Formula:KLS:14.12:25

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lim q → 1 ⁡ p n ⁡ ( 1 2 ⁢ ( 1 - q ) ⁢ x ; q α , - q β ; q ) = L n α ⁡ ( x ) L n α ⁡ ( 0 ) subscript → 𝑞 1 little-q-Jacobi-polynomial-p 𝑛 1 2 1 𝑞 𝑥 superscript 𝑞 𝛼 superscript 𝑞 𝛽 𝑞 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}p_{n}\!\left(% \textstyle\frac{1}{2}(1-q)x;q^{\alpha},-q^{\beta};q\right)=\frac{L^{\alpha}_{n% }\left(x\right)}{L^{\alpha}_{n}\left(0\right)}}}}

Proof

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Symbols List

p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : little q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -Jacobi polynomial : http://drmf.wmflabs.org/wiki/Definition:littleqJacobi
L n ( α ) superscript subscript 𝐿 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}  : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27

Bibliography

Equation in Section 14.12 of KLS.

URL links

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