连续q雅可比多项式

连续q雅可比多项式定义如下

${\displaystyle P_{n}^{(\alpha ;\beta )}(x|q)=}$${\displaystyle {\frac {(q^{\alpha +1};q)_{n}}{(q;q;)_{n}}}}$

• ${\displaystyle _{4}\Phi _{3}}$${\displaystyle (q^{-n},q^{n+\alpha +\beta +1};q^{\alpha /2+1/4}e^{i\theta };q^{\alpha /2+1/4}e^{-i\theta })}$

连续q雅可比多项式→连续q拉盖尔多项式

${\displaystyle \lim _{\beta \to \infty }P_{n}^{(\alpha ;\beta )}(x|q)=P_{n}^{(\alpha )}(x|q)}$

连续q雅可比多项式→雅科比多项式

${\displaystyle \lim _{q\to 1}P_{n}^{(\alpha ;\beta )}(x|q)=P_{n}^{(\alpha ;\beta )}(x|q)}$

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