Hahn

From DRMF
Jump to navigation Jump to search

Hahn

Hypergeometric representation

Q n ⁡ ( x ; α , β , N ) = \HyperpFq ⁢ 32 ⁢ @ ⁢ @ - n , n + α + β + 1 , - x ⁢ α + 1 , - N ⁢ 1 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 \HyperpFq 32 @ @ 𝑛 𝑛 𝛼 𝛽 1 𝑥 𝛼 1 𝑁 1 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right% )=\HyperpFq{3}{2}@@{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}{1}}}} {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\HyperpFq{3}{2}@@{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}{1} }

Constraint(s): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


Orthogonality relation(s)

∑ x = 0 N \binomial ⁢ α + x ⁢ x ⁢ \binomial ⁢ β + N - x ⁢ N - x ⁢ Q m ⁡ ( x ; α , β , N ) ⁢ Q n ⁡ ( x ; α , β , N ) = ( - 1 ) n ⁢ ( n + α + β + 1 ) N + 1 ⁢ ( β + 1 ) n ⁢ n ! ( 2 ⁢ n + α + β + 1 ) ⁢ ( α + 1 ) n ⁢ ( - N ) n ⁢ N ! ⁢ δ m , n superscript subscript 𝑥 0 𝑁 \binomial 𝛼 𝑥 𝑥 \binomial 𝛽 𝑁 𝑥 𝑁 𝑥 Hahn-polynomial-Q 𝑚 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑁 1 Pochhammer-symbol 𝛽 1 𝑛 𝑛 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝑁 𝑛 𝑁 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\binomial{\alpha+x}{x}% \binomial{\beta+N-x}{N-x}Q_{m}\!\left(x;\alpha,\beta,N\right)Q_{n}\!\left(x;% \alpha,\beta,N\right){}=\frac{(-1)^{n}{\left(n+\alpha+\beta+1\right)_{N+1}}{% \left(\beta+1\right)_{n}}n!}{(2n+\alpha+\beta+1){\left(\alpha+1\right)_{n}}{% \left(-N\right)_{n}}N!}\,\delta_{m,n}}}} {\displaystyle \sum_{x=0}^N\binomial{\alpha +x}{x}\binomial{\beta+N-x}{N-x}\Hahn{m}@{x}{\alpha}{\beta}{N}\Hahn{n}@{x}{\alpha}{\beta}{N} {}=\frac{(-1)^n\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!}{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}\,\Kronecker{m}{n} }

Recurrence relation

- x ⁢ Q n ⁡ ( x ) = A n ⁢ Q n + 1 ⁡ ( x ) - ( A n + C n ) ⁢ Q n ⁡ ( x ) + C n ⁢ Q n - 1 ⁡ ( x ) 𝑥 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐶 𝑛 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle-xQ_{n}\!\left(x\right)=A_{n}Q_{n+1}% \!\left(x\right)-\left(A_{n}+C_{n}\right)Q_{n}\!\left(x\right)+C_{n}Q_{n-1}\!% \left(x\right)}}} {\displaystyle -x\Hahn{n}@@{x}{\alpha}{\beta}{N}=A_n\Hahn{n+1}@@{x}{\alpha}{\beta}{N}-\left(A_n+C_n\right)\Hahn{n}@@{x}{\alpha}{\beta}{N}+C_n\Hahn{n-1}@@{x}{\alpha}{\beta}{N} }

Substitution(s): C n = n ⁢ ( n + α + β + N + 1 ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝛼 𝛽 𝑁 1 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta+N+1)(n% +\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}}} &
A n = ( n + α + β + 1 ) ⁢ ( n + α + 1 ) ⁢ ( N - n ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 𝑛 𝛼 𝛽 1 𝑛 𝛼 1 𝑁 𝑛 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+\beta+1)(n+% \alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}}


Q n ⁡ ( x ) := Q n ⁡ ( x ; α , β , N ) assign Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x\right):=Q_{n}\!\left% (x;\alpha,\beta,N\right)}}} {\displaystyle \Hahn{n}@@{x}{\alpha}{\beta}{N}:=\Hahn{n}@{x}{\alpha}{\beta}{N} }

