查看“︁Q查理耶多项式”︁的源代码


'''q查理耶多项式'''是一个以[[基本超几何函数]]定义的[[正交多项式]]

:<math>\displaystyle c_n(x;a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)</math>

==极限关系==
令Q查理耶多项式 a→a*(1-q),并令q→1，即得[[查理耶多项式]]

<math>lim_{q \to 1}C_{n}(q^{-n};a(1-q);q)=C_{n}(x;a)</math>

;验证Q查理耶多项式→查理耶多项式

Q查理耶多项式之第4项(k=4):


<math>{\frac { \left( 1-{q}^{-n} \right)  \left( 1-{q}^{-n}q \right) 
 \left( 1-{q}^{-n}{q}^{2} \right)  \left( 1-{q}^{-n}{q}^{3} \right) 
 \left( 1-{q}^{-x} \right)  \left( 1-{q}^{-x}q \right)  \left( 1-{q}^{
-x}{q}^{2} \right)  \left( 1-{q}^{-x}{q}^{3} \right)  \left( {q}^{n}
 \right) ^{4}{q}^{4}}{{a}^{4} \left( 1-q \right) ^{5} \left( 1-{q}^{2}
 \right)  \left( 1-{q}^{3} \right)  \left( 1-{q}^{4} \right) }}
    </math>
展开之：
<math>\frac{1}{24}\,{\frac {36\,nx-66\,n{x}^{2}+36\,n{x}^{3}-6\,n{x}^{4}-66\,{n}^{2}
x+121\,{n}^{2}{x}^{2}-66\,{n}^{2}{x}^{3}+11\,{n}^{2}{x}^{4}+36\,{n}^{3
}x-66\,{n}^{3}{x}^{2}+36\,{n}^{3}{x}^{3}-6\,{n}^{3}{x}^{4}-6\,{n}^{4}x
+11\,{n}^{4}{x}^{2}-6\,{n}^{4}{x}^{3}+{n}^{4}{x}^{4}}{{a}^{4}}}
</math>

另一方面
[[查理耶多项式]]的k=4项为

<math>\frac{1}{24}\,{\frac {{\it pochhammer} \left( -n,4 \right) {\it pochhammer}
 \left( -x,4 \right) }{{a}^{4}}}</math>

展开之

<math>\frac{1}{24}\,{\frac {nx \left( 36-66\,x+36\,{x}^{2}-6\,{x}^{3}-66\,n+121\,nx-
66\,n{x}^{2}+11\,n{x}^{3}+36\,{n}^{2}-66\,{n}^{2}x+36\,{n}^{2}{x}^{2}-
6\,{n}^{2}{x}^{3}-6\,{n}^{3}+11\,{n}^{3}x-6\,{n}^{3}{x}^{2}+{n}^{3}{x}
^{3} \right) }{{a}^{4}}}
</math>

二者显然相等  QED

==图集==
{|
|[[File:Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT]]
|[[File:Q-CHARLIER IM COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER IM COMPLEX 3D MAPLE PLOT]]
|[[File:Q-CHARLIER RE COMPLEX 3D MAPLE PLOT.gif|thumb|Q-CHARLIER RE COMPLEX 3D MAPLE PLOT]]
|}
{|
|[[File:Q-CHARLIER ABS DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER ABS DENSITY MAPLE PLOT]]
|[[File:Q-CHARLIER IM DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER IM DENSITY MAPLE PLOT]]
|[[File:Q-CHARLIER RE DENSITY MAPLE PLOT.gif|thumb|Q-CHARLIER RE DENSITY MAPLE PLOT]]
|}

==参考文献==
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
*{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}

[[Category:正交多项式]]
[[Category:基本超几何函数]]
{{q超几何函数}}
[[Category:Q超几何多项式]]