## Some arithmetic properties of short random walk integrals.(English)Zbl 1233.60024

Summary: We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequence of a more general evaluation, a closed form is obtained for the average distance traveled in three steps. This evaluation, as well as its proof, rely on explicit combinatorial properties, such as recurrence equations of the even moments (which are lifted to functional equations). The corresponding general combinatorial and analytic features are collected and made explicit in the case of 3 and 4 steps. Explicit hypergeometric expressions are given for the moments of a 3-step and 4-step walk and a general conjecture for even length walks is made.

### MSC:

 60G50 Sums of independent random variables; random walks 33C20 Generalized hypergeometric series, $${}_pF_q$$ 05A10 Factorials, binomial coefficients, combinatorial functions

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### References:

 [1] 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/$$\sim$$jb616/combat.pdf [2] 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) · Zbl 1152.33003 [3] 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) · Zbl 0126.28602 [4] Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987) · Zbl 0611.10001 [5] 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/$$\sim$$jb616/walks2.pdf · Zbl 1268.33005 [6] 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/$$\sim$$jb616/densities.pdf , http://arxiv.org/abs/1103.2995v1 · Zbl 1296.33011 [7] Broadhurst, D.: Bessel moments, random walks and Calabi-Yau equations. Preprint, November 2009 [8] Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162–2188 (2006) · Zbl 1126.65002 [9] Crandall, R.E.: Analytic representations for circle-jump moments. Preprint, October 2009 [10] Donovan, P.: The flaw in the JN-25 series of ciphers, II. Cryptologia (2010, in press) [11] Fettis, F.E.: On a conjecture of Karl Pearson, pp. 39–54 in Rider Anniversary Volume (1963) [12] Hughes, B.D.: Random Walks and Random Environments, vol. 1. Oxford University Press, London (1995) · Zbl 0820.60053 [13] Kluyver, J.C.: A local probability problem. Ned. Acad. Wet. Proc. 8, 341–350 (1906) [14] Mattner, L.: Complex differentiation under the integral. Nieuw Arch. Wiskd. IV Ser. 5/2(1), 32–35 (2001) · Zbl 1239.30025 [15] 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) [16] Pearson, K.: The random walk. Nature 72, 294 (1905) [17] Pearson, K.: The problem of the random walk. Nature 72, 342 (1905) [18] Pearson, K.: A Mathematical Theory of Random Migration. Mathematical Contributions to the Theory of Evolution XV. Drapers (1906) · JFM 38.0285.04 [19] Petkovsek, M., Wilf, H., Zeilberger, D.: A=B, 3rd edn. AK Peters, Wellesley (2006) [20] Rayleigh, L.: The problem of the random walk. Nature 72, 318 (1905) · JFM 36.0891.02 [21] Richmond, L.B., Shallit, J.: Counting abelian squares. Electron. J. Comb. 16, R72 (2009) · Zbl 1191.68479 [22] Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/sequences , 2009 · Zbl 1274.11001 [23] Titchmarsh, E.: The Theory of Functions, 2nd edn. Oxford University Press, London (1939) · Zbl 0022.14602 [24] Verrill, H.A.: Some congruences related to modular forms. Preprint MPI-1999-26, Max-Planck-Institut (1999) · Zbl 1209.11047 [25] Verrill, H.A.: Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations (2004). arXiv:math/0407327v1 [math.CO] [26] Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1941) · Zbl 0028.20605 [27] Wilf, H.: Generatingfunctionology, 2nd edn. Academic Press, San Diego (1993)
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