斯蒂尔吉斯-维格特多项式

Stieltjes-Wigert polynomails

斯蒂尔吉斯-维格特多项式是一个以基本超几何函数定义的正交多项式

$\displaystyle S_{n}(x;q)={\frac {1}{(q;q)_{n})}}{}_{1}\phi _{1}(q^{-n},0;q,-q^{n+1}x)$

极限关系

Q拉盖尔多项式→斯蒂尔吉斯-维格特多项式

$\lim _{\alpha \to \infty }(xq^{-\alpha };q)=S_{n}(x;q)$

Q贝塞尔多项式→斯蒂尔吉斯-维格特多项式

Q查理耶多项式→斯蒂尔吉斯-维格特多项式

斯蒂尔吉斯-维格特多项式→埃尔米特多项式

参考文献

• 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., Ch. 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 
• Szegő, Gábor, Orthogonal Polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, 1975, ISBN 978-0-8218-1023-1, MR 0372517 
• Stieltjes, T. -J., Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 1894, VIII: 1–122 [2015-02-07], JFM 25.0326.01, MR 1344720, （原始内容存档于2015-02-07） （法语） 
• Wigert, S., Sur les polynomes orthogonaux et l'approximation des fonctions continues, Arkiv för matematik, astronomi och fysik, 1923, 17: 1–1, JFM 49.0296.01 （法语）