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11: 18.8 Differential Equations
Table 18.8.1: Classical OPs: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
# f ( x ) A ( x ) B ( x ) C ( x ) λ n
12: 18.33 Polynomials Orthogonal on the Unit Circle
Simon (2005a, b) gives the general theory of these OPs in terms of monic OPs Φ n ( x ) , see §18.33(vi). …
§18.33(iii) Connection with OPs on the Line
Let { p n ( x ) } and { q n ( x ) } , n = 0 , 1 , , be OPs with weight functions w 1 ( x ) and w 2 ( x ) , respectively, on ( 1 , 1 ) . … Instead of (18.33.9) one might take monic OPs { q n ( x ) } with weight function ( 1 + x ) w 1 ( x ) , and then express q n ( 1 2 ( z + z 1 ) ) in terms of ϕ 2 n ( z ± 1 ) or ϕ 2 n + 1 ( z ± 1 ) . After a quadratic transformation (18.2.23) this would express OPs on [ 1 , 1 ] with an even orthogonality measure in terms of the ϕ n . …
13: 18.25 Wilson Class: Definitions
For the Wilson class OPs p n ( x ) with x = λ ( y ) : if the y -orthogonality set is { 0 , 1 , , N } , then the role of the differentiation operator d / d x in the Jacobi, Laguerre, and Hermite cases is played by the operator Δ y followed by division by Δ y ( λ ( y ) ) , or by the operator y followed by division by y ( λ ( y ) ) . …
Table 18.25.1: Wilson class OPs: transformations of variable, orthogonality ranges, and parameter constraints.
OP p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
Table 18.25.2: Wilson class OPs: leading coefficients.
p n ( x ) k n
14: 18.30 Associated OP’s
§18.30 Associated OPs
§18.30(vi) Corecursive Orthogonal Polynomials
Note that this is the same recurrence as in (18.2.8) for the traditional OPs, but with a different initialization. …
Associated Monic OPs
Relationship of Monic Corecursive and Monic Associated OPs
15: 18.9 Recurrence Relations and Derivatives
For the other classical OPs see Table 18.9.1; compare also §18.2(iv).
Table 18.9.1: Classical OPs: recurrence relations (18.9.1).
p n ( x ) A n B n C n
For the other classical OPs see Table 18.9.2.
Table 18.9.2: Classical OPs: recurrence relations (18.9.2_1).
p n ( x ) a n b n c n
They imply the recurrence coefficients for the orthonormal versions of the classical OPs as well, see again §3.5(vi). …
16: 18.6 Symmetry, Special Values, and Limits to Monomials
Table 18.6.1: Classical OPs: symmetry and special values.
p n ( x ) p n ( x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
17: 18.27 q -Hahn Class
The q -hypergeometric OPs comprise the q -Hahn class (or q -linear lattice class) OPs and the Askey–Wilson class (or q -quadratic lattice class) OPs18.28). … A (nonexhaustive) classification of such systems of OPs was made by Hahn (1949). There are 18 families of OPs of q -Hahn class. … All these systems of OPs have orthogonality properties of the form …Some of the systems of OPs that occur in the classification do not have a unique orthogonality property. …
18: 18.20 Hahn Class: Explicit Representations
Table 18.20.1: Krawtchouk, Meixner, and Charlier OPs: Rodrigues formulas (18.20.1).
p n ( x ) F ( x ) κ n
19: DLMF Project News
error generating summary
20: 18.28 Askey–Wilson Class
The Askey–Wilson class OPs comprise the four-parameter families of Askey–Wilson polynomials and of q -Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OPs { p n ( x ) } , n = 0 , 1 , 2 , , that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. … In the remainder of this section the Askey–Wilson class OPs are defined by their q -hypergeometric representations, followed by their orthogonal properties. … Leonard (1982) classified all (finite or infinite) discrete systems of OPs p n ( x ) on a set { x ( m ) } for which there is a system of discrete OPs q m ( y ) on a set { y ( n ) } such that p n ( x ( m ) ) = q m ( y ( n ) ) . … Bannai and Ito (1984) introduced OPs, called the Bannai–Ito polynomials which are the limit for q 1 of the q -Racah polynomials. …