Monic recurrence relation

x ⁢ Q ^ n ⁡ ( x ) = Q ^ n + 1 ⁡ ( x ) + ( A n + C n ) ⁢ Q ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ Q ^ n - 1 ⁡ ( x ) 𝑥 Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Hahn-polynomial-monic-p 𝑛 1 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)=% {\widehat{Q}}_{n+1}\!\left(x\right)+\left(A_{n}+C_{n}\right){\widehat{Q}}_{n}% \!\left(x\right)+A_{n-1}C_{n}{\widehat{Q}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicHahn{n}@@{x}{\alpha}{\beta}{N}=\monicHahn{n+1}@@{x}{\alpha}{\beta}{N}+\left(A_n+C_n\right)\monicHahn{n}@@{x}{\alpha}{\beta}{N}+A_{n-1}C_n\monicHahn{n-1}@@{x}{\alpha}{\beta}{N} }

Substitution(s): C n = n ⁢ ( n + α + β + N + 1 ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝛼 𝛽 𝑁 1 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+\alpha+\beta+N+1)(n% +\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}}} &
A n = ( n + α + β + 1 ) ⁢ ( n + α + 1 ) ⁢ ( N - n ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) subscript 𝐴 𝑛 𝑛 𝛼 𝛽 1 𝑛 𝛼 1 𝑁 𝑛 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+\alpha+\beta+1)(n+% \alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}}


Q n ⁡ ( x ; α , β , N ) = ( n + α + β + 1 ) n ( α + 1 ) n ⁢ ( - N ) n ⁢ Q ^ n ⁡ ( x ) Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝑁 𝑛 Hahn-polynomial-monic-p 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right% )=\frac{{\left(n+\alpha+\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left% (-N\right)_{n}}}{\widehat{Q}}_{n}\!\left(x\right)}}} {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}}\monicHahn{n}@@{x}{\alpha}{\beta}{N} }

Difference equation

n ⁢ ( n + α + β + 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + 1 ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - 1 ) 𝑛 𝑛 𝛼 𝛽 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 1 delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 1 {\displaystyle{\displaystyle{\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-% \left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}} {\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1) }

Substitution(s): D ⁢ ( x ) = x ⁢ ( x - β - N - 1 ) 𝐷 𝑥 𝑥 𝑥 𝛽 𝑁 1 {\displaystyle{\displaystyle{\displaystyle D(x)=x(x-\beta-N-1)}}} &

B ⁢ ( x ) = ( x + α + 1 ) ⁢ ( x - N ) 𝐵 𝑥 𝑥 𝛼 1 𝑥 𝑁 {\displaystyle{\displaystyle{\displaystyle B(x)=(x+\alpha+1)(x-N)}}} &

y ⁢ ( x ) = Q n ⁡ ( x ; α , β , N ) 𝑦 𝑥 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle y(x)=Q_{n}\!\left(x;\alpha,\beta,N% \right)}}}


Forward shift operator

Q n ⁡ ( x + 1 ; α , β , N ) - Q n ⁡ ( x ; α , β , N ) = - n ⁢ ( n + α + β + 1 ) ( α + 1 ) ⁢ N ⁢ Q n - 1 ⁡ ( x ; α + 1 , β + 1 , N - 1 ) Hahn-polynomial-Q 𝑛 𝑥 1 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 𝑁 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 1 𝛽 1 𝑁 1 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x+1;\alpha,\beta,N% \right)-Q_{n}\!\left(x;\alpha,\beta,N\right){}=-\frac{n(n+\alpha+\beta+1)}{(% \alpha+1)N}Q_{n-1}\!\left(x;\alpha+1,\beta+1,N-1\right)}}} {\displaystyle \Hahn{n}@{x+1}{\alpha}{\beta}{N}-\Hahn{n}@{x}{\alpha}{\beta}{N} {}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1} }
Δ ⁢ Q n ⁡ ( x ; α , β , N ) = - n ⁢ ( n + α + β + 1 ) ( α + 1 ) ⁢ N ⁢ Q n - 1 ⁡ ( x ; α + 1 , β + 1 , N - 1 ) Δ Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 𝑁 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 1 𝛽 1 𝑁 1 {\displaystyle{\displaystyle{\displaystyle\Delta Q_{n}\!\left(x;\alpha,\beta,N% \right)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}\!\left(x;\alpha+1,% \beta+1,N-1\right)}}} {\displaystyle \Delta \Hahn{n}@{x}{\alpha}{\beta}{N}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1} }

