User:HowiAuckland/sandbox

Project for Roberto Costas-Santos

• Regular world

1. Wilson 3 free paramers connection coefficients
2. Continuous dual Hahn connection coefficients with three free parameters
3. Continuous Hahn connection coefficients to Meixner Pollaczek with 2 free parameters
4. Meixner-Pollaczek with two free parameters

• $q$-world

1. Expressions of the Askey-Wilson polynomials as terminating hypergeometric series
2. Describe all the algebraic/analytic properties of a generalisation of the Askey-Wilson polynomials with 4N parameters
3. Continuous q-Jacobi connection coefficients (collapse to single sum)
4. Connection coefficients Askey Wilson with one free parameter
5. Connection coefficients for Big q-Jacobi with one free parameter
6. Connection coefficients for little q-Jacobi with one free parameter

Project for Janelle Williams -- NIST DRMF project

New DRMF wiki located at

• https://github.com/DRMF/DRMF/wiki

Project for Rebekah Mae Wheatley -- Expansions in hyperspherical geometry

• Project title: Fourier and Gegenbauer expansions for a fundamental solution of Laplace's equation in hyperspherical geometry
  • Abstract: This project is the positive constant curvature analogue project akin to the negative constant curvature project which has already been completed and published by Howard Cohl and Ernie Kalnins. Rebekah will be computing azimuthal Fourier expansions for a fundamental solution of Laplace's equation in rotationally-invariant coordinate systems on the d-dimensional R-radius hypersphere. She will also compute Gegenbauer polynomial expansions for this fundamental solution of Laplace's equation in Vilenkin's polyspherical coordinate systems.

We have proven that on the $R$-radius hypersphere ${\mathbf {S} }_{R}^{d}$, a Green's function for the Laplace operator (fundamental solution of Laplace's equation) can be given as follows. Let $d\in \{2,3,\ldots \}.$ Define ${\mathcal {I}}_{d}:(0,\pi )\to {\mathbf {R} }$ as

${\mathcal {I}}_{d}(\theta ):=\int _{\theta }^{\pi /2}{\frac {dx}{\sin ^{d-1}x}},$

${\mathbf {x} },{{\mathbf {x} }^{\prime }}\in {\mathbf {S} }_{R}^{d}$, and ${\mathcal {S}}_{R}^{d}:({\mathbf {S} }_{R}^{d}\times {\mathbf {S} }_{R}^{d})\setminus \{({\mathbf {x} },{\mathbf {x} }):{\mathbf {x} }\in {\mathbf {S} }_{R}^{d}\}\to {\mathbf {R} }$ defined such that

${\mathcal {S}}_{R}^{d}({\mathbf {x} },{\mathbf {x} }^{\prime }):={\displaystyle {\frac {\Gamma \left(d/2\right)}{2\pi ^{d/2}R^{d-2}}}{\mathcal {I}}_{d}(\theta )},$

where $\theta :=\cos ^{-1}\left([{\widehat {\mathbf {x} }},{{\widehat {\mathbf {x} }}^{\prime }}]\right)$ is the geodesic distance between ${\widehat {\mathbf {x} }}$ and ${{\widehat {\mathbf {x} }}^{\prime }}$ on the unit radius hypersphere ${\mathbf {S} }^{d}$, with ${\widehat {\mathbf {x} }}={\mathbf {x} }/R,$ ${{\widehat {\mathbf {x} }}^{\prime }}={{\mathbf {x} }^{\prime }}/R$, then ${\mathcal {S}}_{R}^{d}$ is a fundamental solution for $-\Delta$ where $\Delta$ is the Laplace-Beltrami operator on ${\mathbf {S} }_{R}^{d}$. Moreover,

