小q拉盖尔多项式

小q拉盖尔多项式是一个以基本超几何函数定义的正交多项式

4th order Little q-Laguerre polynomials

极限关系

大q拉盖尔多项式→小q拉盖尔多项式

在大q拉盖尔多项式中,令 ,并令 即得小q拉盖尔多项式

 

仿射Q克拉夫楚克多项式→ 小q拉盖尔多项式:

  令小q拉盖尔多项式    ,然后令q→1 即得拉盖尔多项式

 

验证 9阶小q拉盖尔多项式→拉盖尔多项式

作上述代换,


     




求q→1极限得  

令a=3,得  

另一方面

  = 

二者显然相等 QED

图集

 
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

参考文献

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  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., http://dlmf.nist.gov/18 |contribution-url=缺少标题 (帮助), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (编), 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: 102–108, ISSN 0002-9890, JSTOR 2303599, MR 0003641