18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.10 Integral Representations 18.12 Generating Functions

§18.11 Relations to Other Functions

Permalink:: http://dlmf.nist.gov/18.11

§18.11(i) Explicit Formulas
§18.11(ii) Formulas of Mehler–Heine Type

§18.11(i) Explicit Formulas

Notes:: For (18.11.1) see Andrews et al. (1999, (9.6.7)). (18.11.2) is a rewriting of (18.5.12). For (18.11.3) see Temme (1990a, (3.1)).
Permalink:: http://dlmf.nist.gov/18.11.i

See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.

¶ Ultraspherical

Keywords:: Ferrers functions, ultraspherical polynomials

18.11.1 $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\left(\tfrac{1}{2}% \right)_{{m}}(-2)^{m}(1-x^{2})^{{\frac{1}{2}m}}\mathop{C^{{(m+\frac{1}{2})}}_{% {n-m}}\/}\nolimits\!\left(x\right)=\left(n+1\right)_{{m}}(-2)^{{-m}}(1-x^{2})^% {{\frac{1}{2}m}}\mathop{P^{{(m,m)}}_{{n-m}}\/}\nolimits\!\left(x\right),$ $0\leq m\leq n$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer, : nonnegative integer and : real variable
Referenced by:: §14.3(iv), §18.11(i)
Permalink:: http://dlmf.nist.gov/18.11.E1
Encodings:: TeX, pMML, png

For the Ferrers function $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)$ , see §14.3(i).

Compare also (14.3.21) and (14.3.22).

¶ Laguerre

Keywords:: Laguerre polynomials

18.11.2 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(\alpha+1% \right)_{{n}}}{n!}\mathop{M\/}\nolimits\!\left(-n,\alpha+1,x\right)=\frac{(-1)% ^{n}}{n!}\mathop{U\/}\nolimits\!\left(-n,\alpha+1,x\right)=\frac{\left(\alpha+% 1\right)_{{n}}}{n!}x^{{-\frac{1}{2}(\alpha+1)}}e^{{\frac{1}{2}x}}\mathop{M_{{n% +\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}}\/}\nolimits\!\left(x\right)=\frac{(% -1)^{n}}{n!}x^{{-\frac{1}{2}(\alpha+1)}}e^{{\frac{1}{2}x}}\mathop{W_{{n+\frac{% 1}{2}(\alpha+1),\frac{1}{2}\alpha}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function, : base of exponential function, : : factorial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : nonnegative integer and : real variable
Referenced by:: §13.6(v), §18.11(i)
Permalink:: http://dlmf.nist.gov/18.11.E2
Encodings:: TeX, pMML, png

For the confluent hypergeometric functions $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,x\right)$ , see §13.2(i), and for the Whittaker functions $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)$ and $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(x\right)$ see §13.14(i).

¶ Hermite

Keywords:: Hermite polynomials, confluent hypergeometric functions

18.11.3 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=2^{n}\mathop{U\/}\nolimits\!\left(% -\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}x\mathop{U\/}\nolimits\!\left(-% \tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{{\frac{1}{2}n}}e^{{% \frac{1}{2}x^{2}}}\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2},2^{{\frac{1}{2}% }}x\right).$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : nonnegative integer and : real variable
A&S Ref:: 22.5.55, 22.5.58 ((factor missing on RHS of 22.5.55))
Referenced by:: §18.11(i), §18.15(v)
Permalink:: http://dlmf.nist.gov/18.11.E3
Encodings:: TeX, pMML, png

18.11.4 $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)=2^{{\frac{1}{2}n}}% \mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x^{2}% \right)=2^{{\frac{1}{2}(n-1)}}x\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n+% \tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)=e^{{\tfrac{1}{4}x^{2}}}% \mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2},x\right).$

Symbols:: $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : nonnegative integer and : real variable
A&S Ref:: 22.5.59
Permalink:: http://dlmf.nist.gov/18.11.E4
Encodings:: TeX, pMML, png

For the parabolic cylinder function $\mathop{U\/}\nolimits\!\left(a,z\right)$ , see §12.2.

§18.11(ii) Formulas of Mehler–Heine Type

Notes:: For (18.11.5) see Szegö (1975, Theorem 8.1.1). For (18.11.6) see Szegö (1975, Theorem 8.22.4). For (18.11.7) and (18.11.8) see Szegö (1975, Theorem 8.22.8).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.11.ii

¶ Jacobi

Keywords:: Jacobi polynomials

18.11.5 $\lim_{{n\to\infty}}\frac{1}{n^{{\alpha}}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}% \nolimits\!\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{{n\to\infty}}\frac{1}{n^{% {\alpha}}}\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}\mathop{J_{{\alpha}}% \/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : complex variable and : nonnegative integer
A&S Ref:: 22.15.1
Referenced by:: ¶ ‣ §18.11(ii), §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E5
Encodings:: TeX, pMML, png

¶ Laguerre

Keywords:: Laguerre polynomials

18.11.6 $\lim_{{n\to\infty}}\frac{1}{n^{{\alpha}}}\mathop{L^{{(\alpha)}}_{{n}}\/}% \nolimits\!\left(\frac{z}{n}\right)=\frac{1}{z^{{\frac{1}{2}\alpha}}}\mathop{J% _{{\alpha}}\/}\nolimits\!\left(2z^{{\frac{1}{2}}}\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : complex variable and : nonnegative integer
A&S Ref:: 22.15.2
Referenced by:: §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E6
Encodings:: TeX, pMML, png

¶ Hermite

Keywords:: Hermite polynomials

18.11.7 $\lim_{{n\to\infty}}\frac{(-1)^{n}n^{{\frac{1}{2}}}}{2^{{2n}}n!}\mathop{H_{{2n}% }\/}\nolimits\!\left(\frac{z}{2n^{{\frac{1}{2}}}}\right)=\frac{1}{\pi^{{\frac{% 1}{2}}}}\mathop{\cos\/}\nolimits z,$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, : complex variable and : nonnegative integer
A&S Ref:: 22.15.3
Referenced by:: §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E7
Encodings:: TeX, pMML, png

18.11.8 $\lim_{{n\to\infty}}\frac{(-1)^{n}}{2^{{2n}}n!}\mathop{H_{{2n+1}}\/}\nolimits\!% \left(\frac{z}{2n^{{\frac{1}{2}}}}\right)=\frac{2}{\pi^{{\frac{1}{2}}}}\mathop% {\sin\/}\nolimits z.$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : nonnegative integer
A&S Ref:: 22.15.4
Referenced by:: ¶ ‣ §18.11(ii), §18.11(ii)
Permalink:: http://dlmf.nist.gov/18.11.E8
Encodings:: TeX, pMML, png

For the Bessel function $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ , see §10.2(ii). The limits (18.11.5)–(18.11.8) hold uniformly for in any bounded subset of $\Complex$ .

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 18.10 Integral Representations 18.12 Generating Functions

§18.11 Relations to Other Functions

Contents

§18.11(i) Explicit Formulas

¶ Ultraspherical

¶ Laguerre

¶ Hermite

§18.11(ii) Formulas of Mehler–Heine Type

¶ Jacobi

¶ Laguerre

¶ Hermite