Formula:KLS:14.10:86

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lim q ↑ 1 ⁡ C 2 ⁢ m + 1 ⁡ ( x ; - q λ | - q ) = 2 ⁢ ( λ + 1 ) m ( 1 2 ⁢ λ + 1 ) m ⁢ x ⁢ P m ( 1 2 ⁢ λ , 1 2 ⁢ λ ) ⁡ ( 2 ⁢ x 2 - 1 ) subscript ↑ 𝑞 1 continuous-q-ultraspherical-Rogers-polynomial 2 𝑚 1 𝑥 superscript 𝑞 𝜆 𝑞 2 Pochhammer-symbol 𝜆 1 𝑚 Pochhammer-symbol 1 2 𝜆 1 𝑚 𝑥 Jacobi-polynomial-P 1 2 𝜆 1 2 𝜆 𝑚 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle\lim_{q\uparrow 1}C_{2m+1}\!\left(x;% -q^{\lambda}\,|\,-q\right)=2\frac{{\left(\lambda+1\right)_{m}}}{{\left(\frac{1% }{2}\lambda+1\right)_{m}}}xP^{(\frac{1}{2}\lambda,\frac{1}{2}\lambda)}_{m}% \left(2x^{2}-1\right)}}}

Proof

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Symbols List

C n subscript 𝐶 𝑛 {\displaystyle{\displaystyle{\displaystyle C_{n}}}}  : continuous q 𝑞 {\displaystyle{\displaystyle{\displaystyle q}}} -ultraspherical/Rogers polynomial : http://dlmf.nist.gov/18.28#E13
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
P n ( α , β ) subscript superscript 𝑃 𝛼 𝛽 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}  : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3

Bibliography

Equation in Section 14.10 of KLS.

URL links

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