Cayley identity for Schwarzian derivatives

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1: 1.13 Differential Equations

… ►Then the following relation is known as Abel’s identity … ►

Elimination of First Derivative by Change of Independent Variable

… ►Here dots denote differentiations with respect to $\zeta$, and $\{z,\zeta \}$ is the Schwarzian derivative: … ►

Cayley’s Identity

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

2: 19 Elliptic Integrals

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3: Bibliography C

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An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:

Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.

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Referenced by:

§19.12, Ch.19, A. Cayley (1961).

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An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:

Second edition of Cayley (1895), unabridged and corrected

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Referenced by:

§19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.

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ISSN 0009-2614, Document, MathReview (J. S. Joel)

Referenced by:

§36.14(iii).

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The quintuple product identity. Int. J. Number Theory 2 (1), pp. 115–161.

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ISSN 1793-0421, Document, Link, MathReview (Andrew V. Sills)

Referenced by:

§17.8.

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4: Bibliography B

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Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.

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ISSN 0022-4715, Document, MathReview, ZentralBlatt

Referenced by:

§17.17.

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Rogers-Ramanujan Identities: A Century of Progress from Mathematics to Physics. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 163–172.

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ISSN 1431-0643, FullText, MathReview (Anne Schilling), ZentralBlatt

Referenced by:

§17.17.

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Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.

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ISSN 0305-4470, Document, MathReview (J. S. Joel), ZentralBlatt

Referenced by:

§36.9.

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A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

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ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§15.8(v).

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The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.

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Document, MathReview (N. A. Hall), ZentralBlatt

Referenced by:

§15.16.

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5: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

6: Bibliography M

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:

A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.

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Magma

Referenced by:

4th item.

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Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).

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An elementary proof of the Macdonald identities for $A_l^{(1)}$ . Adv. in Math. 57 (1), pp. 34–70.

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ISSN 0001-8708, Document, MathReview (Y. Lehrer-Ilamed), ZentralBlatt

Referenced by:

§17.15.

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Calculation of zeros of cylindrical functions and their derivatives. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (2), pp. 63–64, 71 (Russian).

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Notes:

In Russian

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ISSN 0137-0782, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.

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Notes:

Single-precision Fortran.

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FullText, Code, Document, ZentralBlatt

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4th item.

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7: 21.7 Riemann Surfaces

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§21.7(ii) Fay’s Trisecant Identity

… ►where again all integration paths are identical for all components. Generalizations of this identity are given in Fay (1973, Chapter 2). … ►

§21.7(iii) Frobenius’ Identity

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8: 1.1 Special Notation

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$x,y$	real variables.
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primes	derivatives with respect to the variable, except where indicated otherwise.
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$𝐈$	identity matrix
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9: 20.11 Generalizations and Analogs

… ►This is the discrete analog of the Poisson identity (§1.8(iv)). … ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►Similar identities can be constructed for ${}_2{}^{}F_1^{}(\frac{1}{3},\frac{2}{3};1;k^2)$, ${}_2{}^{}F_1^{}(\frac{1}{4},\frac{3}{4};1;k^2)$, and ${}_2{}^{}F_1^{}(\frac{1}{6},\frac{5}{6};1;k^2)$. … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

10: 9.18 Tables

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Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv \sqrt{\pi }\mathrm{Bi}\left(x\right)$, $U^{\prime }(x)=\sqrt{\pi }{\mathrm{Bi}}^{\prime }\left(x\right)$, $V(x)\equiv \sqrt{\pi }\mathrm{Ai}\left(x\right)$, $V^{\prime }(x)=\sqrt{\pi }{\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-9(.001)9$. Precision is 7S.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Nosova and Tumarkin (1965) tabulates $e_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$, $e_0^{\prime }(x)=-\pi {\mathrm{Hi}}^{\prime }\left(-x\right)$, ${\tilde{e}}_0(-x)\equiv -\pi \mathrm{Gi}\left(x\right)$, ${\tilde{e}}_0^{\prime }(-x)=\pi {\mathrm{Gi}}^{\prime }\left(x\right)$ for $x=-1(.01)10$; 7D. Also included are the real and imaginary parts of $e_0(z)$ and $\mathrm{i}{e}_0^{\prime }(z)$, where $z=\mathrm{i}y$ and $y=0(.01)9$; 6-7D.

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