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 p n ( x ; a | q ) = 2 ϕ 1 ( q − n , 0 ; a q ; q , q x ) = 1 ( a − 1 q − n ; q ) n 2 ϕ 0 ( q − n , x − 1 ; ; q , x / a ) {\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)} 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 x → b q x {\displaystyle x\to bqx} ,并令 b → ∞ {\displaystyle b\to \infty } 即得小q拉盖尔多项式 lim b → ∞ P n ( b q x ; a , b ; q ) = p n ( x ; a | q ) {\displaystyle \lim _{b\to \infty }P_{n}(bqx;a,b;q)=p_{n}(x;a|q)} 仿射Q克拉夫楚克多项式→ 小q拉盖尔多项式： lim a → 1 = K n a f f ( q x − N ; p , N | q ) = p n ( q x ; p , q ) {\displaystyle \lim _{a\to 1}=K_{n}^{aff}(q^{x-N};p,N|q)=p_{n}(q^{x};p,q)} 令小q拉盖尔多项式 a = q a {\displaystyle a=q^{a}} x = ( 1 − q ) ∗ x {\displaystyle x=(1-q)*x} ,然后令q→1 即得拉盖尔多项式 lim q → 1 P a ( 1 − q ) x ; q a | q ) = L n ( a ) ( x ) L n ( a ) ( 0 ) {\displaystyle \lim _{q\to 1}P_{a}(1-q)x;q^{a}|q)={\frac {L_{n}^{(a)}(x)}{L_{n}^{(a)}(0)}}} 验证 9阶小q拉盖尔多项式→拉盖尔多项式 作上述代换， P a ( 1 − q ) x ; q a | q ) = 1 + q x 1 − q α q − x q 8 ( 1 − q α q ) {\displaystyle P_{a}(1-q)x;q^{a}|q)=1+{\frac {qx}{1-{q}^{\alpha }q}}-{\frac {x}{{q}^{8}\left(1-{q}^{\alpha }q\right)}}} + ( 1 − q − 9 ) ( 1 − q − 8 ) q 2 ( 1 − q ) x 2 ( 1 − q 2 ) − 1 ( 1 − q α q ) − 1 ( 1 − q α q 2 ) − 1 {\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}} + ( 1 − q − 9 ) ( 1 − q − 8 ) ( 1 − q − 7 ) q 3 ( 1 − q ) 2 x 3 ( 1 − q 2 ) − 1 {\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}} ( 1 − q 3 ) − 1 ( 1 − q α q ) − 1 ( 1 − q α q 2 ) − 1 ( 1 − q α q 3 ) − 1 + ⋯ {\displaystyle \left(1-{q}^{3}\right)^{-1}\left(1-{q}^{\alpha }q\right)^{-1}\left(1-{q}^{\alpha }{q}^{2}\right)^{-1}\left(1-{q}^{\alpha }{q}^{3}\right)^{-1}+\cdots } 求q→1极限得 令a=3,得 ( 1 − 9 4 x + 9 5 x 2 − 7 10 x 3 + 3 20 x 4 − 3 160 x 5 + 1 720 x 6 − 1 16800 x 7 + 1 739200 x 8 − 1 79833600 x 9 ) {\displaystyle (1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})} 另一方面 L n ( 3 ) ( x ) L n ( 3 ) ( 0 ) {\displaystyle {\frac {L_{n}^{(3)}(x)}{L_{n}^{(3)}(0)}}} = ( 1 − 9 4 x + 9 5 x 2 − 7 10 x 3 + 3 20 x 4 − 3 160 x 5 + 1 720 x 6 − 1 16800 x 7 + 1 739200 x 8 − 1 79833600 x 9 ) {\displaystyle (1-{\frac {9}{4}}x+{\frac {9}{5}}{x}^{2}-{\frac {7}{10}}{x}^{3}+{\frac {3}{20}}{x}^{4}-{\frac {3}{160}}{x}^{5}+{\frac {1}{720}}{x}^{6}-{\frac {1}{16800}}{x}^{7}+{\frac {1}{739200}}{x}^{8}-{\frac {1}{79833600}}{x}^{9})} 二者显然相等 QED 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. (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 [2024-06-17], ISSN 0036-1410, MR 1080161, doi:10.1137/0522019, （原始内容存档于2024-07-21）  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