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1: Bibliography

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Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt

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§32.2(v).

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Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.

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ISSN 0010-485X, Document, MathReview, ZentralBlatt

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§13.29(iii).

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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.

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ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

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Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.

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Notes:

In memory of Gian-Carlo Rota

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ISSN 0097-3165, Document, MathReview (Alessandro Di Bucchianico), ZentralBlatt

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§17.12.

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Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.

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ISBN 0-7923-7119-4, MathReview, ZentralBlatt

Referenced by:

§17.12.

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2: 10.73 Physical Applications

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§10.73(i) Bessel and Modified Bessel Functions

►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …

3: 1.4 Calculus of One Variable

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Chain Rule

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L’Hôpital’s Rule

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4: 1.5 Calculus of Two or More Variables

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Chain Rule

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5: 3.8 Nonlinear Equations

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§3.8(ii) Newton’s Rule

… ► … ►Newton’s rule is given by … ►Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …

6: 17.12 Bailey Pairs

… ►When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … ►The Bailey pair and Bailey chain concepts have been extended considerably. …

7: 18.40 Methods of Computation

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►

Derivative Rule Approach

►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … ►

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Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. … Magnify

►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

8: 20 Theta Functions

Chapter 20 Theta Functions

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9: Bibliography R

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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arXiv:1812.02196

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§18.40(ii).

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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ISSN 1065-2469, Document, MathReview, ZentralBlatt

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§10.60(iii).

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10: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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

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§28.8(iv).

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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

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§17.12.

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

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Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

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3rd item.

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

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