# applied to generalized hypergeometric functions

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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 $\mathbf{O}(m)$ applied to a generalization of the integral (35.5.8). …
##### 3: 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. …
##### 4: 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. …
##### 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 $|\operatorname{ph}\left(-z\right)|\leq(p+1-q-\delta)\pi/2$, 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
###### Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
##### 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 $p\leq 2$ and $q\leq 1$. … 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. …
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
For the functions $K_{b-1}$ and ${{}_{2}{\mathbf{F}}_{1}}$ see §10.25(ii) and §§15.1, 15.2(i). … Similar conventions also apply to the remaining integrals in this subsection. …At the point where the contour crosses the interval $(1,\infty)$, $t^{-b}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values; compare §§15.1 and 15.2(i). … Again, $t^{-c}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values where the contour intersects the positive real axis. …