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Faà di Bruno formula

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1: 1.4 Calculus of One Variable
Leibniz’s Formula
Faà Di Bruno’s Formula
2: Bibliography Z
  • R. Zanovello (1975) Sul calcolo numerico della funzione di Struve H ν ( z ) . Rend. Sem. Mat. Univ. e Politec. Torino 32, pp. 251–269 (Italian. English summary).
  • R. Zanovello (1977) Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).
  • R. Zanovello (1978) Su un integrale definito del prodotto di due funzioni di Struve. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 112 (1-2), pp. 63–81 (Italian).
  • W. Zudilin (2007) Approximations to -, di- and tri-logarithms. J. Comput. Appl. Math. 202 (2), pp. 450–459.
  • M. I. Žurina and L. N. Karmazina (1966) Tables and formulae for the spherical functions P - 1 / 2 + i τ m ( z ) . Translated by E. L. Albasiny, Pergamon Press, Oxford.
  • 3: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
  • W. Gautschi (1979c) Un procedimento di calcolo per le funzioni gamma incomplete. Rend. Sem. Mat. Univ. Politec. Torino 37 (1), pp. 1–9 (Italian).
  • J. S. Geronimo, O. Bruno, and W. Van Assche (2004) WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 4: DLMF Project News
    error generating summary
    5: Bibliography T
  • Y. Takei (1995) On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
  • P. G. Todorov (1991) Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.
  • F. G. Tricomi (1947) Sugli zeri delle funzioni di cui si conosce una rappresentazione asintotica. Ann. Mat. Pura Appl. (4) 26, pp. 283–300 (Italian).
  • F. G. Tricomi (1949) Sul comportamento asintotico dell’ n -esimo polinomio di Laguerre nell’intorno dell’ascissa 4 n . Comment. Math. Helv. 22, pp. 150–167.
  • 6: 32.16 Physical Applications
    For applications in 2D quantum gravity and related aspects of the enumerative topology see Di Francesco et al. (1995). …
    7: Peter Paule
     1958 in Ried im Innkreis, Austria) is Professor of Mathematics (successor to Bruno Buchberger), Director of the Research Institute for Symbolic Computation (RISC), and Director of the Doctoral Program on Computational Mathematics at the Johannes Kepler University, Linz, Austria. …
    8: Bibliography D
  • P. Di Francesco, P. Ginsparg, and J. Zinn-Justin (1995) 2 D gravity and random matrices. Phys. Rep. 254 (1-2), pp. 1–133.
  • A. M. Din (1981) A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.
  • L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.
  • 9: Bibliography
  • M. Abramowitz and I. A. Stegun (Eds.) (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
  • K. Aomoto (1987) Special value of the hypergeometric function F 2 3 and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
  • T. M. Apostol (1985a) Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.
  • T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
  • 10: 27.20 Methods of Computation: Other Number-Theoretic Functions
    The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …