generalized Mehler–Fock transformation

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21: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …

22: 18.7 Interrelations and Limit Relations

… ►

§18.7(i) Linear Transformations

… ►

§18.7(ii) Quadratic Transformations

… ►

18.7.19 $$H_{2n}\left(x\right)={(-1)}^n{2}^{2n}n!L_n^{(-\frac{1}{2})}\left(x^2\right),$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: 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:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

►

18.7.20 $$H_{2n+1}\left(x\right)={(-1)}^n{2}^{2n+1}n!x{L}_n^{(\frac{1}{2})}\left(x^2\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: 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), §18.7(ii)

Permalink:

http://dlmf.nist.gov/18.7.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

… ► See §18.11(ii) for limit formulas of Mehler–Heine type.

23: 18.11 Relations to Other Functions

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

18.11.2 $$L_n^{(\alpha )}\left(x\right)=\frac{{\left(\alpha +1\right)}_n}{n!}M(-n,\alpha +1,x)=\frac{{(-1)}^n}{n!}U(-n,\alpha +1,x)=\frac{{\left(\alpha +1\right)}_n}{n!}{x}^{-\frac{1}{2}(\alpha +1)}{\mathrm{e}}^{\frac{1}{2}x}{M}_{n+\frac{1}{2}(\alpha +1),\frac{1}{2}\alpha }\left(x\right)=\frac{{(-1)}^n}{n!}{x}^{-\frac{1}{2}(\alpha +1)}{\mathrm{e}}^{\frac{1}{2}x}{W}_{n+\frac{1}{2}(\alpha +1),\frac{1}{2}\alpha }\left(x\right).$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§13.6(v), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

… ►

§18.11(ii) Formulas of Mehler–Heine Type

… ►

18.11.6 $$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{\alpha }}{L}_n^{(\alpha )}\left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha }}{J}_{\alpha }\left(2z^{\frac{1}{2}}\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

22.15.2

Referenced by:

§18.11(ii)

Permalink:

http://dlmf.nist.gov/18.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(ii), §18.11(ii), §18.11 and Ch.18

…

24: 7.16 Generalized Error Functions

§7.16 Generalized Error Functions

►Generalizations of the error function and Dawson’s integral are ${\int }_0^x{\mathrm{e}}^{-t^p}dt$ and ${\int }_0^x{\mathrm{e}}^{t^p}dt$. …

25: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. … ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

26: 16.4 Argument Unity

… ► … ►

§16.4(iii) Identities

… ►A different type of transformation is that of Whipple: … ►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. … ► …

27: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

… ►

§22.7(ii) Ascending Landen Transformation

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§22.7(iii) Generalized Landen Transformations

…

28: 16.26 Approximations

§16.26 Approximations

►For discussions of the approximation of generalized hypergeometric functions and the Meijer $G$-function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).

29: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. … ►Lastly, when $p>q+1$ the right-hand side of (16.5.1) can be regarded as the definition of the (customarily undefined) left-hand side. … ►Laplace transforms and inverse Laplace transforms of generalized hypergeometric functions are given in Prudnikov et al. (1992a, §3.38) and Prudnikov et al. (1992b, §3.36). …

30: 24.16 Generalizations

§24.16 Generalizations

… ►For $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, Bernoulli and Euler polynomials of order $\mathrm{\ell }$ are defined respectively by … ► $B_n^{(x)}$ is a polynomial in $x$ of degree $n$. … ►

§24.16(ii) Character Analogs

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§24.16(iii) Other Generalizations

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