Misplaced Pages

Racah polynomials

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
This article includes inline citations, but they are not properly formatted. Please improve this article by correcting them. (May 2024) (Learn how and when to remove this message)

In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients.

The Racah polynomials were first defined by Wilson (1978) and are given by

p n ( x ( x + γ + δ + 1 ) ) = 4 F 3 [ − n n + α + β + 1 − x x + γ + δ + 1 α + 1 γ + 1 β + δ + 1 ; 1 ] . {\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left.}

Orthogonality

∑ y = 0 N R n ⁡ ( x ; α , β , γ , δ ) R m ⁡ ( x ; α , β , γ , δ ) γ + δ + 1 + 2 y γ + δ + 1 + y ω y = h n δ n , m , {\displaystyle \sum _{y=0}^{N}\operatorname {R} _{n}(x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{m}(x;\alpha ,\beta ,\gamma ,\delta ){\frac {\gamma +\delta +1+2y}{\gamma +\delta +1+y}}\omega _{y}=h_{n}\operatorname {\delta } _{n,m},}
when α + 1 = − N {\displaystyle \alpha +1=-N} ,
where R {\displaystyle \operatorname {R} } is the Racah polynomial,
x = y ( y + γ + δ + 1 ) , {\displaystyle x=y(y+\gamma +\delta +1),}
δ n , m {\displaystyle \operatorname {\delta } _{n,m}} is the Kronecker delta function and the weight functions are
ω y = ( α + 1 ) y ( β + δ + 1 ) y ( γ + 1 ) y ( γ + δ + 2 ) y ( − α + γ + δ + 1 ) y ( − β + γ + 1 ) y ( δ + 1 ) y y ! , {\displaystyle \omega _{y}={\frac {(\alpha +1)_{y}(\beta +\delta +1)_{y}(\gamma +1)_{y}(\gamma +\delta +2)_{y}}{(-\alpha +\gamma +\delta +1)_{y}(-\beta +\gamma +1)_{y}(\delta +1)_{y}y!}},}
and
h n = ( − β ) N ( γ + δ + 1 ) N ( − β + γ + 1 ) N ( δ + 1 ) N ( n + α + β + 1 ) n n ! ( α + β + 2 ) 2 n ( α + δ − γ + 1 ) n ( α − δ + 1 ) n ( β + 1 ) n ( α + 1 ) n ( β + δ + 1 ) n ( γ + 1 ) n , {\displaystyle h_{n}={\frac {(-\beta )_{N}(\gamma +\delta +1)_{N}}{(-\beta +\gamma +1)_{N}(\delta +1)_{N}}}{\frac {(n+\alpha +\beta +1)_{n}n!}{(\alpha +\beta +2)_{2n}}}{\frac {(\alpha +\delta -\gamma +1)_{n}(\alpha -\delta +1)_{n}(\beta +1)_{n}}{(\alpha +1)_{n}(\beta +\delta +1)_{n}(\gamma +1)_{n}}},}
( ⋅ ) n {\displaystyle (\cdot )_{n}} is the Pochhammer symbol.

Rodrigues-type formula

ω ( x ; α , β , γ , δ ) R n ⁡ ( λ ( x ) ; α , β , γ , δ ) = ( γ + δ + 1 ) n ∇ n ∇ λ ( x ) n ω ( x ; α + n , β + n , γ + n , δ ) , {\displaystyle \omega (x;\alpha ,\beta ,\gamma ,\delta )\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )=(\gamma +\delta +1)_{n}{\frac {\nabla ^{n}}{\nabla \lambda (x)^{n}}}\omega (x;\alpha +n,\beta +n,\gamma +n,\delta ),}
where ∇ {\displaystyle \nabla } is the backward difference operator,
λ ( x ) = x ( x + γ + δ + 1 ) . {\displaystyle \lambda (x)=x(x+\gamma +\delta +1).}

