en>Monkbot: Task 3: Fix CS1 deprecated coauthor parameter errors

2014-05-07T01:20:24Z

Task 3: Fix CS1 deprecated coauthor parameter errors

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	~~In [[mathematics]], '''poly-Bernoulli numbers''', denoted as <math>B_{n}^{(k)}</math>, were defined by M. Kaneko as~~		The title of the author is Jayson. To play lacross is the thing I adore most of all. Alaska is exactly where he's always been living. Credit authorising is how he makes cash.<br><br>Feel free to visit my web page; [http://165.132.39.93/xe/visitors/372912 free psychic]

	~~:<math>{Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}</math>~~

	~~where ''Li'' is the [[polylogarithm]].~~ The ~~<math>B_{n}^{(1)}</math> are~~ the ~~usual [[Bernoulli number]]s.~~


	~~Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined by H. Jolany as follows~~
	~~:<math>{Li_{k}(1-(ab)^{-x})\over b^x-a^{-x}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(t;a,b,c){x^{n}\over n!}</math>~~

	~~where ''Li''~~ is ~~the [[polylogarithm]]~~.
	~~Kaneko also gave two combinatorial formulas:~~

	~~:<math>B_{n}^{(-k)}=\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},</math>~~

	~~:<math>B_{n}^{(-k)}=\sum_{j=0}^{\min(n,k)} (j!)^{2}S(n+1,j+1)S(k+1,j+1),</math>~~

	~~where <math>S(n,k)</math>~~ is the ~~number~~ of ~~ways to partition a size <math>n</math> set into <math>k</math> non-empty subsets (the [[Stirling number of the second kind]])~~.

	~~A combinatorial interpretation~~ is ~~that the poly-Bernoulli numbers of negative index enumerate the set of <math>n</math> by <math>k</math> [[binary matrix\|(0,1)-matrices]] uniquely reconstructible from their row and column sums.~~

	~~For a positive integer ''n'' and a prime number ''p'', the poly-Bernoulli numbers satisfy~~

	~~:<math>B_n^{(-p)} \equiv 2^n \pmod p,</math>~~

	~~which can be seen as an analog of [[Fermat~~'s ~~little theorem]]~~. ~~Further, the equation~~

	:<~~math~~>~~B_x^{(-n)} + B_y^{(-n)} = B_z^{(-n)}~~<~~/math~~>

	~~has no solution for integers ''x'', ''y'', ''z'', ''n'' > 2~~; ~~an analog of [[Fermat's last theorem]].~~

	~~==References==~~
	* Hassan Jolany, Explicit formula for generalization of Poly-Bernoulli numbers and polynomials with a,b,c parameters, ''[http://arxiv.org/abs/1109.1387 ]'',2012
	* M. Kaneko, ''Poly-Bernoulli numbers'', Journal de Theorie des Nombres de Bordeaux, 9:221-228, 1997
	* Chad Brewbaker, ''[http://~~www~~.~~public~~.~~iastate~~.~~edu~~/~~~crb002~~/~~thesis.pdf Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index]'', Master's thesis, Iowa State University, 2005~~
	* Chad Brewbaker, A Combinatorial Interpretation of the Poly-Bernoulli Numbers and Two Fermat Analogues, INTEGERS, [http://www.integers-ejcnt.org/~~vol8.html VOL 8], A3, 2008~~

	~~[[Category:Integer sequences]]~~
	~~[[Category:Enumerative combinatorics]~~]

en>Manoguru at 12:56, 12 December 2013

2013-12-12T12:56:33Z

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2012-05-06T18:52:07Z

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en>Monkbot: Task 3: Fix CS1 deprecated coauthor parameter errors

en>Manoguru at 12:56, 12 December 2013

en>Helpful Pixie Bot: ISBNs (Build KC)