Backward shift operator

( x + α ) ⁢ ( N + 1 - x ) ⁢ Q n ⁡ ( x ; α , β , N ) - x ⁢ ( β + N + 1 - x ) ⁢ Q n ⁡ ( x - 1 ; α , β , N ) = α ⁢ ( N + 1 ) ⁢ Q n + 1 ⁡ ( x ; α - 1 , β - 1 , N + 1 ) 𝑥 𝛼 𝑁 1 𝑥 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 𝑥 𝛽 𝑁 1 𝑥 Hahn-polynomial-Q 𝑛 𝑥 1 𝛼 𝛽 𝑁 𝛼 𝑁 1 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 1 𝛽 1 𝑁 1 {\displaystyle{\displaystyle{\displaystyle(x+\alpha)(N+1-x)Q_{n}\!\left(x;% \alpha,\beta,N\right)-x(\beta+N+1-x)Q_{n}\!\left(x-1;\alpha,\beta,N\right){}=% \alpha(N+1)Q_{n+1}\!\left(x;\alpha-1,\beta-1,N+1\right)}}} {\displaystyle (x+\alpha)(N+1-x)\Hahn{n}@{x}{\alpha}{\beta}{N}-x(\beta+N+1-x)\Hahn{n}@{x-1}{\alpha}{\beta}{N} {}=\alpha(N+1)\Hahn{n+1}@{x}{\alpha-1}{\beta-1}{N+1} }
∇ ⁡ [ ω ⁢ ( x ; α , β , N ) ⁢ Q n ⁡ ( x ; α , β , N ) ] = N + 1 β ⁢ ω ⁢ ( x ; α - 1 , β - 1 , N + 1 ) ⁢ Q n + 1 ⁡ ( x ; α - 1 , β - 1 , N + 1 ) ∇ 𝜔 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 𝑁 1 𝛽 𝜔 𝑥 𝛼 1 𝛽 1 𝑁 1 Hahn-polynomial-Q 𝑛 1 𝑥 𝛼 1 𝛽 1 𝑁 1 {\displaystyle{\displaystyle{\displaystyle\nabla\left[\omega(x;\alpha,\beta,N)% Q_{n}\!\left(x;\alpha,\beta,N\right)\right]{}=\frac{N+1}{\beta}\omega(x;\alpha% -1,\beta-1,N+1)Q_{n+1}\!\left(x;\alpha-1,\beta-1,N+1\right)}}} {\displaystyle \nabla\left[\omega(x;\alpha,\beta,N)\Hahn{n}@{x}{\alpha}{\beta}{N}\right] {}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)\Hahn{n+1}@{x}{\alpha-1}{\beta-1}{N+1} }

Substitution(s): ω ⁢ ( x ; α , β , N ) = \binomial ⁢ α + x ⁢ x ⁢ \binomial ⁢ β + N - x ⁢ N - x 𝜔 𝑥 𝛼 𝛽 𝑁 \binomial 𝛼 𝑥 𝑥 \binomial 𝛽 𝑁 𝑥 𝑁 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)=\binomial{% \alpha+x}{x}\binomial{\beta+N-x}{N-x}}}}


