In mathematics, the Hahn polynomials are a family of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...

in the

Askey scheme In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , ...

of hypergeometric orthogonal polynomials, introduced by

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...

in 1875 and rediscovered by

Wolfgang Hahn Wolfgang Hahn (April 30, 1911 – January 10, 1998) was a German mathematician who worked on special functions, in particular orthogonal polynomials. He introduced Hahn polynomials, Hahn difference, Hahn q-addition (or Jackson-Hahn-Cigler q-add ...

. The Hahn class is a name for special cases of Hahn polynomials, including Hahn polynomials,

Meixner polynomials In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol In mathematics, t ...

Krawtchouk polynomials Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname ) are discrete orthogonal polynomials associated with the binomial distribution, introduced by . The first few polynomials a ...

, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the

classical orthogonal polynomials In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomi ...

. Hahn polynomials are defined in terms of

generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...

s by :

Q_n(x;\alpha,\beta,N)= _3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).\

give a detailed list of their properties. If

\alpha = \beta = 0

, these polynomials are identical to the

discrete Chebyshev polynomials In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to v ...

except for a scale factor. Closely related polynomials include the

dual Hahn polynomials In mathematics, the dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice x(s)=s(s+1) and are defined as :w_n^ (s,a,b)=\frac _3F_2(-n,a-s, ...

''R''_''n''(''x'';γ,δ,''N''), the

continuous Hahn polynomials 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\frac_3F_ ...

''p''_''n''(''x'',''a'',''b'', , ), and the

continuous dual Hahn polynomials In mathematics, the continuous dual 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 :S_n(x^2;a,b,c)= _3F_2 ...

''S''_''n''(''x'';''a'',''b'',''c''). These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the

q-Hahn polynomials In mathematics, the ''q''-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. Definition The polynomials are given in terms of basic hypergeometric ...

''Q''_''n''(''x'';α,β, ''N'';''q''), and so on.

Orthogonality

\sum_^ Q_n(x)Q_m(x)\rho(x)=\frac\delta_,

\sum_^Q_n(x)Q_n(y)\pi_n=\frac\delta_

where ''δ_x,y'' is the Kronecker delta function and the weight functions are :

\rho(x)=\rho(x;\alpha;\beta,N)=\binom\binom/\binom

and :

\pi_n=\pi_n(\alpha,\beta,N)=\binom\frac
  		
\frac/\binom

Relation to other polynomials

Racah polynomials 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 and are giv ...

are a generalization of Hahn polynomials

References

* * * *{{dlmf, id=18.19, title=Hahn Class: Definitions, first=Tom H. , last=Koornwinder, first2=Roderick S. C., last2= Wong, first3=Roelof , last3=Koekoek, , first4=René F. , last4=Swarttouw Special hypergeometric functions Orthogonal polynomials