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1: 19.2 Definitions

… ►where $p_j$ is a polynomial in $t$ while $\rho$ and $\sigma$ are rational functions of $t$. … ►Here $a,b,p$ are real parameters, and $k_c$ and $x$ are real or complex variables, with $p\ne 0$, $k_c\ne 0$. … ►If , then $k_c$ is pure imaginary. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►For the special cases of $R_C(x,x)$ and $R_C(0,y)$ see (19.6.15). …

2: 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 $q\le 1$. … ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). … ►For applications of the integral representation (35.5.3) see McFarland and Richards (2001, 2002) (statistical estimation of misclassification probabilities for discriminating between multivariate normal populations). The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). …

3: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). … ►The function $I_x(a,b)$ appears in: Monte Carlo sampling in statistical mechanics (Kofke (2004)); analysis of packings of soft or granular objects (Prellberg and Owczarek (1995)); growth formulas in cosmology (Hamilton (2001)). … ►The function $E_p\left(x\right)$, with $p>0$, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). ►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)).

4: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. In Wagstaff (2002) these results are extended to $n=60(2)152$ and $n=40(2)88$, respectively, with further complete and partial factorizations listed up to $n=300$ and $n=200$, respectively. …

5: 17 q-Hypergeometric and Related Functions

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6: 32.9 Other Elementary Solutions

… ►Elementary nonrational solutions of ${\text{P}}_{\text{III}}$ are … ►Then ${\text{P}}_{\text{III}}$ has algebraic solutions iff … ►Elementary nonrational solutions of ${\text{P}}_{\text{V}}$ are … ►An elementary algebraic solution of ${\text{P}}_{\text{VI}}$ is … ►For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b). …

7: 16.24 Physical Applications

… ►For an extension to two-loop integrals see Moch et al. (2002). … ►They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. …These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. Lastly, special cases of the $\mathit{9}j$ symbols are ${}_5{}^{}F_4^{}$ functions with unit argument. …

8: 24.18 Physical Applications

… ►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). …

9: Vadim B. Kuznetsov

… ►He was also editor of The Kowalevski Property, CRM Proceedings and Lecture Notes 32, published by the American Mathematical Society in 2002. …

10: 32.10 Special Function Solutions

… ►with $C_1$, $C_2$ arbitrary constants. … ►For determinantal representations see Forrester and Witte (2002) and Okamoto (1987c). … ► ►For determinantal representations see Forrester and Witte (2002), Masuda (2004), and Okamoto (1987b). … ► …