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
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}}}
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
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