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multiple orthogonal polynomials


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1: 18.36 Miscellaneous Polynomials
These are OP’s on the interval ( 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at 1 and 1 to the weight function for the Jacobi polynomials. …
§18.36(iii) Multiple Orthogonal Polynomials
2: 18.18 Sums
§18.18(iii) Multiplication Theorems
3: 31.9 Orthogonality
§31.9 Orthogonality
§31.9(i) Single Orthogonality
For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).
§31.9(ii) Double Orthogonality
For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). …
4: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
5: 18.27 q -Hahn Class
§18.27(i) Introduction
All these systems of OP’s have orthogonality properties of the form …
§18.27(ii) q -Hahn Polynomials
§18.27(iii) Big q -Jacobi Polynomials
Limit Relations
6: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as n , with x and other parameters fixed, for continuous q -ultraspherical, big and little q -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson p n ( cos θ ; a , b , c , d | q ) the leading term is given by … For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q 1 -Hermite polynomials see Chen and Ismail (1998).
7: Bibliography D
  • P. A. Deift (1998) Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York.
  • P. Deift, T. Kriecherbauer, K. T. McLaughlin, S. Venakides, and X. Zhou (1999a) Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (12), pp. 1491–1552.
  • K. Dilcher (2008) On multiple zeros of Bernoulli polynomials. Acta Arith. 134 (2), pp. 149–155.
  • G. C. Donovan, J. S. Geronimo, and D. P. Hardin (1999) Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.
  • K. Driver and K. Jordaan (2013) Inequalities for extreme zeros of some classical orthogonal and q -orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.
  • 8: 31.11 Expansions in Series of Hypergeometric Functions
    Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … The case α = n for nonnegative integer n corresponds to the Heun polynomial 𝐻𝑝 n , m ( z ) . The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse . …
    §31.11(v) Doubly-Infinite Series
    Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions. …
    9: 18.28 Askey–Wilson Class
    §18.28 Askey–Wilson Class
    §18.28(ii) Askey–Wilson Polynomials
    §18.28(viii) q -Racah Polynomials
    10: Errata
    The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials. …We have significantly expanded the section on associated orthogonal polynomials, including expanded properties of associated Laguerre, Hermite, Meixner–Pollaczek, and corecursive orthogonal and numerator and denominator orthogonal polynomials. …We also discuss non-classical Laguerre polynomials and give much more details and examples on exceptional orthogonal polynomials. We have also completely expanded our discussion on applications of orthogonal polynomials in the physical sciences, and also methods of computation for orthogonal polynomials. …
  • Chapter 18 Orthogonal Polynomials

    The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).