q查理耶多項式是一個以基本超幾何函數定義的正交多項式
![{\displaystyle \displaystyle c_{n}(x;a;q)={}_{2}\phi _{1}(q^{-n},q^{-x};0;q,-q^{n+1}/a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fccb65e05807b273dac084eb5880d7a7a0c53c8b)
極限關係
令Q查理耶多項式 a→a*(1-q),並令q→1,即得查理耶多項式
- 驗證Q查理耶多項式→查理耶多項式
Q查理耶多項式之第4項(k=4):
展開之:
另一方面
查理耶多項式的k=4項為
展開之
二者顯然相等 QED
圖集
Q-CHARLIER ABS COMPLEX 3D MAPLE PLOT
|
Q-CHARLIER IM COMPLEX 3D MAPLE PLOT
|
Q-CHARLIER RE COMPLEX 3D MAPLE PLOT
|
Q-CHARLIER ABS DENSITY MAPLE PLOT
|
Q-CHARLIER IM DENSITY MAPLE PLOT
|
Q-CHARLIER RE DENSITY MAPLE PLOT
|
參考文獻
- Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574
- Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F., Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, 2010, ISBN 978-3-642-05013-8, MR 2656096, doi:10.1007/978-3-642-05014-5
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248