Formula:KLS:09.04:07

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x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + i ⁢ ( A n + C n + a ) ⁢ p ^ n ⁡ ( x ) - A n - 1 ⁢ C n ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 continuous-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 imaginary-unit subscript 𝐴 𝑛 subscript 𝐶 𝑛 𝑎 continuous-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 continuous-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)=% {\widehat{p}}_{n+1}\!\left(x\right)+\mathrm{i}(A_{n}+C_{n}+a){\widehat{p}}_{n}% \!\left(x\right)-A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+b+c-1)(n+b+d-1)}{(2% n+a+b+c+d-2)(2n+a+b+c+d-1)}}}} &
A n = - ( n + a + b + c + d - 1 ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle A_{n}=-\frac{(n+a+b+c+d-1)(n+a+c)(n% +a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}}


Proof

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Symbols List

& : logical and
p ^ n subscript ^ 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{p}}_{n}}}}  : monic continuous Hahn polynomial : http://drmf.wmflabs.org/wiki/Definition:monicctsHahn
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.4 of KLS.

URL links

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