# multiple orthogonal polynomials

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## 1—10 of 21 matching pages

##### 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. …
##### 3: 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
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
##### 5: 18.27 $q$-Hahn Class
###### §18.27(i) Introduction
All these systems of OP’s have orthogonality properties of the form …
##### 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 $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ 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 $\alpha=-n$ for nonnegative integer $n$ corresponds to the Heun polynomial $\mathit{Hp}_{n,m}\left(z\right)$. 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 $\mathcal{E}$. …
###### §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. …