|
|
| Line 1: |
Line 1: |
| {{lowercase}}
| | Andrew Simcox is the name his parents gave him and he totally enjoys this name. As a lady what she really likes is style and she's been performing it for quite a whilst. Since he was eighteen he's been working as [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ online psychic readings] an information officer but he ideas on changing it. Some time ago she chose to reside in Alaska [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ tarot readings] and her mothers and fathers reside nearby.<br><br>Here is my web-site - love psychics ([http://www.onbizin.co.kr/xe/?document_srl=320614 http://www.onbizin.co.kr/xe/?document_srl=320614]) |
| In [[combinatorics|combinatorial]] [[mathematics]], the '''q-difference polynomials''' or '''q-harmonic polynomials''' are a [[polynomial sequence]] defined in terms of the [[q-derivative]]. They are a type of [[Brenke polynomial]], and generalize the [[Appell polynomial]]s. See also [[Sheffer sequence]].
| |
| | |
| ==Definition==
| |
| The q-difference polynomials satisfy the relation
| |
| | |
| :<math>\left(\frac {d}{dz}\right)_q p_n(z) =
| |
| \frac{p_n(qz)-p_n(z)} {qz-z} = p_{n-1}(z)</math>
| |
| | |
| where the derivative symbol on the left is the q-derivative. In the limit of <math>q\to 1</math>, this becomes the definition of the Appell polynomials:
| |
| | |
| :<math>\frac{d}{dz}p_n(z) = p_{n-1}(z).</math>
| |
| | |
| ==Generating function==
| |
| The [[generating function]] for these polynomials is of the type of generating function for Brenke polynomials, namely
| |
| | |
| :<math>A(w)e_q(zw) = \sum_{n=0}^\infty p_n(z) w^n</math>
| |
| | |
| where <math>e_q(t)</math> is the [[q-exponential]]:
| |
| :<math>e_q(t)=\sum_{n=0}^\infty \frac{t^n}{[n]_q!}=
| |
| \sum_{n=0}^\infty \frac{t^n (1-q)^n}{(q;q)_n}.</math>
| |
| | |
| Here, <math>[n]_q!</math> is the [[q-factorial]] and
| |
| | |
| :<math>(q;q)_n=(1-q^n)(1-q^{n-1})\cdots (1-q)</math>
| |
| | |
| is the [[q-Pochhammer symbol]]. The function <math>A(w)</math> is arbitrary but assumed to have an expansion
| |
| | |
| :<math>A(w)=\sum_{n=0}^\infty a_n w^n \mbox{ with } a_0 \ne 0. </math> | |
| | |
| Any such <math>A(w)</math> gives a sequence of q-difference polynomials.
| |
| | |
| ==References==
| |
| * A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", ''Riv. Mat. Univ. Parma'', '''5''' (1954) 325-337.
| |
| * 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. ''(Provides a very brief discussion of convergence.)''
| |
| | |
| [[Category:Q-analogs]]
| |
| [[Category:Polynomials]]
| |
Andrew Simcox is the name his parents gave him and he totally enjoys this name. As a lady what she really likes is style and she's been performing it for quite a whilst. Since he was eighteen he's been working as online psychic readings an information officer but he ideas on changing it. Some time ago she chose to reside in Alaska tarot readings and her mothers and fathers reside nearby.
Here is my web-site - love psychics (http://www.onbizin.co.kr/xe/?document_srl=320614)