generalized Mehler%E2%80%93Fock transformation

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21: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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§22.7(iii) Generalized Landen Transformations

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22: 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).

23: 16.4 Argument Unity

… ► … ►

§16.4(iii) Identities

… ► … ►A different type of transformation is that of Whipple: … ► …

24: 18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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§18.7(ii) Quadratic Transformations

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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

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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.

25: 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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26: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(iii) Generalized Exponential Integral

… ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

27: 9.1 Special Notation

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$k$	nonnegative integer, except in §9.9(iii).
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… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

28: 4.44 Other Applications

§4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). ►For an application of the Lambert $W$-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). …

29: 4.12 Generalized Logarithms and Exponentials

§4.12 Generalized Logarithms and Exponentials

►A generalized exponential function $\varphi (x)$ satisfies the equations …Its inverse $\psi (x)$ is called a generalized logarithm. It, too, is strictly increasing when $0\le x\le 1$, and … ►For analytic generalized logarithms, see Kneser (1950).

30: 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). …