Generating functions

There are three generating functions for x ∈ { 0 , 1 , 2 , . . . , N } {\displaystyle x\in \{0,1,2,...,N\}}

when β + δ + 1 = − N {\displaystyle \beta +\delta +1=-N\quad } or γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,}
2 F 1 ( − x , − x + α − γ − δ ; α + 1 ; t ) 2 F 1 ( x + β + δ + 1 , x + γ + 1 ; β + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x+\alpha -\gamma -\delta ;\alpha +1;t){}_{2}F_{1}(x+\beta +\delta +1,x+\gamma +1;\beta +1;t)}
= ∑ n = 0 N ( β + δ + 1 ) n ( γ + 1 ) n ( β + 1 ) n n ! R n ⁡ ( λ ( x ) ; α , β , γ , δ ) t n , {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\beta +\delta +1)_{n}(\gamma +1)_{n}}{(\beta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},}
when α + 1 = − N {\displaystyle \alpha +1=-N\quad } or γ + 1 = − N , {\displaystyle \quad \gamma +1=-N,}
2 F 1 ( − x , − x + β − γ ; β + δ + 1 ; t ) 2 F 1 ( x + α + 1 , x + γ + 1 ; α − δ + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x+\beta -\gamma ;\beta +\delta +1;t){}_{2}F_{1}(x+\alpha +1,x+\gamma +1;\alpha -\delta +1;t)}
= ∑ n = 0 N ( α + 1 ) n ( γ + 1 ) n ( α − δ + 1 ) n n ! R n ⁡ ( λ ( x ) ; α , β , γ , δ ) t n , {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\gamma +1)_{n}}{(\alpha -\delta +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n},}
when α + 1 = − N {\displaystyle \alpha +1=-N\quad } or β + δ + 1 = − N , {\displaystyle \quad \beta +\delta +1=-N,}
2 F 1 ( − x , − x − δ ; γ + 1 ; t ) 2 F 1 ( x + α + 1 ; x + β + γ + 1 ; α + β − γ + 1 ; t ) {\displaystyle {}_{2}F_{1}(-x,-x-\delta ;\gamma +1;t){}_{2}F_{1}(x+\alpha +1;x+\beta +\gamma +1;\alpha +\beta -\gamma +1;t)}
= ∑ n = 0 N ( α + 1 ) n ( β + δ + 1 ) n ( α + β − γ + 1 ) n n ! R n ⁡ ( λ ( x ) ; α , β , γ , δ ) t n . {\displaystyle \quad =\sum _{n=0}^{N}{\frac {(\alpha +1)_{n}(\beta +\delta +1)_{n}}{(\alpha +\beta -\gamma +1)_{n}n!}}\operatorname {R} _{n}(\lambda (x);\alpha ,\beta ,\gamma ,\delta )t^{n}.}

Connection formula for Wilson polynomials

When α = a + b − 1 , β = c + d − 1 , γ = a + d − 1 , δ = a − d , x → − a + i x , {\displaystyle \alpha =a+b-1,\beta =c+d-1,\gamma =a+d-1,\delta =a-d,x\rightarrow -a+ix,}

R n ⁡ ( λ ( − a + i x ) ; a + b − 1 , c + d − 1 , a + d − 1 , a − d ) = W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ( a + c ) n ( a + d ) n , {\displaystyle \operatorname {R} _{n}(\lambda (-a+ix);a+b-1,c+d-1,a+d-1,a-d)={\frac {\operatorname {W} _{n}(x^{2};a,b,c,d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}}},}
where W {\displaystyle \operatorname {W} } are Wilson polynomials.

q-analog

Askey & Wilson (1979) introduced the q-Racah polynomials defined in terms of basic hypergeometric functions by

p n ( q − x + q x + 1 c d ; a , b , c , d ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x q x + 1 c d a q b d q c q ; q ; q ] . {\displaystyle p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi _{3}\left.}

They are sometimes given with changes of variables as

W n ( x ; a , b , c , N ; q ) = 4 ϕ 3 [ q − n a b q n + 1 q − x c q x − n a q b c q q − N ; q ; q ] . {\displaystyle W_{n}(x;a,b,c,N;q)={}_{4}\phi _{3}\left.}

References

  1. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Wilson Class: Definitions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  2. Koekoek, Roelof; Swarttouw, René F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue


Stub icon

This algebra-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: