### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 042, 26 pages      arXiv:1707.05218      https://doi.org/10.3842/SIGMA.2018.042

### Higher Derivatives of Airy Functions and of their Products

Eugeny G. Abramochkin and Evgeniya V. Razueva
Coherent Optics Lab, Lebedev Physical Institute, Samara, 443011, Russia

Received October 13, 2017, in final form April 26, 2018; Published online May 05, 2018

Abstract
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.

Key words: Airy functions; Gegenbauer polynomials; hypergeometric function.

pdf (505 kb)   tex (85 kb)

References

1. Airy G.B., On the intensity of light in the neighbourhood of a caustic, Trans. Cambridge Phil. Soc. 6 (1838), 379-401.
2. Airy G.B., Supplement to the paper ''On the intensity of light in the neighbourhood of a caustic'', Trans. Cambridge Phil. Soc. 8 (1849), 595-599.
3. Aspnes D.E., Electric-field effects on optical absorption near thresholds in solids, Phys. Rev. 147 (1966), 554-566.
4. Bailey W.N., Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, Vol. 32, Stechert-Hafner, Inc., New York, 1964.
5. Borwein D., Borwein J.M., Glasser M.L., Wan J.G., Moments of Ramanujan's generalized elliptic integrals and extensions of Catalan's constant, J. Math. Anal. Appl. 384 (2011), 478-496.
6. Brychkov Y.A., Handbook of special functions. Derivatives, integrals, series and other formulas, CRC Press, Boca Raton, FL, 2008.
7. Brychkov Y.A., On higher derivatives of the Bessel and related functions, Integral Transforms Spec. Funct. 24 (2013), 607-612.
8. Ebisu A., Special values of the hypergeometric series, Mem. Amer. Math. Soc. 248 (2017), v+96 pages, arXiv:1308.5588.
9. Englert B.G., Schwinger J., Statistical atom: some quantum improvements, Phys. Rev. A 29 (1984), 2339-2352.
10. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, McGraw-Hill Book Co., New York, 1953.
11. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, McGraw-Hill Book Company, New York, 1953.
12. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. III, McGraw-Hill Book Company, New York, 1955.
13. Gessel I., Stanton D., Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), 295-308.
14. Graham R.L., Knuth D.E., Patashnik O., Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994.
15. Laurenzi B.J., Polynomials associated with the higher derivatives of the Airy functions ${\rm Ai}(z)$ and ${\rm Ai}'(z)$, arXiv:1110.2025.
16. Leal Ferreira P., Castilho Alcarás J.A., $S$-wave radial excitations for a linear potential, Lett. Nuovo Cimento 14 (1975), 500-504.
17. Maurone P.A., Phares A.J., On the asymptotic behavior of the derivatives of Airy functions, J. Math. Phys. 20 (1979), 2191.
18. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010, available at http://dlmf.nist.gov.
19. Petkovšek M., Wilf H.S., Zeilberger D., $A=B$, A.K. Peters, Ltd., Wellesley, MA, 1996, available at http://www.math.upenn.edu/~wilf/AeqB.pdf.
20. Pólya G., Induction and analogy in mathematics. Mathematics and plausible reasoning, Vol. I, Princeton University Press, Princeton, N.J., 1954.
21. Prudnikov A.P., Brychkov Y.A., Marichev O.I., Integrals and series, Vol. 3, More special functions, Gordon and Breach Science Publishers, New York, 1990.
22. Szegő G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1959.
23. Titchmarsh E.C., The theory of functions, 2nd ed., Oxford University Press, Oxford, 1939.
24. Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.
25. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
26. WolframAlpha, $n$-th derivative of Airy function, available at http://www.wolframalpha.com/input/?i=n-th+derivative+of+Airy+function.