en>Frietjes: /* VMP Algorithm */

2014-01-16T00:22:48Z

VMP Algorithm

← Older revision		Revision as of 02:22, 16 January 2014
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	~~The author~~ is ~~known by~~ the ~~title of Figures Wunder~~. ~~He utilized~~ to ~~be unemployed but now he~~ is a ~~pc operator but his marketing by no means comes~~. ~~To gather coins is 1~~ of the ~~issues I love most~~. ~~South Dakota is where me~~ and ~~my spouse reside~~.<br><br>~~Feel free to surf to my website -~~ [~~http:~~//~~www~~.~~hotporn123~~.~~com~~/~~user~~/~~RPocock http~~://~~www~~.~~hotporn123~~.~~com~~/~~user~~/~~RPocock~~]		In [[mathematics]], a [[polynomial sequence]] <math>\{p_n(z) \}</math> has a '''generalized Appell representation''' if the [[generating function]] for the [[polynomial]]s takes on a certain form:

			:<math>K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n
			</math>
			where the generating function or [[kernel (category theory)\|kernel]] <math>K(z,w)</math> is composed of the series

			:<math>A(w)= \sum_{n=0}^\infty a_n w^n \quad</math> with <math>a_0 \ne 0 </math>

			and
			:<math>\Psi(t)= \sum_{n=0}^\infty \Psi_n t^n \quad</math> and all <math>\Psi_n \ne 0 </math>

			and
			:<math>g(w)= \sum_{n=1}^\infty g_n w^n \quad</math> with <math>g_1 \ne 0.</math>

			Given the above, it is not hard to show that <math>p_n(z)</math> is a [[Degree of a polynomial\|polynomial of degree]] <math>n</math>.

			[[Boas–Buck polynomials]] are a slightly more general class of polynomials.

			==Special cases==
			* The choice of <math>g(w)=w</math> gives the class of [[Brenke polynomials]].
			* The choice of <math>\Psi(t)=e^t</math> results in the [[Sheffer sequence]] of polynomials, which include the [[general difference polynomials]], such as the [[Newton polynomials]].
			* The combined choice of <math>g(w)=w</math> and <math>\Psi(t)=e^t</math> gives the [[Appell sequence]] of polynomials.

			==Explicit representation==
			The generalized Appell polynomials have the explicit representation

			:<math>p_n(z) = \sum_{k=0}^n z^k \Psi_k h_k.</math>

			The constant is

			:<math>h_k=\sum_{P} a_{j_0} g_{j_1} g_{j_2} \cdots g_{j_k} </math>

			where this sum extends over all [[partition (number theory)\|partitions]] of <math>n</math> into <math>k+1</math> parts; that is, the sum extends over all <math>\{j\}</math> such that

			:<math>j_0+j_1+ \cdots +j_k = n.\,</math>

			For the Appell polynomials, this becomes the formula

			:<math>p_n(z) = \sum_{k=0}^n \frac {a_{n-k} z^k} {k!}.</math>

			==Recursion relation==
			Equivalently, a necessary and sufficient condition that the kernel <math>K(z,w)</math> can be written as <math>A(w)\Psi(zg(w))</math> with <math>g_1=1</math> is that

			:<math>\frac{\partial K(z,w)}{\partial w} =
			c(w) K(z,w)+\frac{zb(w)}{w} \frac{\partial K(z,w)}{\partial z}</math>

			where <math>b(w)</math> and <math>c(w)</math> have the power series

			:<math>b(w) = \frac{w}{g(w)} \frac {d}{dw} g(w)
			= 1 + \sum_{n=1}^\infty b_n w^n</math>

			and

			:<math>c(w) = \frac{1}{A(w)} \frac {d}{dw} A(w)
			= \sum_{n=0}^\infty c_n w^n.</math>

			Substituting

			:<math>K(z,w)= \sum_{n=0}^\infty p_n(z) w^n</math>

			immediately gives the [[recursion relation]]

			:<math> z^{n+1} \frac {d}{dz} \left[ \frac{p_n(z)}{z^n} \right]=
			-\sum_{k=0}^{n-1} c_{n-k-1} p_k(z)
			-z \sum_{k=1}^{n-1} b_{n-k} \frac{d}{dz} p_k(z).
			</math>

			For the special case of the Brenke polynomials, one has <math>g(w)=w</math> and thus all of the <math>b_n=0</math>, simplifying the recursion relation significantly.

			==See also==
			{{portal\|Mathematics}}
			* [[q-difference polynomial]]s

			==References==
			* Ralph P. Boas, Jr. and R. Creighton Buck, ''Polynomial Expansions of Analytic Functions (Second Printing Corrected)'', (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.
			* William C. Brenke, ''On generating functions of polynomial systems'', (1945) American Mathematical Monthly, '''52''' pp. 297–301.
			* W. N. Huff, ''The type of the polynomials generated by f(xt) φ(t)'' (1947) Duke Mathematical Journal, '''14''' pp. 1091–1104.

			[[Category:Polynomials]]

94.245.127.14: /* References */

2010-09-13T15:16:06Z

References

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The author is known by the title of Figures Wunder. He utilized to be unemployed but now he is a pc operator but his marketing by no means comes. To gather coins is 1 of the issues I love most. South Dakota is where me and my spouse reside.<br><br>Feel free to surf to my website - [http://www.hotporn123.com/user/RPocock http://www.hotporn123.com/user/RPocock]

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94.245.127.14: /* References */