Gold 73

Continuous Hahn polynomials

Gold ID: 73
Link: https://sigir21.wmflabs.org/wiki/Continuous_Hahn_polynomials#math.125.15
Formula: $P_{n}^{(α, β)} = \lim_{t \to \infty} t^{- n} p_{n} (\frac{1}{2} x t; \frac{1}{2} (α + 1 - i t), \frac{1}{2} (β + 1 + i t), \frac{1}{2} (α + 1 + i t), \frac{1}{2} (β + 1 - i t))$
TeX Source: P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations
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Translation: P_n^{(\alpha,\beta)} = \lim_{t\to\infty} t^{-n} \contHahnpolyp{n}@{\tfrac12 xt}{\tfrac12(\alpha + 1 - \iunit t)}{\tfrac12(\beta + 1 + \iunit t)}{\tfrac12(\alpha + 1 + \iunit t)}{\tfrac12(\beta + 1 - \iunit t)}
Expected (Gold Entry): \JacobipolyP{\alpha}{\beta}{n}@{x} = \lim_{t\to\infty} t^{-n} \contHahnpolyp{n}@{\tfrac12 xt}{\tfrac12(\alpha + 1 - \iunit t)}{\tfrac12(\beta + 1 + \iunit t)}{\tfrac12(\alpha + 1 + \iunit t)}{\tfrac12(\beta + 1 - \iunit t)}

Translation: (Subscript[P, n])^(\[Alpha], \[Beta]) == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t)}, 1], t -> Infinity, GenerateConditions->None]
Expected (Gold Entry): JacobiP[n, \[Alpha], \[Beta], x] == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Alpha]+ 1 + I*t), Divide[1,2]*(\[Alpha]+ 1 - I*t) + Divide[1,2]*(\[Beta]+ 1 - I*t)}, 1], t -> Infinity]