${\mathcal {I}}_{d}(\theta )={\begin{cases}\displaystyle {\frac {(d-3)!!}{(d-2)!!}}{\biggl [}\log \cot {\frac {\theta }{2}}+\cos \theta \sum _{k=1}^{d/2-1}{\frac {(2k-2)!!}{(2k-1)!!}}{\frac {1}{\sin ^{2k}\theta }}{\biggr ]},&\mathrm {if} \ d\ \mathrm {even} ,\\[0.6cm]{\begin{cases}{\displaystyle \left({\frac {d-3}{2}}\right)!\sum _{k=1}^{(d-1)/2}{\frac {\cot ^{2k-1}\theta }{(2k-1)(k-1)!((d-2k-1)/2)!}}},\\\mathrm {or} \\[0.0cm]\displaystyle {\frac {(d-3)!!}{(d-2)!!}}\cos \theta \sum _{k=1}^{(d-1)/2}{\frac {(2k-3)!!}{(2k-2)!!}}{\frac {1}{\sin ^{2k-1}\theta }},\end{cases}},&{\text{if }}n{\text{ is odd}}\end{cases}}$

$={\begin{cases}{\displaystyle {}_{2}F_{1}\left({\frac {1}{2}},{\frac {d}{2}};{\frac {3}{2}};\cos ^{2}\theta \right),}\\{\displaystyle {\frac {\cos \theta }{\sin ^{d-2}\theta }}\,{}_{2}F_{1}\left(1,{\frac {3-d}{2}};{\frac {3}{2}};\cos ^{2}\theta \right),}\\{\displaystyle {\frac {(d-2)!}{\displaystyle \Gamma \left(d/2\right)2^{d/2-1}}}{\frac {1}{(\sin \theta )^{d/2-1}}}{\mathrm {Q} }_{d/2-1}^{1-d/2}(\cos \theta ).}\end{cases}}$

Project for Michael Baeder -- Askey scheme generalized generating functions

• Project title: Generalizations of generating functions for hypergeometric orthogonal polynomials in the Askey scheme
  • Abstract: Michael will be developing proofs and computing generalizations of generating functions for hypergeometric orthogonal polynomials in the Askey Scheme. These generalized hypergeometric orthogonal polynomial are of continuous or discrete type orthogonality and include Wilson, Racah, Continuous dual Hahn, Continuous Hahn, Hahn, Dual Hahn, Meixner-Pollaczek, Jacobi, Meixner and Krawtchouk, Laguerre and Charlier polynomials. Many generating functions are known for these orthogonal polynomials and generalizations are produced by combining these generating functions with connection formulae with one-free parameter. We will start with the Wilson polynomials and see how far we can get.

Project for Philbert Hwang-- q-Askey scheme generalized generating functions

• Project title: Generalizations of generating functions for basic hypergeometric orthogonal polynomials in the q-Askey scheme
  • Abstract: Philbert will be developing proofs and computing generalizations of generating functions for basic hypergeometric orthogonal polynomials in the q-Askey Scheme. These basic hypergeometric orthogonal polynomial are of continuous or discrete type orthogonality and include Askey-Wilson polynomials, q-Racah, Continuous dual q-Hahn polynomials, Continuous q-Hahn polynomials, Big q-Jacobi polynomials, q-Hahn polynomials, dual q-Hahn polynomials, Al-Salam–Chihara, q-Meixner–Pollaczek, Continuous q-Jacobi, Big q-Laguerre, Little q-Jacobi, q-Meixner, Quantum q-Krawtchouk, q-Krawtchouk, Affine q-Krawtchouk, Dual q-Krawtchouk, Continuous big q-Hermite, Continuous q-Laguerre, Little q-Laguerre, q-Laguerre, q-Bessel, q-Charlier, Al-Salam-Carlitz I, Al-Salam-Carlitz II polynomials. Many generating functions are known for these orthogonal polynomials and generalizations are produced by combining these generating functions with connection formulae with one-free parameter. We will start with easy classes like little q-Jacobi, q-Ultraspherical and little q-Laguerre polynomials and see how far we can get.

HowiAuckland (talk) 16:45, 9 May 2013 (UTC)