4th order Little q-Laguerre polynomials
小q拉蓋爾多項式是一個以基本超幾何函數定義的正交多項式
![{\displaystyle \displaystyle p_{n}(x;a|q)={}_{2}\phi _{1}(q^{-n},0;aq;q,qx)={\frac {1}{(a^{-1}q^{-n};q)_{n}}}{}_{2}\phi _{0}(q^{-n},x^{-1};;q,x/a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf44f9d8c59253bf07c89d757ccb77edf384e9aa)
極限關係[編輯]
- 大q拉蓋爾多項式→小q拉蓋爾多項式
在大q拉蓋爾多項式中,令
,並令
即得小q拉蓋爾多項式
仿射Q克拉夫楚克多項式→ 小q拉蓋爾多項式:
令小q拉蓋爾多項式
,然後令q→1
即得拉蓋爾多項式
- 驗證 9階小q拉蓋爾多項式→拉蓋爾多項式
作上述代換,
![{\displaystyle +\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right){q}^{2}\left(1-q\right){x}^{2}\left(1-{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7324e47b114a7fb3723811048bbd9e46bf205ba9)
![{\displaystyle +\left(1-{q}^{-9}\right)\left(1-{q}^{-8}\right)\left(1-{q}^{-7}\right){q}^{3}\left(1-q\right)^{2}{x}^{3}\left(1-{q}^{2}\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26b08b777abcc1c652cdf95851bce3ea46739cfb)
求q→1極限得
令a=3,得
另一方面
=
二者顯然相等 QED
LITTLE Q-LAGUERRE POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT
|
LITTLE Q-LAGUERRE POLYNOMIALS IM COMPLEX 3D MAPLE PLOT
|
LITTLE Q-LAGUERRE POLYNOMIALS RE COMPLEX 3D MAPLE PLOT
|
LITTLE Q-LAGUERRE POLYNOMIALS ABS DENSITY MAPLE PLOT
|
LITTLE Q-LAGUERRE POLYNOMIALS IM DENSITY MAPLE PLOT
|
LITTLE Q-LAGUERRE POLYNOMIALS RE DENSITY MAPLE PLOT
|
參考文獻[編輯]
- Chihara, Theodore Seio, An introduction to orthogonal polynomials, Mathematics and its Applications 13, New York: Gordon and Breach Science Publishers, 1978, ISBN 978-0-677-04150-6, MR 0481884, Reprinted by Dover 2011
- 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
- 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., Chapter 18: Orthogonal Polynomials, 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
- Van Assche, Walter; Koornwinder, Tom H., Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials, SIAM Journal on Mathematical Analysis, 1991, 22 (1): 302–311, ISSN 0036-1410, MR 1080161, doi:10.1137/0522019
- Wall, H. S., A continued fraction related to some partition formulas of Euler, The American Mathematical Monthly, 1941, 48 (2): 102–108, ISSN 0002-9890, JSTOR 2303599, MR 0003641, doi:10.1080/00029890.1941.11991074