Continuous q-Jacobi polynomials explained

In mathematics, the continuous q-Jacobi polynomials P(x|q), introduced by, are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by

P ^{(\alpha,\beta)} _{n} (x;q) = ((q ^{n+1} ;q) _{n})/((q;q) _{n}) {} _{4\phi} _{3\left[\begin{matrix}} q ^{-n} ,q ^{n+\alpha+\beta+1} ,q ^{(12 \alpha + (14 e)/(^{i\theta}))/()} ,q ^{(12 \alpha + (14 e)/(^{-i\theta}))/()} \ q ^{n+1} , - q ^{(12 (\alpha + \beta +1 ))/()} , - q ^{(12 (\alpha + \beta +2 ))/()} \end{matrix} ;q,q\right] x = \cos\theta.

References

• • • • • • Sadjang . Patrick Njionou . Moments of Classical Orthogonal Polynomials . Ph.D. . Universität Kassel . 10.1.1.643.3896 .