雙重哈恩多項式 (Dual Hahn polynomials)是一個正交多項式，定義如下[1]
Dual Hahn polynomials plot Dual Hahn polynomials plot
R
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{\displaystyle R_{n}(\lambda (x);\gamma ,\delta ,N)={}_{3}F_{2}(-n,-x,x+\gamma +\delta +1;\gamma +1,-N;1).}
n ≤N 雙重哈恩多項式的前幾個：
雙重哈恩多項式滿足下列正交關係：[2]
∑
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{\displaystyle \sum _{x=0}^{N}{\frac {(2x+\gamma +\delta +1)(\gamma +1)_{x}(-N)_{x}N!}{(-1)^{x}(x+\gamma +\delta +1)_{N+1}(\delta +1)_{x}x!}}}
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{\displaystyle R_{m}(\lambda (x);\gamma ,\delta ,N)R_{n}(\lambda (x);\gamma ,\delta ,N)={\frac {\delta _{mn}}{{\gamma +n \choose n}{\delta +N-n \choose N-n}}}}
極限關係
編輯
拉卡多項式 →雙重哈恩多項式
lim
β
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∞
R
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{\displaystyle \lim _{\beta \to \infty }R_{n}(\lambda (x);-N-1,\beta ,\gamma ,\delta )=R_{n}(\lambda (\lim _{\beta \to \infty }R_{n}(\lambda (x);-x);\gamma ,\delta ,N)}
雙重哈恩多項式 →梅西納多項式
lim
N
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∞
R
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M
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{\displaystyle \lim _{N\to \infty }R_{n}(\lambda (x);\beta -1,N(1-c)c^{-1},N)=M_{n}(x;\beta ,c)}
參考文獻
編輯
^ Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York
^ KoeKoef p209
