Little q-Jacobi polynomials
{{DISPLAYTITLE:Little q-Jacobi polynomials}}{{No footnotes|date=May 2022}}
In mathematics, the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by {{harvtxt|Hahn|1949}}. {{harvs|txt | 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|loc=14}} give a detailed list of their properties.
Definition
The little q-Jacobi polynomials are given in terms of basic hypergeometric functions by
:
Gallery
The following are a set of animation plots for Little q-Jacobi polynomials, with varying q;
three density plots of imaginary, real and modulus in complex space; three set of complex 3D plots
of imaginary, real and modulus of the said polynomials.
| File:LITTLE Q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT.gif
|File:LITTLE Q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT.gif |File:LITTLE Q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT.gif |
| File:LITTLE Q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT.gif |
References
- {{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=Hahn | first1=Wolfgang | title=Über Orthogonalpolynome, die q-Differenzengleichungen genügen | doi=10.1002/mana.19490020103 | mr=0030647 | year=1949 | journal=Mathematische Nachrichten | issn=0025-584X | volume=2 | issue=1–2 | pages=4–34}}
- {{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}}