Some arithmetic properties of short random walk integrals

Bailey, D.H., Borwein, J.M.: Hand-to-hand combat: Experimental mathematics with multi-thousand-digit integrals. J. Comput. Sci. (2010). doi: 10.1016/j.jocs.2010.12.004 . Available at: http://www.carma.newcastle.edu.au/~jb616/combat.pdf

Bailey, D.H., Borwein, J.M., Broadhurst, D.J., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A, Math. Theor. 41, 5203–5231 (2008)

Barrucand, P.: Sur la somme des puissances des coefficients multinomiaux et les puissances successives d’une fonction de Bessel. C. R. Hebd. Séances Acad. Sci. 258, 5318–5320 (1964)

Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987)

Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals. Exp. Math. (2011, in press). Available at: http://www.carma.newcastle.edu.au/~jb616/walks2.pdf

Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks. Canad. J. Math. (2011, in press). Available at: http://www.carma.newcastle.edu.au/~jb616/densities.pdf , http://arxiv.org/abs/1103.2995v1

Broadhurst, D.: Bessel moments, random walks and Calabi-Yau equations. Preprint, November 2009

Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162–2188 (2006)

Crandall, R.E.: Analytic representations for circle-jump moments. Preprint, October 2009

Donovan, P.: The flaw in the JN-25 series of ciphers, II. Cryptologia (2010, in press)

Fettis, F.E.: On a conjecture of Karl Pearson, pp. 39–54 in Rider Anniversary Volume (1963)

Hughes, B.D.: Random Walks and Random Environments, vol. 1. Oxford University Press, London (1995)

Kluyver, J.C.: A local probability problem. Ned. Acad. Wet. Proc. 8, 341–350 (1906)

Mattner, L.: Complex differentiation under the integral. Nieuw Arch. Wiskd. IV Ser. 5/2(1), 32–35 (2001)

Merzbacher, E., Feagan, J.M., Wu, T.-H.: Superposition of the radiation from N independent sources and the problem of random flights. Am. J. Phys. 45(10), 964–969 (1977)

Pearson, K.: The random walk. Nature 72, 294 (1905)

Pearson, K.: The problem of the random walk. Nature 72, 342 (1905)

Pearson, K.: A Mathematical Theory of Random Migration. Mathematical Contributions to the Theory of Evolution XV. Drapers (1906)

Petkovsek, M., Wilf, H., Zeilberger, D.: A=B, 3rd edn. AK Peters, Wellesley (2006)

Rayleigh, L.: The problem of the random walk. Nature 72, 318 (1905)

Richmond, L.B., Shallit, J.: Counting abelian squares. Electron. J. Comb. 16, R72 (2009)

Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/sequences , 2009

Titchmarsh, E.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939)

Verrill, H.A.: Some congruences related to modular forms. Preprint MPI-1999-26, Max-Planck-Institut (1999)

Verrill, H.A.: Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations (2004). arXiv:math/0407327v1 [math.CO]

Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1941)

Wilf, H.: Generatingfunctionology, 2nd edn. Academic Press, San Diego (1993)