applied to generalized hypergeometric functions

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1: 16.4 Argument Unity

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2: 35.10 Methods of Computation

… ►See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). …

3: 17.17 Physical Applications

§17.17 Physical Applications

… ►Quantum groups also apply $q$-series extensively. …They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics. See Kassel (1995). … ►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …

4: 16.23 Mathematical Applications

§16.23 Mathematical Applications

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … ►

§16.23(ii) Random Graphs

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§16.23(iv) Combinatorics and Number Theory

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5: 16.5 Integral Representations and Integrals

§16.5 Integral Representations and Integrals

… ►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. In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. … ►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). …

6: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

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Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

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7: 35.9 Applications

§35.9 Applications

►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 $p\le 2$ and $q\le 1$. … ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). ►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. …

8: Richard A. Askey

… ►One of his most influential papers Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials (with J. …Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. … ►He was elected Fellow of the Society for Industrial and Applied Mathematics (SIAM) in 2009 and Fellow of the American Mathematical Society (AMS) in 2012. … ►

5 Gamma Function

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16 Generalized Hypergeometric Functions & Meijer G-Function

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9: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

… ►Systems of partial differential equations for the ${}_0{}^{}F_1^{}$ (defined in §35.8) and ${}_1{}^{}F_1^{}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). … ►Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …

10: Joris Van der Jeugt

… ► 1959 in Lokeren, Belgium) is Professor Emeritus in the Department of Applied Mathematics, Computer Science and Statistics at Ghent University. His research interests are in the following areas: Group theoretical methods in physics; Representation theory of Lie algebras, Lie superalgebras and quantum groups with applications in mathematical physics; 3$nj$-symbols and their relations to special functions and orthogonal polynomials; Quantum theory, finite quantum systems, quantum oscillator models, Wigner quantum systems; and Parabosons, parafermions and generalized quantum statistics. … ►As postdoc he went to Queen Mary College (London) and to the University of Southampton. … ►His results for Lie superalgebra representations continue to inspire many scientists. His publications on Clebsch-Gordan coefficients, Racah coefficients, 3$nj$-coefficients and their relation to hypergeometric series are considered as standard and a review is part of the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. …