“Pólya's Random Walk Constants”的概念、定义、习题解题方法及翻译-数学辞典

单词

Pólya's Random Walk Constants

释义

Pólya's Random Walk Constants

N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.

Let be the probability that a Random Walk on a -D lattice returns to the origin. Pólya (1921) provedthat

(1)

but

(2)

for

. Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that

(3)

where

where

is a complete Elliptic Integral of the First Kind and

is the Gamma Function. Closedforms for

are not known, but Montroll (1956) showed that

(9)

where


			(10)

and

is a Modified Bessel Function of the First Kind. Numerical values from Montroll (1956) and Flajolet(Finch) are

See also Random Walk

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/polya/polya.html

Domb, C. ``On Multiple Returns in the Random-Walk Problem.'' Proc. Cambridge Philos. Soc. 50, 586-591, 1954.

Glasser, M. L. and Zucker, I. J. ``Extended Watson Integrals for the Cubic Lattices.'' Proc. Nat. Acad. Sci. U.S.A. 74, 1800-1801, 1977.

McCrea, W. H. and Whipple, F. J. W. ``Random Paths in Two and Three Dimensions.'' Proc. Roy. Soc. Edinburgh 60, 281-298, 1940.

Montroll, E. W. ``Random Walks in Multidimensional Spaces, Especially on Periodic Lattices.'' J. SIAM 4, 241-260, 1956.

Watson, G. N. ``Three Triple Integrals.'' Quart. J. Math., Oxford Ser. 2 10, 266-276, 1939.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。