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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument

{{}_{p}F_{q}}

, with

p\leq 2

and

q\leq 1

. … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

33: 16.4 Argument Unity

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Watson’s Sum

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34: 14.14 Continued Fractions

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14.14.1

\tfrac{1}{2}\left(x^{2}-1\right)^{1/2}\frac{P^{\mu}_{\nu}\left(x\right)}{P^{% \mu-1}_{\nu}\left(x\right)}=\cfrac{x_{0}}{y_{0}+\cfrac{x_{1}}{y_{1}+\cfrac{x_{% 2}}{y_{2}+\cdots}}},

ⓘ

Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Referenced by:: §14.14
Permalink:: http://dlmf.nist.gov/14.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.14 and Ch.14

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14.14.3

(\nu-\mu)\frac{Q^{\mu}_{\nu}\left(x\right)}{Q^{\mu}_{\nu-1}\left(x\right)}=% \cfrac{x_{0}}{y_{0}-\cfrac{x_{1}}{y_{1}-\cfrac{x_{2}}{y_{2}-\cdots}}},

\nu\neq\mu

ⓘ

Symbols:: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $x$ : real variable, $\mu$ : general order, $\nu$ : general degree, $x_{k}$ : numerator and $x_{k}$ : denominator
Referenced by:: §14.14
Permalink:: http://dlmf.nist.gov/14.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.14 and Ch.14

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35: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

36: 16.9 Zeros

§16.9 Zeros

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37: Publications

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B. V. Saunders and Q. Wang (1999) Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology.

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B. V. Saunders and Q. Wang (2000) From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces, Proceedings of the 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, September 25-28, 2000.

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B. V. Saunders and Q. Wang (2005) Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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B. V. Saunders and Q. Wang (2006) From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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A. Youssef (2007) Methods of Relevance Ranking and Hit-content Generation in Math Search, Proceedings of Mathematical Knowledge Management (MKM2007), RISC, Hagenberg, Austria, June 27–30, 2007.

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38: Howard S. Cohl

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and

q

-series. …

39: 14.10 Recurrence Relations and Derivatives

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14.10.1

{\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.10
Permalink:: http://dlmf.nist.gov/14.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

►

14.10.2

{\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

►

14.10.3

{(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.5.3 (modified)
Referenced by:: §14.10, §14.14, §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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14.10.4

\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.5(i)
Permalink:: http://dlmf.nist.gov/14.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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14.10.5

\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}=(\nu+\mu)\mathsf{P}^{\mu}_{\nu-1}\left(x\right)-\nu x\mathsf{P}^{% \mu}_{\nu}\left(x\right).

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.5.4 (modified)
Referenced by:: §14.10
Permalink:: http://dlmf.nist.gov/14.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §14.10 and Ch.14

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40: Barry I. Schneider

… ►He is also a General Editor for the DLMF project. …Before coming to NIST in 2014, he was a postdoctoral research associate at the University of Southern California (1969-1970), and a staff member of the General Telephone and Electronics Laboratory (1970-1972). …In early 2014, he came to NIST as General Editor of the DLMF project. …