}\nolimits\!\left(x\right)=x\mathop{W_{{n}}\/}\nolimits\!\left(2x^{2}-1\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.29
Referenced by:: §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E18
Encodings:: TeX, pMML, png

18.7.19 $\mathop{H_{{2n}}\/}\nolimits\!\left(x\right)=(-1)^{n}2^{{2n}}n!\mathop{L^{{(-\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x^{2}\right),$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.40
Referenced by:: §18.16(v), §18.18(iii), §18.18(iv), §18.18(viii), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E19
Encodings:: TeX, pMML, png

18.7.20 $\mathop{H_{{2n+1}}\/}\nolimits\!\left(x\right)=(-1)^{n}2^{{2n+1}}n!\, x\mathop{L^{{(\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x^{2}\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.41
Referenced by:: §18.16(v), §18.18(iii), §18.18(iv), §18.7(ii)
Permalink:: http://dlmf.nist.gov/18.7.E20
Encodings:: TeX, pMML, png

§18.7(iii) Limit Relations

Notes:: For (18.7.21) see Szegö (1975, (5.3.4)). (18.7.22) follows from (18.7.21) and the symmetry in Row 2 of Table 18.6.1. (18.7.23) follows from (18.7.24) and (18.7.1). For (18.7.24) see Szegö (1975, (5.6.3)). For (18.7.25) see Szegö (1975, (4.7.8)). For (18.7.26) see Calogero (1978).
Keywords:: classical orthogonal polynomials
Referenced by:: ¶ ‣ §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.7.iii

¶ Jacobi $\to$ Laguerre

Keywords:: Laguerre polynomials

18.7.21 $\lim _{{\beta\to\infty}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(1-(\ifrac{2x}{\beta})\right)=\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.15.5
Referenced by:: §18.10(ii), §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E21
Encodings:: TeX, pMML, png

18.7.22 $\lim _{{\alpha\to\infty}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left((2x/\alpha)-1\right)=(-1)^{n}\mathop{L^{{(\beta)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E22
Encodings:: TeX, pMML, png

¶ Jacobi $\to$ Hermite

18.7.23 $\lim _{{\alpha\to\infty}}\alpha^{{-\frac{1}{2}n}}\mathop{P^{{(\alpha,\alpha)}}_{{n}}\/}\nolimits\!\left(\alpha^{{-\frac{1}{2}}}x\right)=\frac{\mathop{H_{{n}}\/}\nolimits\!\left(x\right)}{2^{n}n!}.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.10(ii), §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E23
Encodings:: TeX, pMML, png

¶ Ultraspherical $\to$ Hermite

18.7.24 $\lim _{{\lambda\to\infty}}\lambda^{{-\frac{1}{2}n}}\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(\lambda^{{-\frac{1}{2}}}x\right)=\frac{\mathop{H_{{n}}\/}\nolimits\!\left(x\right)}{n!}.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.15.6
Referenced by:: §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E24
Encodings:: TeX, pMML, png

18.7.25 $\lim _{{\lambda\to 0}}\frac{1}{\lambda}\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{2}{n}\mathop{T_{{n}}\/}\nolimits\!\left(x\right),$ $n\geq 1$ .

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E25
Encodings:: TeX, pMML, png

¶ Laguerre $\to$ Hermite

Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, ultraspherical polynomials

18.7.26 $\lim _{{\alpha\to\infty}}\left(\frac{2}{\alpha}\right)^{{\frac{1}{2}n}}\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left((2\alpha)^{{\frac{1}{2}}}x+\alpha\right)=\frac{(-1)^{n}}{n!}\mathop{H_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $!$ : $n!$ : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.7(iii)
Permalink:: http://dlmf.nist.gov/18.7.E26
Encodings:: TeX, pMML, png

See Figure 18.21.1 for the Askey schematic representation of most of these limits.