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21: 17.12 Bailey Pairs

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${\beta }_n={\displaystyle \sum _{j=0}^n}{\alpha }_j{u}_{n-j}{v}_{n+j},$

… ►A sequence of pairs of rational functions of several variables $({\alpha }_n,{\beta }_n)$, $n=0,1,2,\mathrm{\dots }$, is called a Bailey pair provided that for each $n\geqq 0$ … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then … ►If $({\alpha }_n,{\beta }_n)$ is a Bailey pair, then so is $({\alpha }_n^{\prime },{\beta }_n^{\prime })$, where … ►See Andrews (2000, 2001), Andrews and Berkovich (1998), Andrews et al. (1999), Milne and Lilly (1992), Spiridonov (2002), and Warnaar (1998).

22: 18.16 Zeros

… ►Let $j_{\alpha ,m}$ be the $m$th positive zero of the Bessel function $J_{\alpha }\left(x\right)$ (§10.21(i)). … ►Let ${\varphi }_m=j_{\alpha ,m}/\rho$. … ►For three additional terms in this expansion see Gatteschi (2002). … ►In the notation of this reference $x_{n,m}=u_{a,m}$, $\mu =\sqrt{2n+1}$, and $\alpha ={\mu }^{-\frac{4}{3}}{a}_m$. … ►For further information on the zeros of the classical orthogonal polynomials, see Szegő (1975, Chapter VI), Erdélyi et al. (1953b, §§10.16 and 10.17), Gatteschi (1987, 2002), López and Temme (1999a), and Temme (1990a). …

23: 31.8 Solutions via Quadratures

… ►Denote $𝐦=(m_0,m_1,m_2,m_3)$ and $\lambda =-4q$. …Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …Lastly, ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$, are the zeros of the Wronskian of $w_{+}(𝐦;\lambda ;z)$ and $w_{-}(𝐦;\lambda ;z)$. ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …For more details see Smirnov (2002). …

24: Bibliography G

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Asymptotics and bounds for the zeros of Laguerre polynomials: A survey. J. Comput. Appl. Math. 144 (1-2), pp. 7–27.

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ISSN 0377-0427, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.16(iv), §18.16(iv), §18.16(vi).

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Exponential integral ${\int }_1^{\mathrm{\infty }}{e}^{-xt}{t}^{-n}𝑑t$ for large values of $n$ . J. Res. Nat. Bur. Standards 62, pp. 123–125.

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MathReview (W. Wasow), ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii).

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Von Staudt for ${𝐅}_q[T]$ . Duke Math. J. 45 (4), pp. 885–910.

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ISSN 0012-7094, Document, MathReview (Neal Koblitz), ZentralBlatt

Referenced by:

§24.16(iii).

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Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.

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ISSN 0025-5718, Document, MathReview (David A. Richter), ZentralBlatt

Referenced by:

§19.29(ii).

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Painlevé Differential Equations in the Complex Plane. Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin-New York.

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ISBN 3-11-017379-4, MathReview, ZentralBlatt

Referenced by:

§32.10(i), §32.10(iii), §32.10(iv), §32.10(v), §32.10(vi), §32.2(iv), §32.7(iii), §32.7(iv), §32.7(v), §32.7(vi), §32.7(vii), §32.8(ii), §32.8(iii), §32.8(iv), §32.8(v), §32.9(i), §32.9(ii), §32.9(iii).

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25: Bibliography H

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La $q$-conjecture de Macdonald-Morris pour $G_2$ . C. R. Acad. Sci. Paris Sér. I Math. 303 (6), pp. 211–213 (French).

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ISSN 0249-6291, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.14.

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Analytic evaluation of two-center STO electron repulsion integrals via ellipsoidal expansion. Internat. J. Quantum Chem. 88 (6), pp. 701–734.

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Document

Referenced by:

§8.24(iii).

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25D Table of the First One Hundred Values of $j_{0,s}$,$J_1(j_{0,s})$, $j_{1,s}$,$J_0(j_{1,s})=J_0(j_{0,s+1}^{\prime })$,$j_{1,s}^{\prime }$,$J_1(j_{1,s}^{\prime })$ . Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

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Ms. of six pages deposited in the UMT file

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8th item.

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Algorithm 571: Statistics for von Mises’ and Fisher’s distributions of directions: $I_1(x)/I_0(x)$, $I_{1.5}(x)/I_{0.5}(x)$ and their inverses [S14]. ACM Trans. Math. Software 7 (2), pp. 233–238.

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Notes:

Single-precision Fortran, minimum accuracy 8S.

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ISSN 0098-3500, Document, GAMS (module 571; package TOMS)

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5th item.

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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

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ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt

Referenced by:

§1.6(i).

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26: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

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Uniform asymptotic expansion of $J_{\nu }(\nu a)$ via a difference equation. Numer. Math. 91 (1), pp. 147–193.

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ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.20(ii).

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Rational Chebyshev approximations for the Bessel functions $J_0(x)$, $J_1(x)$, $Y_0(x)$, $Y_1(x)$ . Math. Comp. 39 (160), pp. 617–623.

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Notes:

Tables on microfiche

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

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9th item.

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Asymptotics for the ${}_4{}^{}F_3^{}$ polynomials. J. Approx. Theory 66 (1), pp. 58–71.

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ISSN 0021-9045, Document, MathReview (R. G. Buschman), ZentralBlatt

Referenced by:

§18.26(v).

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An asymptotic expansion of $W_{k,m}(z)$ with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

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FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§13.19.

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27: 32.7 Bäcklund Transformations

… ►Let $w_j(z_j)=w(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2$, be solutions of ${\text{P}}_{\text{V}}$ with … ►satisfies ${\text{P}}_{\text{V}}$ with … ►Let $w_j(z_j)=w_j(z_j;{\alpha }_j,{\beta }_j,{\gamma }_j,{\delta }_j)$, $j=0,1,2,3$, be solutions of ${\text{P}}_{\text{VI}}$ with … ► ${\text{P}}_{\text{VI}}$ also has quadratic and quartic transformations. …Also, …

28: Bibliography S

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Adaptive Quasi-Monte Carlo Integration Based on MISER and VEGAS. In Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 393–406.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.5(x).

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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Elliptic Solitons and Heun’s Equation. In The Kowalevski Property (Leeds, UK, 2000), V. B. Kuznetsov (Ed.), CRM Proc. Lecture Notes, Vol. 32, pp. 287–306.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.8.

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An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 2002 (37), pp. 1945–1977.

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ISSN 1073-7928, Document, MathReview (S. Ole Warnaar), ZentralBlatt

Referenced by:

§17.12.

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On the roots of $f(z)=J_0(z)-i{J}_1(z)$ . Quart. Appl. Math. 46 (1), pp. 105–107.

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ISSN 0033-569X, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(ix).

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29: 9.16 Physical Applications

… ►The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). …

30: 30.16 Methods of Computation

… ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements …and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. … ►which yields ${\lambda }_4^2\left(10\right)=\mathrm{13.97907\hspace{0.33em}345}$. … ►Form the eigenvector ${[e_{1,d},e_{2,d},\mathrm{\dots },e_{d,d}]}^{\mathrm{T}}$ of $𝐀$ associated with the eigenvalue ${\alpha }_{p,d}$, $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$, normalized according to … ►For other methods see Van Buren and Boisvert (2002, 2004).