relations%20to%20theta%20functions

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11: Bibliography D

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

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ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

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ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§10.49(ii), §18.34(iii).

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12: Bibliography M

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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Tata Lectures on Theta. II. Birkhäuser Boston Inc., Boston, MA.

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Notes:

Jacobian theta functions and differential equations

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ISBN 0-8176-3110-0, MathReview (M. Kh. Gizatullin), ZentralBlatt

Referenced by:

§21.7(i), §21.7(ii), §21.7(iii), Ch.21.

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13: Bibliography I

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Theta Functions. Springer-Verlag, New York.

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Notes:

Die Grundlehren der mathematischen Wissenschaften, Band 194

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ISBN 3540056998, MathReview (H. Klingen), ZentralBlatt

Referenced by:

§21.5(ii), §21.8, Ch.21.

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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Two families of orthogonal polynomials related to Jacobi polynomials. Rocky Mountain J. Math. 21 (1), pp. 359–375.

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ISSN 0035-7596, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.30(i).

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Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2 (1), pp. 1–21.

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ISSN 1073-2772, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(v).

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14: 14.30 Spherical and Spheroidal Harmonics

… ► $Y_{l,m}(\theta ,\varphi )$ are known as spherical harmonics. …Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ►Most mathematical properties of $Y_{l,m}(\theta ,\varphi )$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. … ►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►has solutions $W(\rho ,\theta ,\varphi )={\rho }^l{Y}_{l,m}(\theta ,\varphi )$, which are everywhere one-valued and continuous. …

15: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Cyclic identities for Jacobi elliptic and related functions. J. Math. Phys. 44 (4), pp. 1822–1841.

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ISSN 0022-2488, Document, MathReview (Zhi Guo Liu), ZentralBlatt

Referenced by:

§22.9(iv), §22.9(v).

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Theta relations and projective normality of Abelian varieties. Amer. J. Math. 98 (4), pp. 865–889.

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FullText, MathReview (T. Oda), ZentralBlatt

Referenced by:

§21.6(ii).

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The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.

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ISSN 0377-0427, Document, Link, MathReview Entry

Referenced by:

(18.9.18_5).

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16: 18.5 Explicit Representations

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Chebyshev

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Laguerre

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Hermite

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17: Bibliography C

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt

Referenced by:

§20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i), §25.18(i).

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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18: 1.11 Zeros of Polynomials

… ►A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►Addition of $-\frac{1}{3}a$ to each of these roots gives the roots of $f(z)=0$. … ►For the roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ of $g(w)=0$ and the roots ${\theta }_1,{\theta }_2,{\theta }_3$ of the resolvent cubic equation …Add $-\frac{1}{4}a$ to the roots of $g(w)=0$ to get those of $f(z)=0$. … ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …