In mathematics, the continuous Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by
p n ( x ; a , b , c , d ) = i n ( a + c ) n ( a + d ) n n ! 3 F 2 ( − n , n + a + b + c + d − 1 , a + i x a + c , a + d ; 1 ) {\displaystyle p_{n}(x;a,b,c,d)=i^{n}{\frac {(a+c)_{n}(a+d)_{n}}{n!}}{}_{3}F_{2}\left({\begin{array}{c}-n,n+a+b+c+d-1,a+ix\\a+c,a+d\end{array}};1\right)} Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010 , 14) give a detailed list of their properties.
Closely related polynomials include the dual Hahn polynomials R n (x ;γ,δ,N ), the Hahn polynomials Q n (x ;a ,b ,c ), and the continuous dual Hahn polynomials S n (x ;a ,b ,c ). These polynomials all have q -analogs with an extra parameter q , such as the q-Hahn polynomials Q n (x ;α,β, N ;q ), and so on.
The continuous Hahn polynomials p n (x ;a ,b ,c ,d ) are orthogonal with respect to the weight function
w ( x ) = Γ ( a + i x ) Γ ( b + i x ) Γ ( c − i x ) Γ ( d − i x ) . {\displaystyle w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix).} In particular, they satisfy the orthogonality relation[ 1] [ 2] [ 3]
1 2 π ∫ − ∞ ∞ Γ ( a + i x ) Γ ( b + i x ) Γ ( c − i x ) Γ ( d − i x ) p m ( x ; a , b , c , d ) p n ( x ; a , b , c , d ) d x = Γ ( n + a + c ) Γ ( n + a + d ) Γ ( n + b + c ) Γ ( n + b + d ) n ! ( 2 n + a + b + c + d − 1 ) Γ ( n + a + b + c + d − 1 ) δ n m {\displaystyle {\begin{aligned}&{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{m}(x;a,b,c,d)\,p_{n}(x;a,b,c,d)\,dx\\&\qquad \qquad ={\frac {\Gamma (n+a+c)\,\Gamma (n+a+d)\,\Gamma (n+b+c)\,\Gamma (n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma (n+a+b+c+d-1)}}\,\delta _{nm}\end{aligned}}} for ℜ ( a ) > 0 {\displaystyle \Re (a)>0} , ℜ ( b ) > 0 {\displaystyle \Re (b)>0} , ℜ ( c ) > 0 {\displaystyle \Re (c)>0} , ℜ ( d ) > 0 {\displaystyle \Re (d)>0} , c = a ¯ {\displaystyle c={\overline {a}}} , d = b ¯ {\displaystyle d={\overline {b}}} .
Recurrence and difference relations [ edit ] The sequence of continuous Hahn polynomials satisfies the recurrence relation[ 4]
x p n ( x ) = p n + 1 ( x ) + i ( A n + C n ) p n ( x ) − A n − 1 C n p n − 1 ( x ) , {\displaystyle xp_{n}(x)=p_{n+1}(x)+i(A_{n}+C_{n})p_{n}(x)-A_{n-1}C_{n}p_{n-1}(x),} where p n ( x ) = n ! ( n + a + b + c + d − 1 ) ! ( 2 n + a + b + c + d − 1 ) ! p n ( x ; a , b , c , d ) , A n = − ( n + a + b + c + d − 1 ) ( n + a + c ) ( n + a + d ) ( 2 n + a + b + c + d − 1 ) ( 2 n + a + b + c + d ) , and C n = n ( n + b + c − 1 ) ( n + b + d − 1 ) ( 2 n + a + b + c + d − 2 ) ( 2 n + a + b + c + d − 1 ) . {\displaystyle {\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}} The continuous Hahn polynomials are given by the Rodrigues-like formula[ 5]
Γ ( a + i x ) Γ ( b + i x ) Γ ( c − i x ) Γ ( d − i x ) p n ( x ; a , b , c , d ) = ( − 1 ) n n ! d n d x n ( Γ ( a + n 2 + i x ) Γ ( b + n 2 + i x ) Γ ( c + n 2 − i x ) Γ ( d + n 2 − i x ) ) . {\displaystyle {\begin{aligned}&\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{n}(x;a,b,c,d)\\&\qquad ={\frac {(-1)^{n}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(\Gamma \left(a+{\frac {n}{2}}+ix\right)\,\Gamma \left(b+{\frac {n}{2}}+ix\right)\,\Gamma \left(c+{\frac {n}{2}}-ix\right)\,\Gamma \left(d+{\frac {n}{2}}-ix\right)\right).\end{aligned}}} Generating functions [ edit ] The continuous Hahn polynomials have the following generating function:[ 6]