Rodrigues-type formula

ω ⁢ ( x ; α , β , N ) ⁢ Q n ⁡ ( x ; α , β , N ) = ( - 1 ) n ⁢ ( β + 1 ) n ( - N ) n ⁢ ∇ n ⁡ [ ω ⁢ ( x ; α + n , β + n , N - n ) ] 𝜔 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝑁 𝑛 superscript ∇ 𝑛 𝜔 𝑥 𝛼 𝑛 𝛽 𝑛 𝑁 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)Q_{n}\!\left% (x;\alpha,\beta,N\right){}=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(-N% \right)_{n}}}\nabla^{n}\left[\omega(x;\alpha+n,\beta+n,N-n)\right]}}} {\displaystyle \omega(x;\alpha,\beta,N)\Hahn{n}@{x}{\alpha}{\beta}{N} {}=\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{-N}{n}}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right] }

Substitution(s): ω ⁢ ( x ; α , β , N ) = \binomial ⁢ α + x ⁢ x ⁢ \binomial ⁢ β + N - x ⁢ N - x 𝜔 𝑥 𝛼 𝛽 𝑁 \binomial 𝛼 𝑥 𝑥 \binomial 𝛽 𝑁 𝑥 𝑁 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)=\binomial{% \alpha+x}{x}\binomial{\beta+N-x}{N-x}}}}


Generating functions

\HyperpFq ⁢ 11 ⁢ @ ⁢ @ - x ⁢ α + 1 - t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ x - N ⁢ β + 1 ⁢ t = ∑ n = 0 N ( - N ) n ( β + 1 ) n ⁢ n ! ⁢ Q n ⁡ ( x ; α , β , N ) ⁢ t n \HyperpFq 11 @ @ 𝑥 𝛼 1 𝑡 \HyperpFq 11 @ @ 𝑥 𝑁 𝛽 1 𝑡 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝑁 𝑛 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{-x}{\alpha+1}{-t}% \,\HyperpFq{1}{1}@@{x-N}{\beta+1}{t}=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}% }{{\left(\beta+1\right)_{n}}n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}} {\displaystyle \HyperpFq{1}{1}@@{-x}{\alpha+1}{-t}\,\HyperpFq{1}{1}@@{x-N}{\beta+1}{t} =\sum_{n=0}^N\frac{\pochhammer{-N}{n}}{\pochhammer{\beta+1}{n}n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }
\HyperpFq 20 @ @ - x , - x + β + N + 1 - - t \HyperpFq 20 @ @ x - N , x + α + 1 - t = ∑ n = 0 N ( - N ) n ⁢ ( α + 1 ) n n ! Q n ⁡ ( x ; α , β , N ) t n fragments \HyperpFq 20 @ @ x , x β N 1 t \HyperpFq 20 @ @ x N , x α 1 t superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝑁 𝑛 Pochhammer-symbol 𝛼 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{0}@@{-x,-x+\beta+N+1}{-% }{-t}\,\HyperpFq{2}{0}@@{x-N,x+\alpha+1}{-}{t}{}=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\alpha+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N% \right)t^{n}}}} {\displaystyle \HyperpFq{2}{0}@@{-x,-x+\beta+N+1}{-}{-t}\,\HyperpFq{2}{0}@@{x-N,x+\alpha+1}{-}{t} {}=\sum_{n=0}^N\frac{\pochhammer{-N}{n}\pochhammer{\alpha+1}{n}}{n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }
[ ( 1 - t ) - α - β - 1 ⁢ \HyperpFq ⁢ 32 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) , - x ⁢ α + 1 , - N - 4 ⁢ t ( 1 - t ) 2 ] N = ∑ n = 0 N ( α + β + 1 ) n n ! ⁢ Q n ⁡ ( x ; α , β , N ) ⁢ t n subscript superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 32 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝑥 𝛼 1 𝑁 4 𝑡 superscript 1 𝑡 2 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝛼 𝛽 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left[(1-t)^{-\alpha-\beta-1}\,% \HyperpFq{3}{2}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x}{% \alpha+1,-N}{-\frac{4t}{(1-t)^{2}}}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(% \alpha+\beta+1\right)_{n}}}{n!}Q_{n}\!\left(x;\alpha,\beta,N\right)t^{n}}}} {\displaystyle \left[(1-t)^{-\alpha-\beta-1}\,\HyperpFq{3}{2}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x}{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N {}=\sum_{n=0}^N\frac{\pochhammer{\alpha+\beta+1}{n}}{n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }

