Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Hahn polynomials
open-in-new
<h2 id="orthogonality">Orthogonality</h2> ∑ x = 0 N − 1 Q n ( x ) Q m ( x ) ρ ( x ) = 1 π n δ m , n , {\displaystyle \sum _{x=0}^{N-1}Q_{n}(x)Q_{m}(x)\rho (x)={\frac {1}{\pi _{n}}}\delta _{m,n},} ∑ n = 0 N − 1 Q n ( x ) Q n ( y ) π n = 1 ρ ( x ) δ x , y {\displaystyle \sum _{n=0}^{N-1}Q_{n}(x)Q_{n}(y)\pi _{n}={\frac {1}{\rho (x)}}\delta _{x,y}} <p>where <i>δx,y</i> is the Kronecker delta function and the weight functions are </p> ρ ( x ) = ρ ( x ; α ; β , N ) = ( α + x x ) ( β + N − 1 − x N − 1 − x ) / ( N + α + β N − 1 ) {\displaystyle \rho (x)=\rho (x;\alpha ;\beta ,N)={\binom {\alpha +x}{x}}{\binom {\beta +N-1-x}{N-1-x}}/{\binom {N+\alpha +\beta }{N-1}}} <p>and </p> π n = π n ( α , β , N ) = ( N − 1 n ) 2 n + α + β + 1 α + β + 1 Γ ( β + 1 , n + α + 1 , n + α + β + 1 ) Γ ( α + 1 , α + β + 1 , n + β + 1 , n + 1 ) / ( N + α + β + n n ) {\displaystyle \pi _{n}=\pi _{n}(\alpha ,\beta ,N)={\binom {N-1}{n}}{\frac {2n+\alpha +\beta +1}{\alpha +\beta +1}}{\frac {\Gamma (\beta +1,n+\alpha +1,n+\alpha +\beta +1)}{\Gamma (\alpha +1,\alpha +\beta +1,n+\beta +1,n+1)}}/{\binom {N+\alpha +\beta +n}{n}}} . <h2 id="relation-to-other-polynomials">Relation to other polynomials</h2> <ul><li><a href="/facts/Racah_polynomials/RryLag1h">Racah polynomials</a> are a generalization of Hahn polynomials</li></ul> <ul><li>Chebyshev, P. (1907), "Sur l'interpolation des valeurs équidistantes", in Markoff, A.; Sonin, N. (eds.), <a href="https://archive.org/stream/uvresdepltcheby01chebgoog#page/n250"><i>Oeuvres de P. L. Tchebychef</i></a>, vol. 2, pp. 219–242, Reprinted by Chelsea</li> <li><a href="/facts/Wolfgang_Hahn/Q4npKg4U">Hahn, Wolfgang</a> (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", <i><a href="/facts/Mathematische_Nachrichten/gFAfDqQq">Mathematische Nachrichten</a></i>, 2: 4–34, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fmana.19490020103">10.1002/mana.19490020103</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-584X">0025-584X</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0030647">0030647</a></li> <li>Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), <i>Hypergeometric orthogonal polynomials and their q-analogues</i>, Springer Monographs in Mathematics, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-05014-5">10.1007/978-3-642-05014-5</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-05013-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2656096">2656096</a></li> <li>Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), <a href="http://dlmf.nist.gov/18.19">"Hahn Class: Definitions"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li></ul>