Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 423-439 (2020); correction ibid. 313, C1 (2020).
The paper is a good demonstration that the theory of random walks has beautiful connections to a large number of exotic topics of mathematics. Those who study random walks will eventually encounter topics like arithmetic differential equations, algebraic groups, Mahler measure, special function theory, modular functions, and so on.
The first part of the paper studies uniform random walks. An $$N$$-step uniform planar random walk starts at the origin and consists of $$N$$ steps of length 1 each taken into a uniformly random direction. Let $$X_N$$ be the distance to the origin after $$N$$ steps. The $$s$$-th moment of $$X_N$$ is denoted by $$W_N(s)$$.
We learn from the third section that $$W_N(s)$$ is a Mahler measure of the polynomial $$x_1+\cdots+x_N$$ at $$s$$, and that $$W_N'(0)$$ is the logarithmic Mahler measure of the same polynomial. Based on this theory, one can deduce interesting closed form expressions for $$W_k'(0)$$, for small integers $$k$$.
The second part studies generic two-step random walks, that is, random walks where $$X=e^{2\pi i\theta_1}X_1+e^{2\pi i\theta_2}X_2$$, where $$\theta_i$$ are uniformly distributed on $$[0,1]$$, and $$X_i$$ are independent random variables on $$[0,\infty[$$ $$(i=1,2)$$. The finding of the $$W_N'(0)$$ values is described in details in Sections 4–7.
The last part of the paper studies additional relations between Mahler measures and the $$W_N'(0)$$ values for small $$N$$.
For the entire collection see [Zbl 1442.00024].

### MSC:

 33E05 Elliptic functions and integrals 33C20 Generalized hypergeometric series, $${}_pF_q$$
Full Text:

### References:

  Adamchik, V.S.: Integral and series representations for Catalan’s constant. Unpublished note. http://www.cs.cmu.edu/ adamchik/articles/catalan.htm · Zbl 1009.33006  Ahlgren, S., Berndt, B.C., Yee, A.Y., Zaharescu, A.: Integrals of Eisenstein series and derivatives of $$L$$-functions. Int. Math. Res. Not. 2002(32), 1723-1738 (2002) · Zbl 1086.11023  Akatsuka, H.: Zeta Mahler measures. J. Number Theory 129(11), 2713-2734 (2009) · Zbl 1235.11096  Almkvist, G., van Straten, D., Zudilin, W.: Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations. Proc. Edinb. Math. Soc. 54(2), 273-295 (2011) · Zbl 1223.33007  Bailey, D.H., Borwein, J.M.: Hand-to-hand combat with multi-thousand-digit integrals. J. Comput. Sci. 3, 77-86 (2012)  Bailey, D.H., Borwein, J.M., Broadhurst, D.J., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A 41(20), 5203-5231 (2008) · Zbl 1152.33003  Berndt, B.C., Zaharescu, A.: An integral of Dedekind eta-functions in Ramanujan’s lost notebook. J. Reine Angew. Math. 551, 33-39 (2002) · Zbl 1001.11017  Borwein, J.M.: A short walk can be beautiful. J. Humanist. Math. 6(1), 86-109 (2016)  Borwein, J.M., Nuyens, D., Straub, A., Wan, J.: Some arithmetic properties of short random walk integrals. Ramanujan J. 26(1), 109-132 (2011) · Zbl 1233.60024  Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks, with an appendix by D. Zagier. Can. J. Math. 64(5), 961-990 (2012) · Zbl 1296.33011  Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals. Exp. Math. 22(1), 1-14 (2013) · Zbl 1268.33005  Boyd, D.: Mahler’s measure and special values of $$L$$-functions. Exp. Math. 7(1), 37-82 (1998) · Zbl 0932.11069  Boyd, D., Lind, D., Rodriguez-Villegas, F., Deninger, C.: The many aspects of Mahler’s measure. Final report of the Banff workshop 03w5035 (26 April-1 May 2003). http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf  Broadhurst, D.: Feynman integrals, $$L$$-series and Kloosterman moments. Commun. Number Theory Phys. 10(3), 527-569 (2016) · Zbl 1362.81072  Chan, H.H., Wan, J., Zudilin, W.: Legendre polynomials and Ramanujan-type series for $$1/\pi$$. Isr. J. Math. 194(1), 183-207 (2013) · Zbl 1357.11123  Cohen, H.: Personal communication (23 March 2018)  Duke, W., Imamo $$\bar{\rm{g}}$$ lu, Ö.: On a formula of Bloch. Funct. Approx. 37(1), 109-117 (2007) · Zbl 1213.11141  Guillera, J.: A family of Ramanujan-Orr formulas for $$1/\pi$$. Integral Transforms Spec. Funct. 26(7), 531-538 (2015) · Zbl 1432.33014  Paşol, V., Zudilin, W.: A study of elliptic gamma function and allies. Res. Math. Sci. 5(4), Art. 39, p 11 (2018) · Zbl 1440.11063  Rogers, M.D.: A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities. J. Number Theory 121, 265-304 (2006) · Zbl 1132.11032  Rogers, M.D., Straub, A.: A solution of Sun’s \$520 challenge concerning $$520/\pi$$. Int. J. Number Theory 9, 1273-1288 (2013) · Zbl 1277.33002  Rogers, M.D., Zudilin, W.: On the Mahler measure of $$1+X+1/X+Y+1/Y$$. Int. Math. Res. Not. 2014(9), 2305-2326 (2014) · Zbl 1378.11091  Shinder, E., Vlasenko, M.: Linear Mahler measures and double $$L$$-values of modular forms. J. Number Theory 142, 149-182 (2014) · Zbl 1298.11100  Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc. 23, 49-63 (1981) · Zbl 0442.10034  Sun, Z.-W.: List of conjectural series for powers of $$\pi$$ and other constants (2014). arXiv:1102.5649v47 [math.CA]  Takloo-Bighash, R.: A remark on a paper of S. Ahlgren, B.C. Berndt, A.J. Yee, and A. Zaharescu: “Integrals of Eisenstein series and derivatives of $$L$$-functions” . Int. J. Number Theory 2(1), 111-114 (2006)  Wan, J.G.: Personal communication (26 July 2011)  Wan, J.G.: Series for $$1/\pi$$ using Legendre’s relation. Integral Transforms Spec. Funct. 25(1), 1-14 (2014) · Zbl 1283.33012  Wan, J.G., Zudilin, W.: Generating functions of Legendre polynomials: a tribute to Fred Brafman. J. Approx. Theory 164, 488-503 (2012) · Zbl 1242.33018  Zhou, Y.: On Borwein’s conjectures for planar uniform random walks. J Austral. Math. Soc. 107(3), 392-411 (2019) · Zbl 1472.60081  Zhou, Y.: Wick rotations, Eichler integrals, and multi-loop Feynman diagrams. Commun. Number Theory Phys. 12(1), 127-192 (2018) · Zbl 1393.81029  Zudilin, W.: Period(d)ness of $$L$$-values. In: Borwein, J.M., et al. (eds.) Number Theory and Related Fields, In Memory of Alf van der Poorten. Springer Proceedings in Mathematics and Statistics, vol. 43, pp. 381-395. Springer, New York (2013) · Zbl 1316.11038  Zudilin, W.: A generating function of the squares of Legendre polynomials. Bull. Aust. Math. Soc. 89(1), 125-131 (2014) · Zbl 1334.33022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.