Limit relations

Racah polynomial to Hahn polynomial

lim δ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , β , - N - 1 , δ ) = Q n ⁡ ( x ; α , β , N ) subscript → 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝑁 1 𝛿 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}% \!\left(\lambda(x);\alpha,\beta,-N-1,\delta\right)=Q_{n}\!\left(x;\alpha,\beta% ,N\right)}}} {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\delta}=\Hahn{n}@{x}{\alpha}{\beta}{N} }
lim γ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; α , β , γ , - β - N - 1 ) = Q n ⁡ ( x ; α , β , N ) subscript → 𝛾 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛽 𝑁 1 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\gamma\rightarrow\infty}R_{n}% \!\left(\lambda(x);\alpha,\beta,\gamma,-\beta-N-1\right)=Q_{n}\!\left(x;\alpha% ,\beta,N\right)}}} {\displaystyle \lim_{\gamma\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{-\beta-N-1}=\Hahn{n}@{x}{\alpha}{\beta}{N} }
lim δ → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; - N - 1 , β + γ + N + 1 , γ , δ ) = Q n ⁡ ( x ; γ , β , N ) subscript → 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝑁 1 𝛽 𝛾 𝑁 1 𝛾 𝛿 Hahn-polynomial-Q 𝑛 𝑥 𝛾 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{\delta\rightarrow\infty}R_{n}% \!\left(\lambda(x);-N-1,\beta+\gamma+N+1,\gamma,\delta\right)=Q_{n}\!\left(x;% \gamma,\beta,N\right)}}} {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{-N-1}{\beta+\gamma+N+1}{\gamma}{\delta}=\Hahn{n}@{x}{\gamma}{\beta}{N} }

Hahn polynomial to Jacobi polynomial

lim N → ∞ ⁡ Q n ⁡ ( N ⁢ x ; α , β , N ) = P n ( α , β ) ⁡ ( 1 - 2 ⁢ x ) P n ( α , β ) ⁡ ( 1 ) subscript → 𝑁 Hahn-polynomial-Q 𝑛 𝑁 𝑥 𝛼 𝛽 𝑁 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_{n}\left(1-2x\right)}{% P^{(\alpha,\beta)}_{n}\left(1\right)}}}} {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{Nx}{\alpha}{\beta}{N}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}}{\Jacobi{\alpha}{\beta}{n}@{1}} }

Hahn polynomial to Meixner polynomial

lim N → ∞ ⁡ Q n ⁡ ( x ; b - 1 , N ⁢ ( 1 - c ) ⁢ c - 1 , N ) = M n ⁡ ( x ; b , c ) subscript → 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝑏 1 𝑁 1 𝑐 superscript 𝑐 1 𝑁 Meixner-polynomial-M 𝑛 𝑥 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(x;b-1,N(1-c)c^{-1},N\right)=M_{n}\!\left(x;b,c\right)}}} {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{x}{b-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{b}{c} }

Hahn polynomial to Krawtchouk polynomial

lim t → ∞ ⁡ Q n ⁡ ( x ; p ⁢ t , ( 1 - p ) ⁢ t , N ) = K n ⁡ ( x ; p , N ) subscript → 𝑡 Hahn-polynomial-Q 𝑛 𝑥 𝑝 𝑡 1 𝑝 𝑡 𝑁 Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}Q_{n}\!% \left(x;pt,(1-p)t,N\right)=K_{n}\!\left(x;p,N\right)}}} {\displaystyle \lim_{t\rightarrow\infty}\Hahn{n}@{x}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }

