克拉夫丘克多項式

克拉夫丘克多項式以超幾何函數定義如下

Kravchuk polynomials animation

Kravchuk polynomials animation

${\displaystyle K_{n}(x;p,N)={N \choose n}^{-1}*}$${\displaystyle \sum _{k=0}^{n}{N-x \choose n-k}*{x \choose k}*(1-1/p)^{k}}$=${\displaystyle _{2}F_{1}(-n,-x;N;1/p)}$

克拉夫丘克多項式的頭幾項是

${\displaystyle K_{0}(x,p,N)=1}$

${\displaystyle K_{1}(x,p,N)=1-{\frac {x}{p*N}}}$

${\displaystyle K_{2}(x,p,N)=1+({\frac {-2}{p*N}}+{\frac {1}{(p^{2}*N*(-N+1)}})*x-{\frac {x^{2}}{(p^{2}*N*(-N+1))}}}$

${\displaystyle K_{3}(x,p,N)=1+({\frac {-3}{(p*N)}}+{\frac {3}{(p^{2}*N*(-N+1))}}-{\frac {2}{(p^{3}*N*(-N+1)*(-N+2))}})*x+(-3/(p^{2}*N*(-N+1))+3/(p^{3}*N*(-N+1)*(-N+2)))*x^{2}-x^{3}/(p^{3}*N*(-N+1)*(-N+2))}$

極限關係

量子Q克拉夫丘克多項式→ 克拉夫丘克多項式：

${\displaystyle \lim _{a\to 1}=K_{n}^{qtm}(q^{-}{x};p,N;q)=K_{n}(x;p^{-1},N)}$

令雙Q克拉夫丘克多項式${\displaystyle c=1=p^{-1}}$,並令q→1,即得克拉夫丘克多項式

參考文獻

• Kravchuk, M., Sur une généralisation des polynomes d'Hermite., Comptes Rendus Mathematique, 1929, 189: 620–622, JFM 55.0799.01 （法語）
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F., Hahn Class: Definitions, Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (編), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010, ISBN 978-0521192255, MR2723248
• Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Berlin: Springer-Verlag, 1991, ISBN 3-540-51123-7, MR 1149380.
• Levenshtein, Vladimir I., Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, IEEE Transactions on Information Theory, 1995, 41 (5): 1303–1321, MR 1366326, doi:10.1109/18.412678.
• F. J. MacWilliams; N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977, ISBN 0-444-85193-3 引文使用過時參數coauthors (幫助)