Affine q-Krawtchouk polynomials: Difference between revisions

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In mathematics, the affine q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions by ^[1]

K_{n}^{\text{aff}}(q^{-x};p;N;q)={}_{3}\phi _{2}\left({\begin{matrix}q^{-n},0,q^{-x}\\pq,q^{-N}\end{matrix}};q,q\right),\qquad n=0,1,2,\ldots ,N.

Relation to other polynomials

affine q-Krawtchouk polynomials → little q-Laguerre polynomials：

\lim _{a\to 1}=K_{n}^{\text{aff}}(q^{x-N};p,N\mid q)=p_{n}(q^{x};p,q)

.

References

^ Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p. 501, Springer, 2010

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
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), "Affine q-Krawtchouk polynomials", 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.
Stanton, Dennis (1981), "Three addition theorems for some q-Krawtchouk polynomials", Geometriae Dedicata, 10 (1): 403–425, doi:10.1007/BF01447435, ISSN 0046-5755, MR 0608153, S2CID 119838893

[1] Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p. 501, Springer, 2010

[1]

@@ Line 3: / Line 3: @@
 ==Definition==
-The  polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by ]]<ref>Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p501,Springer,2010</ref>
+The  polynomials are given in terms of [[basic hypergeometric function]]s by <ref>Roelof Koekoek, Hypergeometric Orthogonal Polynomials and its q-Analogues, p. 501, Springer, 2010</ref>
+: <math> K^{\text{aff}}_n (q^{-x};p;N;q) = {}_3\phi_2\left( \begin{matrix}
+q^{-n},0,q^{-x}\\
+pq,q^{-N}\end{matrix};q,q\right), \qquad n=0,1,2,\ldots, N.</math>
+==Relation to other polynomials==
-<math>K^{aff}_{n}(q^{-x};p;N;q)=\;_{2}\phi_1\left(\begin{matrix}
-q^{-n} &0 & q^{-x}   \\
-pq &q^{-N}   \end{matrix}
-; q,q \right)</math>
+affine q-Krawtchouk polynomials → [[little q-Laguerre polynomials]]：
-<math>n=0,1,2,\cdots N</math>
-:<math>\displaystyle    </math>
+: <math>\lim_{a \to 1}=K_n^\text{aff}(q^{x-N};p,N\mid q)=p_n(q^x;p,q)</math>.
-==Orthogonality==
-{{Empty section|date=September 2011}}
-==Recurrence and difference relations==
-{{Empty section|date=September 2011}}
-==Rodrigues formula==
-{{Empty section|date=September 2011}}
-==Generating function==
-{{Empty section|date=September 2011}}
-==Relation to other polynomials==
-{{Empty section|date=September 2011}}
 ==References==
+{{Reflist}}
-*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
+*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
 *{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
 *{{dlmf|id=18|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
-*{{Citation | last1=Stanton | first1=Dennis | title=Three addition theorems for some q-Krawtchouk polynomials | doi=10.1007/BF01447435 | mr=608153 | year=1981 | journal=Geometriae Dedicata | issn=0046-5755 | volume=10 | issue=1 | pages=403–425}}
+*{{Citation | last1=Stanton | first1=Dennis | title=Three addition theorems for some q-Krawtchouk polynomials | doi=10.1007/BF01447435 | mr=608153 | year=1981 | journal=Geometriae Dedicata | issn=0046-5755 | volume=10 | issue=1 | pages=403–425| s2cid=119838893 }}
 [[Category:Orthogonal polynomials]]