Remark

Q n ⁡ ( x ; α , β , N ) = R x ⁡ ( λ ⁢ ( n ) ; α , β , N ) Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 dual-Hahn-R 𝑥 𝜆 𝑛 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right% )=R_{x}\!\left(\lambda(n);\alpha,\beta,N\right)}}} {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{\lambda(n)}{\alpha}{\beta}{N} }
∑ n = 0 N ( 2 ⁢ n + α + β + 1 ) ⁢ ( α + 1 ) n ⁢ ( - N ) n ⁢ N ! ( - 1 ) n ⁢ ( n + α + β + 1 ) N + 1 ⁢ ( β + 1 ) n ⁢ n ! ⁢ Q n ⁡ ( x ; α , β , N ) ⁢ Q n ⁡ ( y ; α , β , N ) = δ x , y ( α + x x ) ⁢ ( β + N - x N - x ) superscript subscript 𝑛 0 𝑁 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝑁 𝑛 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑁 1 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑦 𝛼 𝛽 𝑁 Kronecker-delta 𝑥 𝑦 binomial 𝛼 𝑥 𝑥 binomial 𝛽 𝑁 𝑥 𝑁 𝑥 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\frac{(2n+\alpha+\beta% +1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}{(-1)^{n}{\left(n+\alpha% +\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}n!}Q_{n}\!\left(x;\alpha,\beta% ,N\right)Q_{n}\!\left(y;\alpha,\beta,N\right){}=\frac{\delta_{x,y}}{\dbinom{% \alpha+x}{x}\dbinom{\beta+N-x}{N-x}}}}} {\displaystyle \sum_{n=0}^N\frac{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}{(-1)^n\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!} \Hahn{n}@{x}{\alpha}{\beta}{N}\Hahn{n}@{y}{\alpha}{\beta}{N} {}=\frac{\Kronecker{x}{y}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}} }

Constraint(s): x , y ∈ { 0 , 1 , 2 , … , N } 𝑥 𝑦 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle x,y\in\{0,1,2,\ldots,N\}}}}


Koornwinder Addendum: Hahn

Hahn: Special values

Q n ⁡ ( 0 ; α , β , N ) = 1 ⁢ Q n ⁡ ( N ; α , β , N ) Hahn-polynomial-Q 𝑛 0 𝛼 𝛽 𝑁 1 Hahn-polynomial-Q 𝑛 𝑁 𝛼 𝛽 𝑁 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(0;\alpha,\beta,N\right% )=1Q_{n}\!\left(N;\alpha,\beta,N\right)}}} {\displaystyle \Hahn{n}@{0}{\alpha}{\beta}{N}=1 \Hahn{n}@{N}{\alpha}{\beta}{N} }
Q n ⁡ ( 0 ; α , β , N ) = ( - 1 ) n ⁢ ( β + 1 ) n ( α + 1 ) n Hahn-polynomial-Q 𝑛 0 𝛼 𝛽 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(0;\alpha,\beta,N\right% )=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}}}}} {\displaystyle \Hahn{n}@{0}{\alpha}{\beta}{N} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+1}{n}} }
Q 2 ⁢ n ⁡ ( N ; α , α , 2 ⁢ N ) = ( 1 2 ) n ⁢ ( N + α + 1 ) n ( - N + 1 2 ) n ⁢ ( α + 1 ) n Hahn-polynomial-Q 2 𝑛 𝑁 𝛼 𝛼 2 𝑁 Pochhammer-symbol 1 2 𝑛 Pochhammer-symbol 𝑁 𝛼 1 𝑛 Pochhammer-symbol 𝑁 1 2 𝑛 Pochhammer-symbol 𝛼 1 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{2n}\!\left(N;\alpha,\alpha,2N% \right)=\frac{{\left(\frac{1}{2}\right)_{n}}{\left(N+\alpha+1\right)_{n}}}{{% \left(-N+\frac{1}{2}\right)_{n}}{\left(\alpha+1\right)_{n}}}}}} {\displaystyle \Hahn{2n}@{N}{\alpha}{\alpha}{2N}=\frac{\pochhammer{\frac12}{n}\pochhammer{N+\alpha+1}{n}}{\pochhammer{-N+\frac12}{n}\pochhammer{\alpha+1}{n}} }
Q N ⁡ ( x ; α , β , N ) = ( - N - β ) x ( α + 1 ) x    ( x fragments Hahn-polynomial-Q 𝑁 𝑥 𝛼 𝛽 𝑁 Pochhammer-symbol 𝑁 𝛽 𝑥 Pochhammer-symbol 𝛼 1 𝑥 italic-   fragments ( x {\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(x;\alpha,\beta,N\right% )=\frac{{\left(-N-\beta\right)_{x}}}{{\left(\alpha+1\right)_{x}}}\qquad(x}}} {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{-N-\beta}{x}}{\pochhammer{\alpha+1}{x}}\qquad(x }
Q N ⁡ ( x ; α , β , N ) = 0 , 1 , … , N ) fragments Hahn-polynomial-Q 𝑁 𝑥 𝛼 𝛽 𝑁 0 , 1 , … , N ) {\displaystyle{\displaystyle{\displaystyle Q_{N}\!\left(x;\alpha,\beta,N\right% )=0,1,\ldots,N)}}} {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N} =0,1,\ldots,N) }

