Askey scheme

Askey scheme for hypergeometric orthogonal polynomials

Koekoek, Lesky & Swarttouw (2010) give the following version of the Askey scheme:

;{}_4F_3(4): Wilson | Racah ;{}_3F_2(3): Continuous dual Hahn | Continuous Hahn | Hahn | dual Hahn ;{}_2F_1(2): Meixner–Pollaczek | Jacobi | Pseudo Jacobi | Meixner | Krawtchouk ;{}_2F_0(1)\ \ / \ \ {}_1F_1(1): Laguerre | Bessel | Charlier ;{}_2F_0(0): Hermite Here {}_pF_q(n) indicates a hypergeometric series representation with n parameters

Askey scheme for basic hypergeometric orthogonal polynomials

Koekoek, Lesky & Swarttouw (2010) give the following scheme for basic hypergeometric orthogonal polynomials:

;4\phi3: Askey–Wilson | q-Racah ;3\phi2: Continuous dual q-Hahn | Continuous q-Hahn | Big q-Jacobi | q-Hahn | dual q-Hahn ;2\phi1: 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 ;2\phi0/1\phi1: Continuous big q-Hermite | Continuous q-Laguerre | Little q-Laguerre | q-Laguerre | q-Bessel | q-Charlier | Al-Salam–Carlitz I | Al-Salam–Carlitz II ;1\phi0: Continuous q-Hermite | Stieltjes–Wigert | Discrete q-Hermite I | Discrete q-Hermite II

Completeness

While there are several approaches to constructing still more general families of orthogonal polynomials, it is usually not possible to extend the Askey scheme by reusing hypergeometric functions of the same form. For instance, one might naively hope to find new examples given by : p_n(x) = {}{q + 1}F_q \left ( \begin{array}{c} -n, n + \mu, a_1(x), \dots, a{q - 1}(x) \ b_1, \dots, b_q \end{array} ; 1 \right ) above q = 3 which corresponds to the Wilson polynomials. This was ruled out in under the assumption that the a_i(x) are degree 1 polynomials such that : \prod_{i = 1}^{q - 1} (a_i(x) + r) = \prod_{i = 1}^{q - 1} a_i(x) + \pi(r) for some polynomial \pi(r).

References

References

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3. (1985). "Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials". Memoirs of the American Mathematical Society.
4. (1998). "The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue". Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.
5. (2010). "Hypergeometric orthogonal polynomials and their q-analogues". [[Springer-Verlag]].
6. {{harvtxt. Koekoek. Lesky. Swarttouw. 2010
7. {{harvtxt. Koekoek. Lesky. Swarttouw. 2010
8. (2009). "On Askey-scheme and d-orthogonality, I: A characterization theorem". Journal of Computational and Applied Mathematics.