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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 ζ , and { z , ζ } 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, λ ; (ii) the corresponding (real) eigenfunctions, u ( x ) and w ( t ) , have the same number of zeros, also called nodes, for t ( 0 , c ) as for x ( 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
3: Bibliography B
  • R. J. Baxter (1981) Rogers-Ramanujan identities in the hard hexagon model. J. Statist. Phys. 26 (3), pp. 427–452.
  • A. Berkovich and B. M. McCoy (1998) 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.
  • M. V. Berry and F. J. Wright (1980) Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.
  • J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
  • J. L. Burchnall and T. W. Chaundy (1948) The hypergeometric identities of Cayley, Orr, and Bailey. Proc. London Math. Soc. (2) 50, pp. 56–74.
  • 4: Bibliography C
  • A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.
  • A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).
  • B. K. Choudhury (1995) The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.
  • J. N. L. Connor and D. Farrelly (1981) 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.
  • A. Cruz and J. Sesma (1982) Zeros of the Hankel function of real order and of its derivative. Math. Comp. 39 (160), pp. 639–645.
  • 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
  • Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
  • J. Miller and V. S. Adamchik (1998) Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.
  • S. C. Milne (1985b) An elementary proof of the Macdonald identities for A l ( 1 ) . Adv. in Math. 57 (1), pp. 34–70.
  • V. P. Modenov and A. V. Filonov (1986) Calculation of zeros of cylindrical functions and their derivatives. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (2), pp. 63–64, 71 (Russian).
  • R. J. Moore (1982) Algorithm AS 187. Derivatives of the incomplete gamma integral. Appl. Statist. 31 (3), pp. 330–335.
  • 7: 21.7 Riemann Surfaces
    §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
    8: 1.1 Special Notation
    x , y real variables.
    primes derivatives with respect to the variable, except where indicated otherwise.
    𝐈 identity matrix
    9: 20.11 Generalizations and Analogs
    This is the discrete analog of the Poisson identity1.8(iv)). … In the case z = 0 identities for theta functions become identities in the complex variable q , with | q | < 1 , 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 F 1 2 ( 1 3 , 2 3 ; 1 ; k 2 ) , F 1 2 ( 1 4 , 3 4 ; 1 ; k 2 ) , and F 1 2 ( 1 6 , 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: 36.10 Differential Equations
    §36.10(ii) Partial Derivatives with Respect to the x n
    K = 1 , fold: (36.10.6) is an identity. …
    §36.10(iv) Partial z -Derivatives