Symmetries

Q n ⁡ ( N - x ; α , β , N ) Q n ⁡ ( N ; α , β , N ) = Q n ⁡ ( x ; β , α , N ) Hahn-polynomial-Q 𝑛 𝑁 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑁 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛽 𝛼 𝑁 {\displaystyle{\displaystyle{\displaystyle\frac{Q_{n}\!\left(N-x;\alpha,\beta,% N\right)}{Q_{n}\!\left(N;\alpha,\beta,N\right)}=Q_{n}\!\left(x;\beta,\alpha,N% \right)}}} {\displaystyle \frac{\Hahn{n}@{N-x}{\alpha}{\beta}{N}}{\Hahn{n}@{N}{\alpha}{\beta}{N}}=\Hahn{n}@{x}{\beta}{\alpha}{N} }
Q N - n ⁡ ( x ; α , β , N ) Q N ⁡ ( x ; α , β , N ) = Q n ⁡ ( x ; - N - β - 1 , - N - α - 1 , N )    ( x fragments Hahn-polynomial-Q 𝑁 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑁 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝑁 𝛽 1 𝑁 𝛼 1 𝑁 italic-   fragments ( x {\displaystyle{\displaystyle{\displaystyle\frac{Q_{N-n}\!\left(x;\alpha,\beta,% N\right)}{Q_{N}\!\left(x;\alpha,\beta,N\right)}=Q_{n}\!\left(x;-N-\beta-1,-N-% \alpha-1,N\right)\qquad(x}}} {\displaystyle \frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}} =\Hahn{n}@{x}{-N-\beta-1}{-N-\alpha-1}{N} \qquad(x }
Q N - n ⁡ ( x ; α , β , N ) Q N ⁡ ( x ; α , β , N ) = 0 , 1 , … , N ) fragments Hahn-polynomial-Q 𝑁 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑁 𝑥 𝛼 𝛽 𝑁 0 , 1 , … , N ) {\displaystyle{\displaystyle{\displaystyle\frac{Q_{N-n}\!\left(x;\alpha,\beta,% N\right)}{Q_{N}\!\left(x;\alpha,\beta,N\right)}=0,1,\ldots,N)}}} {\displaystyle \frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}} =0,1,\ldots,N) }

Duality

Q n ⁡ ( x ; α , β , N ) = R x ⁡ ( n ⁢ ( n + α + β + 1 ) ; α , β , N ) ( n , x ∈ { 0 , 1 , … N } ) fragments Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 dual-Hahn-R 𝑥 𝑛 𝑛 𝛼 𝛽 1 𝛼 𝛽 𝑁 fragments ( n , x fragments { 0 , 1 , … N } ) {\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;\alpha,\beta,N\right% )=R_{x}\!\left(n(n+\alpha+\beta+1);\alpha,\beta,N\right)(n,x\in\{0,1,\ldots N% \})}}} {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{n(n+\alpha+\beta+1)}{\alpha}{\beta}{N} (n,x\in\{0,1,\ldots N\}) }

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Hahn&oldid=1469"