∑ n = 0 ∞ Γ ( n + a + b + c + d ) Γ ( a + c + 1 ) Γ ( a + d + 1 ) Γ ( a + b + c + d ) Γ ( n + a + c + 1 ) Γ ( n + a + d + 1 ) ( − i t ) n p n ( x ; a , b , c , d ) = ( 1 − t ) 1 − a − b − c − d 3 F 2 ( 1 2 ( a + b + c + d − 1 ) , 1 2 ( a + b + c + d ) , a + i x a + c , a + d ; − 4 t ( 1 − t ) 2 ) . {\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }{\frac {\Gamma (n+a+b+c+d)\,\Gamma (a+c+1)\,\Gamma (a+d+1)}{\Gamma (a+b+c+d)\,\Gamma (n+a+c+1)\,\Gamma (n+a+d+1)}}(-it)^{n}p_{n}(x;a,b,c,d)\\&\qquad =(1-t)^{1-a-b-c-d}{}_{3}F_{2}\left({\begin{array}{c}{\frac {1}{2}}(a+b+c+d-1),{\frac {1}{2}}(a+b+c+d),a+ix\\a+c,a+d\end{array}};-{\frac {4t}{(1-t)^{2}}}\right).\end{aligned}}} A second, distinct generating function is given by
∑ n = 0 ∞ Γ ( a + c + 1 ) Γ ( b + d + 1 ) Γ ( n + a + c + 1 ) Γ ( n + b + d + 1 ) t n p n ( x ; a , b , c , d ) = 1 F 1 ( a + i x a + c ; − i t ) 1 F 1 ( d − i x b + d ; i t ) . {\displaystyle \sum _{n=0}^{\infty }{\frac {\Gamma (a+c+1)\,\Gamma (b+d+1)}{\Gamma (n+a+c+1)\,\Gamma (n+b+d+1)}}t^{n}p_{n}(x;a,b,c,d)=\,_{1}F_{1}\left({\begin{array}{c}a+ix\\a+c\end{array}};-it\right)\,_{1}F_{1}\left({\begin{array}{c}d-ix\\b+d\end{array}};it\right).} Relation to other polynomials [ edit ] The Wilson polynomials are a generalization of the continuous Hahn polynomials. The Bateman polynomials F n (x) are related to the special case a =b =c =d =1/2 of the continuous Hahn polynomials by p n ( x ; 1 2 , 1 2 , 1 2 , 1 2 ) = i n n ! F n ( 2 i x ) . {\displaystyle p_{n}\left(x;{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}}\right)=i^{n}n!F_{n}\left(2ix\right).} The Jacobi polynomials P n (α,β) (x) can be obtained as a limiting case of the continuous Hahn polynomials:[ 7] P n ( α , β ) = lim t → ∞ t − n p n ( 1 2 x t ; 1 2 ( α + 1 − i t ) , 1 2 ( β + 1 + i t ) , 1 2 ( α + 1 + i t ) , 1 2 ( β + 1 − i t ) ) . {\displaystyle P_{n}^{(\alpha ,\beta )}=\lim _{t\to \infty }t^{-n}p_{n}\left({\tfrac {1}{2}}xt;{\tfrac {1}{2}}(\alpha +1-it),{\tfrac {1}{2}}(\beta +1+it),{\tfrac {1}{2}}(\alpha +1+it),{\tfrac {1}{2}}(\beta +1-it)\right).} ^ Koekoek, Lesky, & Swarttouw (2010), p. 200. ^ Askey, R. (1985), "Continuous Hahn polynomials", J. Phys. A: Math. Gen. 18 : pp. L1017-L1019. ^ Andrews, Askey, & Roy (1999), p. 333. ^ Koekoek, Lesky, & Swarttouw (2010), p. 201. ^ Koekoek, Lesky, & Swarttouw (2010), p. 202. ^ Koekoek, Lesky, & Swarttouw (2010), p. 202. ^ Koekoek, Lesky, & Swarttouw (2010), p. 203. Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten , 2 : 4–34, doi :10.1002/mana.19490020103 , ISSN 0025-584X , MR 0030647 Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues , Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag , doi :10.1007/978-3-642-05014-5 , ISBN 978-3-642-05013-8 , MR 2656096 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn 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 . Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions , Encyclopedia of Mathematics and its Applications 71, Cambridge: Cambridge University Press , ISBN 978-0-521-62321-6