Three-step and four-step random walk integrals

Experimental Mathematics

Volumn 22, Issue 1, 2013, Pages 1-14

  Borwein, Jonathan M ^a   Straub, Armin ^b   Wan, James ^a  

^a UNIVERSITY OF NEWCASTLE   (Australia)

^b TULANE UNIVERSITY   (United States)

Author keywords

elliptic integrals; hypergeometric functions; Meijer G functions; random walks

Indexed keywords

EID: 84876194996     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2013.748379     Document Type: Article

References (24)

1. [Abramowitz and Stegun 72] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, 1972.
2. [Amdeberhan et al. 12] T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison, V. H. Moll, and A. Straub. "Ramanujan's Master Theorem." The Ramanujan Journal, 29 (2012), 103-120.
3. [Andrews et al. 99] G. E. Andrews, R. Askey, and R. Roy. Special Functions. Cambridge University Press, 1999.
4. [Bailey et al. 08] D. H. Bailey, J. M. Borwein, D. J. Broadhurst, and M. L. Glasser. "Elliptic Integral Evaluations of Bessel Moments and Applications." J. Phys. A: Math. Theor. 41 (2008), 205203.
5. [Bailey 32] W. N. Bailey. "Some Transformations of Generalized Hypergeometric Series, and Contour Integrals of Barnes's Type." Quart. J. Math. Oxford 3 (1932), 168- 182.
6. [Bernardi 12] O. Bernardi. "A Short Proof of Rayleigh's Theorem with Extensions." Preprint, arXiv: 1007.4870, 2012.
7. [Borwein and Borwein 98] J. M. Borwein and P. B. Borwein. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, 1998.
8. [Borwein et al. 04] J. M. Borwein, D. H. Bailey, and R. Girgensohn. Experimentation in Mathematics: Computational Paths to Discovery. A K Peters, 2004.
9. [Borwein et al. 11] J. M. Borwein, D. Nuyens, A. Straub, and J. Wan. "Some Arithmetic Properties of Short Random Walk Integrals." Ramanujan Journal 26:1 (2011), 109- 132.
10. [Borwein et al. 12] J. M. Borwein, A. Straub, J. Wan, and W. Zudilin. "Densities of Short Uniform Random Walks (with an Appendix by Don Zagier)." Canadian Journal of Mathematics 64:5 (2012), 961-990.
11. [Broadhurst 09] D. J. Broadhurst. "Bessel Moments, Random Walks and Calabi-Yau Equations." Preprint, 2009.
12. [Crandall 09] R. E. Crandall. Analytic Representations for Circle-Jump Moments. PSIpress, 2009.
13. [Hardy 78] G. H. Hardy. Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work, 3rd edition. Chelsea Publishing Company, 1978.
14. [Hughes 95] B. D. Hughes. Random Walks and Random Environments, vol. 1. Oxford University Press, 1995.
15. [Kluyver 06] J. C. Kluyver. "A Local Probability Problem." Nederl. Acad. Wetensch. Proc. 8 (1906), 341-350.
16. [Marichev 83] O. I. Marichev. Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables. Ellis Horwood Limited, 1983.
17. [Nesterenko 03] Y. V. Nesterenko. "Integral Identities and Constructions of Approximations to Zeta-Values." J. Theor. Nombres Bordeaux 15:2 (2003), 535-550.
18. [Pearson 05a] K. Pearson. "The Problem of the Random Walk." Nature 72:1866 (1905), 294.
19. [Pearson 05b] K. Pearson. "The Problem of the Random Walk." Nature 72:1867 (1905), 342.
20. [Pearson 06] K. Pearson. "A Mathematical Theory of Random Migration." In Drapers Company Research Memoirs, Biometric Series 3. Cambridge University Press, 1906.
21. [Watson 38] G. N. Watson. "Three Triple Integrals." Physica 5 (1938), 465.
22. [Watson 41] G. N. Watson. A Treatise on the Theory of Bessel Functions, 2nd edition. Cambridge University Press, 1941.
23. [Zudilin 02] W. Zudilin. "Very Well-Poised Hypergeometric Series and Multiple Integrals." Russian Math. Surveys 57:4 (2002), 824-826.
24. [Zudilin 04] W. Zudilin. "Arithmetic of Linear Forms Involving Odd Zeta Values." J. Theorie Nombres Bordeaux 16:1 (2004), 251-291.