SL%282%2CZ%29 bilinear transformation

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21: Bibliography E

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Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

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ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(i).

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

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Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:

Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778, v. 36 (1981), no. 153, p. 315, v. 30 (1976), no. 135, p. 675, v. 29 (1975), no. 130, p. 670, v. 26 (1972), no. 118, p. 598, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239, v. 17 (1963), no. 84, p. 485.

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MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iv), §10.32(i), §10.32(iii), §10.32(iv), §10.43(iv), §10.43(vi), §10.44(iv), §10.60(iv), §10.9(i), §10.9(iii), §10.9(iv), §11.10(vi), §11.4(i), Ch.11, §12.12, §12.12, §12.13(i), §12.5(iv), Ch.12, §18.16(vi), (18.17.21_1), §18.17(iv), §18.17(i), §18.17(viii), (18.35.1), (18.35.2), (18.35.4_5), §18.35(i), (18.5.11_1), (18.5.11_3), §18.5(ii), §18.9(iii), §19.1, §19.10(i), §19.14(ii), (19.25.40), §19.25(vi), §19.5, §19.7(ii), §19.9(i), Ch.19, §20.9(i), §22.15(ii), §23.6(iv), Ch.23, §8.13(i), §8.14, §8.3(i), §8.4, §8.6(iii), §8.7, Ch.8, Notations P, Notations P.

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Painlevé transcendent describes quantum correlation function of the $XXZ$ antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

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ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt

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§32.16.

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On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

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Notes:

With a loose Russian summary

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ISSN 0011-4642, MathReview (D. Hinton)

Referenced by:

§1.13(viii).

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22: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

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§22.7(iii) Generalized Landen Transformations

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23: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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MathReview (L. Berg)

Referenced by:

§7.14(iii).

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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.

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Document, MathReview (E. L. Koh), ZentralBlatt

Referenced by:

§2.5(ii).

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

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Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1991.

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ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.11(v), Ch.3.

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An explicit formula for Bernoulli numbers. Rep. Fac. Sci. Technol. Meijo Univ. 29, pp. 1–6.

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ZentralBlatt

Referenced by:

§24.6.

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24: Bibliography W

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Two tables of partitions. Proc. London Math. Soc. (2) 42, pp. 550–556.

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Document, ZentralBlatt

Referenced by:

§27.21.

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Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

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ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt

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7th item, 10th item.

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

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ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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Document, ZentralBlatt

Referenced by:

§16.6.

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A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

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25: 28.6 Expansions for Small $q$

… ►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). ►The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … ►Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. … ►where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime }=\sqrt{1-k^2}$. … ►For more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). …

26: 18.38 Mathematical Applications

… ►However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … ►For the generalized hypergeometric function ${}_3{}^{}F_2^{}$ see (16.2.1). … ►

Radon Transform

… ►Define a further operator $K_2$ by … ►See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). …

27: Bibliography D

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Sur une classe de nombres rationnels réductibles aux nombres de Bernoulli. Bull. Sci. Math. (2) 28, pp. 29–32 (French).

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ZentralBlatt

Referenced by:

§24.1.

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On the real roots of Euler polynomials. Monatsh. Math. 106 (2), pp. 115–138.

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ISSN 0026-9255, MathReview (H. M. Srivastava), ZentralBlatt

Referenced by:

§24.12(ii).

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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$2$D gravity and random matrices. Phys. Rep. 254 (1-2), pp. 1–133.

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ISSN 0370-1573, Document, MathReview (Yolanda Lozano)

Referenced by:

§32.16.

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The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

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28: 26.2 Basic Definitions

… ►Thus $231$ is the permutation $\sigma (1)=2$, $\sigma (2)=3$, $\sigma (3)=1$. … ►Here $\sigma (1)=2,\sigma (2)=5$, and $\sigma (5)=1$. … ►A lattice path is a directed path on the plane integer lattice $\{0,1,2,\mathrm{\dots }\}\times \{0,1,2,\mathrm{\dots }\}$. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …

29: Bibliography S

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

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The elliptical microstrip antenna with circular polarization. IEEE Trans. Antennas and Propagation 29 (1), pp. 90–94.

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FullText

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5th item.

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Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§18.36(vi), §18.39(i).

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30: 29.21 Tables

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Ince (1940a) tabulates the eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^{m+1}\left(k^2\right)$ (with $a_{\nu }^{2m+1}$ and $b_{\nu }^{2m+1}$ interchanged) for $k^2=0.1,0.5,0.9$, $\nu =-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.