Formula:KLS:09.08:18

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

Proof

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Symbols List

p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
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 9.8 of KLS.

URL links

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