Formula:KLS:09.01:27

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lim t → ∞ ⁡ W n ⁡ ( 1 2 ⁢ ( 1 - x ) ⁢ t 2 ; 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( β + 1 ) + i ⁢ t , 1 2 ( β + 1 ) - i t ) t 2 ⁢ n ⁢ n ! = P n ( α , β ) ⁡ ( x ) subscript → 𝑡 fragments Wilson-polynomial-W 𝑛 1 2 1 𝑥 superscript 𝑡 2 1 2 𝛼 1 1 2 𝛼 1 1 2 𝛽 1 imaginary-unit 𝑡 fragments 1 2 fragments ( β 1 imaginary-unit t ) superscript 𝑡 2 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{W_{n}% \!\left(\frac{1}{2}(1-x)t^{2};\frac{1}{2}(\alpha+1),\frac{1}{2}(\alpha+1),% \frac{1}{2}(\beta+1)+\mathrm{i}t,\frac{1}{2}(\beta+1\right)-\mathrm{i}t)}{t^{2% n}n!}{}=P^{(\alpha,\beta)}_{n}\left(x\right)}}}

Proof

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Symbols List

W n subscript 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}}}}  : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
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 9.1 of KLS.

URL links

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