双哈恩多项式

双重哈恩多项式(Dual Hahn polynomials)是一个正交多项式，定义如下^[1]

R_{n} (λ (x); γ, δ, N) =_{3} F_{2} (- n, - x, x + γ + δ + 1; γ + 1, - N; 1) .

其中0≤n≤N

双重哈恩多项式的前几个：

正交性

双重哈恩多项式满足下列正交关系：^[2]

$\sum_{x = 0}^{N} \frac{(2 x + γ + δ + 1) (γ + 1)_{x} (- N)_{x} N!}{(- 1)^{x} (x + γ + δ + 1)_{N + 1} (δ + 1)_{x} x!}$ * $R_{m} (λ (x); γ, δ, N) R_{n} (λ (x); γ, δ, N) = \frac{δ_{m n}}{(\binom{γ + n}{n}) (\binom{δ + N - n}{N - n})}$

极限关系

拉卡多项式→双重哈恩多项式

$\lim_{β \to \infty} R_{n} (λ (x); - N - 1, β, γ, δ) = R_{n} (λ (\lim_{β \to \infty} R_{n} (λ (x); - x); γ, δ, N)$

双重哈恩多项式→梅西纳多项式

$\lim_{N \to \infty} R_{n} (λ (x); β - 1, N (1 - c) c^{- 1}, N) = M_{n} (x; β, c)$

参考文献

↑ 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

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[1] Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York

[K-2] KoeKoef p209

[1]

[2]

双哈恩多项式

正交性

极限关系

参考文献

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