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Orthogonality

The continuous Hahn polynomials pn(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 relation¹²³

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 relation⁴

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}}}

Rodrigues formula

The continuous Hahn polynomials are given by the Rodrigues-like formula⁵

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

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 Fn(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 Pn(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).}

• 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

References

1. Koekoek, Lesky, & Swarttouw (2010), p. 200. ↩

2. Askey, R. (1985), "Continuous Hahn polynomials", J. Phys. A: Math. Gen. 18: pp. L1017-L1019. ↩

3. Andrews, Askey, & Roy (1999), p. 333. ↩

4. Koekoek, Lesky, & Swarttouw (2010), p. 201. ↩

5. Koekoek, Lesky, & Swarttouw (2010), p. 202. ↩

6. Koekoek, Lesky, & Swarttouw (2010), p. 202. ↩

7. Koekoek, Lesky, & Swarttouw (2010), p. 203. ↩