标题： 关于连续$q$-超球多项式的$q$-差分方程的分解 显示英文标题

标题： On factorization of $q$-difference equation for continuous $q$-ultraspherical polynomials

摘要： 我们证明了连续 $q$-超球多项式 $C_n(x;\beta| q)$的罗杰斯的二阶 $q$-差分方程的常规斯特姆-刘维尔形式可以以某些显式定义的 $q$-差分算子 ${\mathcal D}_x^{\beta, q}$的因子形式表示。 这揭示了一个事实，即连续的$q$-超球多项式$C_n(x;\beta| q)$实际上由$q$-差分方程${\mathcal D}_x^{\beta, q} C_n(x;\beta| q)= (q^{-n/2}+\beta q^{n/2}) C_n(x;\beta| q)$所支配，该方程可以看作是从其原始形式中得到的方程的平方根。 显示英文摘要

摘要： We prove that a customary Sturm-Liouville form of second-order $q$-difference equation for the continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ of Rogers can be written in a factorized form in terms of some explicitly defined $q$-difference operator ${\mathcal D}_x^{\beta, q}$. This reveals the fact that the continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ are actually governed by the $q$-difference equation ${\mathcal D}_x^{\beta, q} C_n(x;\beta| q)= (q^{-n/2}+\beta q^{n/2}) C_n(x;\beta| q)$, which can be regarded as a square root of the equation, obtained from its original form.